L² Approaches in Several Complex Variables Towards the Oka–Cartan Theory with Precise Bounds Second Edition Springer Monographs in Mathematics

Springer Monographs in Mathematics

Takeo Ohsawa L² Approaches in Several Complex Variables Towards the Oka–Cartan Theory with Precise Bounds Second Edition Springer Monographs in Mathematics

Editors-in-Chief Isabelle Gallagher, Paris, France Minhyong Kim, Oxford, UK

Series Editors Sheldon Axler, San Francisco, USA Mark Braverman, Princeton, USA Maria Chudnovsky, Princeton, USA Tadahisa Funaki, Tokyo, Japan Sinan C. Güntürk, New York, USA Claude Le Bris, Marne la Vallée, France Pascal Massart, Orsay, France Alberto Pinto, Porto, Portugal Gabriella Pinzari, Napoli, Italy Ken Ribet, Berkeley, USA René Schilling, Dresden, Germany Panagiotis Souganidis, Chicago, USA Endre Süli, Oxford, UK Shmuel Weinberger, Chicago, USA Boris Zilber, Oxford, UK This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

More information about this series at http://www.springer.com/series/3733 Takeo Ohsawa

L2 Approaches in Several Complex Variables Towards the Oka–Cartan Theory with Precise Bounds

Second Edition

123 Takeo Ohsawa Professor Emeritus Nagoya University Nagoya, Japan

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-4-431-56851-3 ISBN 978-4-431-56852-0 (eBook) https://doi.org/10.1007/978-4-431-56852-0

Library of Congress Control Number: 2018959147

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This Springer imprint is published by the registered company Springer Japan KK part of Springer Nature. The registered company address is: Shiroyama Trust Tower, 4-3-1 Toranomon, Minato-ku, Tokyo 105-6005, Japan Preface

As in the study of complex analysis of one variable, the general theory of several complex variables has manifold aspects. First, it provides a firm ground for systematic studies of special functions such as elliptic functions, theta functions, and modular functions. The general theory plays a role of confirming the existence and uniqueness of functions with prescribed zeros and poles. Another aspect is to give an insight into the connection between two different fields of mathematics by understanding how the tools work. The theory of sheaves bridged analysis and topology in such a way. In the construction of this basic theory of several complex variables, a particularly important contribution was made by two mathematicians, Kiyoshi Oka (1901–1978) and Henri Cartan (1904–2008). The theory of Oka and Cartan is condensed in a statement that the first cohomology of coherent analytic sheaves over Cn is zero. On the other hand, the method of PDE (partial differential equations) had turned out to be essential in the existence of conformal mappings. By this approach, the function theory on Riemann surfaces as one-dimensional complex manifolds was explored by H. Weyl. Weyl’s method was developed on manifolds of higher dimension by K. Kodaira who generalized Riemann’s condition for Abelian varieties by establishing a differential geometric characterization of nonsingular projective algebraic varieties. This PDE method, based on the L2 estimates for the ∂¯-operator, was generalized by J. Kohn, L. Hörmander, A. Andreotti, and E. Vesentini. As a result, it enabled us to see the results of Oka and Cartan in a much higher resolution. In particular, based on such a refinement, existence theorems for holomorphic functions with L2 growth conditions have been obtained by Hörmander, H. Skoda, and others. The purpose of the present monograph is to report on some of the recent results in several complex variables obtained by the L2 method which can be regarded as a continuation of these works. Among various topics including complex geometry, the Bergman kernel, and holomorphic foliations, a special emphasis is put on the extension theorems and its applications. In this topic, highlighted are the recent developments after the solution of a long- standing open question of N. Suita. It is an inequality between the Bergman kernel and the logarithmic capacity on Riemann surfaces, which was first proved by Z. Błocki for plane domains. Q. Guan and X.-Y. Zhou proved generalized variants

v vi Preface and characterized those surfaces on which the inequality is strict. Their work gave the author a decisive impetus to start writing a survey to cover these remarkable achievements. As a result, he could ﬁnd an alternate proof of the inequality, based on hyperbolic geometry, which is presented in Chap. 3. However, the readers are recommended to have a glance at Chap. 4 ﬁrst, where the questions on the Bergman kernels are described more systematically. (The author started to write the monograph from Chap. 4.) Since there have been a lot of subsequent progress concerning the materials in Chaps. 3 and 4 during the preparation of the manuscript, it soon became beyond the author’s ability to give a satisfactory account of the whole development. So he will be happy to have a chance in the future to revise and enlarge this rather brief monograph.

Nagoya, Japan Takeo Ohsawa March 2015 Preface to the Second Edition

Thanks to the goodwill of the publisher, the revision and enlargement have been realized. What made this edition possible was the recent remarkable activity after Błocki’s solution of Suita’s conjecture for plane domains. Among many corrections, the most important one is the replacement of an erroneous proof of Theorem 3.2 by the present one which is hopefully correct. The author is very grateful to Shigeharu Takayama for pointing out the mistake. Additions have been made to focus on the results which appeared in the past 3 years. Some of them are in Sect. 4.4.5 “Berndtsson–Lempert Theory and Beyond” and in the section “A History of Levi Flat Hypersurfaces”in5.3. Besides these, each chapter has been supplemented by a section titled “Notes and Remarks,” in which the author also tried to enhance the depth feeling of complex analysis and convey the atmosphere of several complex variables similar to searching for extraterrestrial intelligence since Hartogs and Oka.

Nagoya, Japan Takeo Ohsawa April 2018

vii Contents

1 Basic Notions and Classical Results...... 1 1.1 Functions and Domains Over Cn ...... 2 1.1.1 Holomorphic Functions and Cauchy’s Formula...... 2 1.1.2 Weierstrass Preparation Theorem ...... 4 1.1.3 Domains of Holomorphy and Plurisubharmonic Functions .. 7 1.2 Complex Manifolds and Convexity Notions ...... 9 1.2.1 Complex Manifolds, Stein Manifolds and Holomorphic Convexity ...... 10 1.2.2 Complex Exterior Derivatives and Levi Form...... 14 1.2.3 Pseudoconvex Manifolds and Oka–Grauert Theory ...... 16 1.3 Oka–Cartan Theory ...... 19 1.3.1 Sheaves and Cohomology ...... 19 1.3.2 Coherent Sheaves, Complex Spaces, and Theorems A and B ...... 25 1.3.3 Coherence of Direct Images and a Theorem of Andreotti and Grauert ...... 30 1.4 ∂¯-Equations on Manifolds ...... 31 1.4.1 Holomorphic Vector Bundles and ∂¯-Cohomology ...... 32 1.4.2 Cohomology with Compact Support...... 36 1.4.3 Serre’s Duality Theorem ...... 38 1.4.4 Fiber Metric and L2 Spaces ...... 41 1.5 Notes and Remarks ...... 42 References ...... 45 2 Analyzing the L2 ∂¯-Cohomology ...... 47 2.1 Orthogonal Decompositions in Hilbert Spaces ...... 47 2.1.1 Basics on Closed Operators ...... 47 2.1.2 Kodaira’s Decomposition Theorem and Hörmander’s Lemma ...... 48 2.1.3 Remarks on the Closedness ...... 50

ix x Contents

2.2 Vanishing Theorems ...... 51 2.2.1 Metrics and L2 ∂¯-Cohomology ...... 51 2.2.2 Complete Metrics and Gaffney’s Theorem ...... 54 2.2.3 Some Commutator Relations...... 55 2.2.4 Positivity and L2 Estimates ...... 58 2.2.5 L2 Vanishing Theorems on Complete Kähler Manifolds ..... 60 2.2.6 Pseudoconvex Cases ...... 66 2.2.7 Sheaf Theoretic Interpretation ...... 67 2.2.8 Application to the Cohomology of Complex Spaces ...... 70 2.3 Finiteness Theorems ...... 77 2.3.1 L2 Finiteness Theorems on Complete Manifolds ...... 77 2.3.2 Approximation and Isomorphism Theorems ...... 79 2.4 Notes on Metrics and Pseudoconvexity ...... 88 2.4.1 Pseudoconvex Manifolds with Positive Line Bundles ...... 89 2.4.2 Geometry of the Boundaries of Complete Kähler Domains .. 91 2.4.3 Curvature and Pseudoconvexity...... 94 2.4.4 Miscellanea on Locally Pseudoconvex Domains...... 95 2.5 Notes and Remarks ...... 99 References ...... 111 3 L2 Oka–Cartan Theory ...... 115 3.1 L2 Extension Theorems ...... 115 3.1.1 Extension by the Twisted Nakano Identity ...... 115 3.1.2 L2 Extension Theorems on Complex Manifolds ...... 121 3.1.3 Application to Embeddings ...... 125 3.1.4 Application to Analytic Invariants ...... 127 3.2 L2 Division Theorems ...... 128 3.2.1 A Gauss–Codazzi-Type Formula ...... 129 3.2.2 Skoda’s Division Theorem ...... 131 3.2.3 From Division to Extension ...... 135 3.2.4 Proof of a Precise L2 Division Theorem ...... 138 3.3 L2 Approaches to Analytic Ideals ...... 140 3.3.1 Briançon–Skoda Theorem...... 140 3.3.2 Nadel’s Coherence Theorem ...... 142 3.3.3 Miscellanea on Multiplier Ideal Sheaves ...... 142 3.4 Notes and Remarks ...... 153 References ...... 161 4 Bergman Kernels...... 165 4.1 Bergman Kernel and Metric ...... 165 4.1.1 Bergman Kernels ...... 166 4.1.2 The Bergman Metric...... 169 4.2 The Boundary Behavior ...... 170 4.2.1 Localization Principle ...... 171 4.2.2 Bergman’s Conjecture and Hörmander’s Theorem ...... 172 Contents xi

4.2.3 Miscellanea on the Boundary Behavior ...... 173 4.2.4 Comparison with a Capacity Function...... 177 4.3 Sequences of Bergman Kernels ...... 181 4.3.1 Weighted Sequences of Bergman Kernels ...... 181 4.3.2 Demailly’s Approximation Theorem ...... 182 4.3.3 Towering Bergman Kernels ...... 183 4.4 Parameter Dependence...... 184 4.4.1 Stability Theorems...... 184 4.4.2 Maitani–Yamaguchi Theorem...... 185 4.4.3 Berndtsson’s Method ...... 188 4.4.4 Guan–Zhou Method ...... 190 4.4.5 Berndtsson–Lempert Theory and Beyond ...... 192 4.5 Notes and Remarks ...... 194 References ...... 201 5 L2 Approaches to Holomorphic Foliations ...... 205 5.1 Holomorphic Foliation...... 205 5.1.1 Foliation and Its Normal Bundle ...... 205 5.1.2 Holomorphic Foliations of Codimension One...... 208 5.2 Applications of the L2 Method ...... 212 5.2.1 Applications to Stable Sets ...... 212 5.2.2 Hartogs-Type Extensions by L2 Method ...... 217 5.3 A History of Levi Flat Hypersurfaces ...... 220 5.4 Levi Flat Hypersurfaces in Tori and Hopf Surfaces ...... 226 5.4.1 Lemmas on Distance Functions ...... 226 5.4.2 A Reduction Theorem in Tori ...... 229 5.4.3 Classiﬁcation in Hopf Surfaces ...... 231 5.5 Notes and Remarks ...... 234 References ...... 235

Bibliography ...... 239

Index ...... 255 Chapter 1 Basic Notions and Classical Results

Abstract As a preliminary, basic properties of holomorphic functions and complex manifolds are recalled. Beginning with the deﬁnitions and characterizations of holomorphic functions, we shall give an overview of the classical theorems in several complex variables, restricting ourselves to extremely important ones for the discussion in later chapters. Most of the materials presented here are contained in well-written textbooks such as Gunning and Rossi (Analytic functions of several complex variables. Prentice-Hall, Inc., Englewood Cliffs, 1965, pp xiv+317), Hörmander (An introduction to complex analysis in several variables, 3rd edn. North-Holland Mathematical Library, vol 7. North-Holland Publishing Co., Ams- terdam, 1990, pp xii+254), Wells (Differential analysis on complex manifolds, 3rd edn. With a new appendix by Oscar Garcia-Prada. Graduate texts in mathematics, vol 65. Springer, New York, 2008), Grauert and Remmert (Theory of Stein spaces. Translated from the German by Alan Huckleberry. Reprint of the 1979 translation. Classics in mathematics. Springer, Berlin, 2004, pp xxii+255; Coherent analytic sheaves. Grundlehren der Mathematischen Wissenschaften, vol 265. Springer, Berlin, 1984, pp xviii+249) and Noguchi (Analytic function theory of several variables—elements of Oka’s coherence, preprint) (see also Demailly, Analytic methods in algebraic geometry. Surveys of modern mathematics, vol 1. International Press, Somerville/Higher Education Press, Beijing, 2012, pp viii+231) and Ohsawa (Analysis of several complex variables. Translated from the Japanese by Shu Gilbert Nakamura. Translations of mathematical monographs. Iwanami series in modern mathematics, vol 211. American Mathematical Society, Providence, 2002, pp xviii+121), so that only sketchy accounts are given for most of the proofs and historical backgrounds. An exception is Serre’s duality theorem. It will be presented after an article of Laurent-Thiébaut and Leiterer (Some applications of Serre duality in CR manifolds. Nagoya Math J 154:141–156, 1999), since none of the above books contains its proof in full generality.

© Springer Japan KK, part of Springer Nature 2018 1 T. Ohsawa, L2 Approaches in Several Complex Variables, Springer Monographs in Mathematics, https://doi.org/10.1007/978-4-431-56852-0_1 2 1 Basic Notions and Classical Results

1.1 Functions and Domains Over Cn

1.1.1 Holomorphic Functions and Cauchy’s Formula

Let n be a positive integer and let C{z} be the convergent power series ring in z = (z1,...,zn) with coefﬁcients in C. Since the nineteenth century, C{z} has been identiﬁed with the set of germs at z = (0,...,0) of functions of a distinguished class in complex variables z1 = x1 + iy1,...,zn = xn + iyn, i.e. the class of holomorphic functions. Recall that a function f on an open subset U of Cn is called a holomorphic function if the values of f are equal to those of a convergent power series in z − a around each point a of U. The set of holomorphic functions on U will be denoted by O(U) and the germ of f ∈ O(U) at a ∈ U by fa.Themost important formula for holomorphic functions is Cauchy’s formula, 1 f(ζ) f(z)= dζ. (1.1) 2πi ∂D ζ − z

Here D is a bounded domain in C with C1-smooth boundary, i.e. the boundary ∂D of D is the disjoint union of ﬁnitely many C1-smooth closed curves, f is holomorphic on a neighborhood of the closure of D, z ∈ D and the orientation of ∂D as a path of the integral is deﬁned to be the direction which sees the interior 1 of D on the left-hand side. Let D1,...,Dn be bounded domains in C with C - n smooth boundary. If f(z) = f(z1,...,zn) is a holomorphic function on U ⊂ C , U ⊃ D1 ×···×Dn and zj ∈ Dj , then (1.1) is generalized to

n 1 n dζj f(z)= f(ζ1,...,ζn). (1.2) 2πi ζj − zj j=1 ∂Dj ˜ Cn \ n C ×···× The right-hand side of (1.2), say f(z), is holomorphic on ( j=1 ∂Dj ×···×C) even if f is only deﬁned on ∂D1 ×···×∂Dn and continuous there. Hence, if further, f is continuously extended to a subset B of D1 ×···×Dn with ◦ ˜ ∂D1 ×···×∂Dn ⊂ B in such a way that (1.2) holds for all z ∈ B , then f |D ×···×D ◦ 1 n is a holomorphic extension of f |B◦ .HereB denotes the set of interior points of B. In particular, letting Dj be the unit disc D ={ζ ∈ C;|ζ | < 1} and choosing B in such a way that

◦ = := { ∈ Dn; {| | | |} | | } B TR1,R2 z max z1 , max R1 zj < 1orR2 z1 > 1 2≤j≤n for R1,R2 > 1, one has: 1.1 Functions and Domains Over Cn 3

Theorem 1.1 (Hartogs’s continuation theorem, cf. [Ht-1]) If n ≥ 2, the natural restriction map

O Dn −→ O ( ) (TR1,R2 ) is surjective. Thus, Cauchy’s integral formula is useful to solve the boundary value problem for holomorphic functions. A remarkable point is that the boundary values have to be given only along a special subset of the topological boundary. In the case of one complex variable, (1.1) is also useful to solve the boundary value problem of this type for harmonic functions, the Dirichlet problem, but only in special cases (e.g., Poisson’s formula). The class of subharmonic functions is useful to solve it in full generality. We recall that a subharmonic function on a domain D ⊂ C is by deﬁnition an upper semicontinuous function u : D →[−∞, ∞) such that, for any disc D(c, r) := {z ∈ C;|z − c|

∞ = j k ∈ C{ } σ ajkz1z2 z1,z2 , j,k=0 the lower envelope r(z˜ 2) of the radii of convergence r(z2) of the series ∞ k j ajkz2 z1 j k=0 in z1 (r(c)˜ := lim 0 inf {r(ζ); 0 < |ζ − c| <}), has the property that − log r(z˜ 2) is a subharmonic function on a neighborhood of 0, because of the subharmonicity 1 | ∞ k| of j log k=0 ajkz2 and the Cauchy-Hadamard formula. A general theory of subharmonic functions including a decomposition of subharmonic functions as a sum of harmonic functions and the integrals of the logarithm was established by F. Riesz around 1930. By the L2 method, it turned out in 1992 by Demailly’s work [Dm-6] that any subharmonic function can be approximated (in an appropriate sense) by a subharmonic function on D ⊂ C of the form 4 1 Basic Notions and Classical Results 2 log |fj | (fj ∈ O(D)) j

(cf. Chap. 4). Cauchy’s formula holds because holomorphic functions locally admit primi- tives, but Stokes’ formula says that (1.2) holds as well if f is of class C1 on D1 ×···×Dn and satisﬁes the Cauchy–Riemann equation

n ¯ ∂f ∂f := dzj = 0onD1 ×···×Dn. (1.3) ∂zj j=1

Here ∂ 1 ∂ ∂ = + i and dzj = dxj − idyj . ∂zj 2 ∂xj ∂yj

Hence, as is well known, any C1 function satisfying the Cauchy–Riemann equation is holomorphic. The following characterization of holomorphic functions is equally important for later purposes. Theorem 1.2 Let f be a measurable function on an open set U ⊂ Cn which is square integrable on every compact subset of U with respect to the Lebesgue measure dλ(= dλn). Suppose that ∂φ f · dλ = 0 for all j U ∂zj holds for any C-valued C∞ function φ on U whose support is a compact subset of U. Then f is almost everywhere equal to a holomorphic function on U. The proof of Theorem 1.2 is done by approximating f locally by taking convolutions with radially symmetric smooth functions with compact support. The same method works to characterize holomorphic functions as those distributions which are weak solutions of the Cauchy–Riemann equation.

1.1.2 Weierstrass Preparation Theorem

In a paper of K.Weierstrass published in 1879, the following is proved.

Theorem 1.3 Let F(z1,...,zn) be a holomorphic function on a neighborhood n of the origin (0,...,0) of C satisfying F(0,...,0) = 0 and F0(z1) := ≡ = p = F(z1, 0,...,0) 0. Let p be the integer such that F0(z1) z1 G(z1), G(0) 0. Then there exist a holomorphic function of the form 1.1 Functions and Domains Over Cn 5

p + p−1 +···+ z1 a1z1 ap, say f(z1; z2,...,zn), where ak are holomorphic functions in (z2,...,zn) satisfying ak(0,...,0) = 0, and a function g(z1,...,zn) holomorphic and nowhere vanishing in a neighborhood of the origin, such that

F = f · g holds in a neighborhood of the origin. Theorem 1.3 is called the Weierstrass preparation theorem. Functions f(z1; z2,...,zn) are called Weierstrass polynomials in z1. Weierstrass polynomials are also called distinguished polynomials because they are polynomials in z1 with “distinguished coefﬁcients” in the ring C{z }. Since the elements of C{z − a} (a ∈ Cn) are the building blocks of holomorphic functions, the algebraic structures of C{z − a} and their relations to those of C{z − b} for nearby b are particulary important in the local theory of holomorphic functions. The Weierstrass preparation theorem is the most basic tool for studying such properties of the convergent power series rings. There are several proofs of Theorem 1.3 including purely algebraic ones (cf. [Ng, p.191]), but Cauchy’s integral formula gives a very straightforward one:

Proof of Theorem 1.3 By assumption, there exist a neighborhood U1 of 0 ∈ C and a neighborhood U of the origin of Cn−1 such that, for any z ∈ U one can ﬁnd s1,...,sp ∈ C satisfying

F(z1,z) = 0 and (z1,z) ∈ U1 × U ⇐⇒ z1 ∈{s1,...,sp}.

Hence it sufﬁces to show that

; := − ··· − = p + p−1 +···+ f(z1 z ) (z1 s1) (z1 sp) z1 a1(z )z1 ap(z ) is holomorphic. By Cauchy’s integral formula, m := m +···+ m = 1 ζ ∂F(ζ,z ) s(m) s1 sp dζ 2πi |ζ|= F(ζ,z) ∂ζ holds for m ∈ N and for sufﬁciently small >0. Hence s(m) is holomorphic in z . By Newton’s identity

m−1 s(m) = a1s(m − 1) − a2s(m − 2) +···+(−1) mam (1

Hence aj is a polynomial in s1,...,sj , so that f is holomorphic. From now on, the origin (0,...,0) will be denoted simply by 0. In a paper published in 1887, L. Stickelberger wrote the following as a lemma. 6 1 Basic Notions and Classical Results

Theorem 1.4 Let F and G be as in Theorem 1.3. Then, for any g ∈ C{z} one can ﬁnd q ∈ C{z} and h ∈ C{z }[z ] such that h ≤ p − and g = qF + h. 1 degz1 1 Proof By Theorem 1.3, it sufﬁces to show the assertion when F is a Weierstrass polynomial. Let us put

1 g(ζ,z )dζ q(z ,z) = . 1 − 2πi |ζ−z1|= F(ζ,z)(ζ z1)

Here is sufﬁciently small and z . Then q is holomorphic in a neighborhood of 0. Letting

1 F(ζ,z) − F(z ,z) g(ζ,z )dζ h(z ,z) = 1 , 1 − 2πi |ζ−z1|= ζ z1 F(ζ,z)

h ∈ C{z }[z ] h ≤ p − g = qF + h one has 1 ,degz1 1 and .

For simplicity, we put C{z − a}=Oa.

Deﬁnition 1.1 An invertible element f of Oa is called a unit. f is called a prime element if it is not the product of two elements g,h ∈ Oa which are both not units. Two elements of Oa,sayf and g are said to be relatively prime to each other if there exist no h ∈ Oa with h(a) = 0 dividing both f and g. The following theorems, which are essentially corollaries of Theorem 1.4, carry the ﬂavor of Euclid’s ΣTOIXEIA.

Theorem 1.5 Oa is a unique factorization domain. Here a commutative ring with the multiplicative identity and without zero divisors is called a unique factorization domain if every nonzero element is decomposed into the product of prime elements uniquely up to multiplication of units. Theorem 1.6 Let D be a domain in Cn and let f, g ∈ O(D).Ifthegermsoff and g are relatively prime to each other at a point c ∈ D, then so are they at all points in a neighborhood of c. Theorem 1.4 is called the Weierstrass division theorem. As one of its important applications, let us mention the following. Theorem 1.7 Let D be a domain in Cn and let F ∈ O(D) \{0}. Then, for every point c ∈ F −1(0), there exist a neighborhood U c and f ∈ O(U) such that, for −1 any d ∈ U and for any g ∈ Od vanishing along F (0) on a neighborhood of d, fd divides g. The set F −1(0) is called a (complex) hypersurface of D and f as above is called a minimal local deﬁning function of F −1(0). 1.1 Functions and Domains Over Cn 7

In view of these classical theorems, a natural question is to extend them to vector-valued holomorphic functions. Namely, what can we say about the local defining functions of the common zeros of holomorphic functions? An answer was given by the Oka–Cartan theory which will be reviewed in the last section of this chapter. In Chap. 3, refinements of Oka–Cartan theory by the L2 method will be given following the development in recent decades. For that the following is important as well as Theorem 1.2. For a proof applying Theorem 1.2,see[Oh-21, Proposition 1.14] for instance. Theorem 1.8 Let D be a domain in Cn,letF ∈ O(D) \{0}, and let g ∈ O(D \ F −1(0)).If |g(z)|2 dλ < ∞, D\F −1(0) then g is holomorphically extendable to D, i.e. there exists g˜ ∈ O(D) such that ˜| = g D\F −1(0) g. In Chap. 3, the reader will find a relation between subharmonicity and the Weierstrass division theorem bound by the L2 theory (cf. Theorem 3.19).

1.1.3 Domains of Holomorphy and Plurisubharmonic Functions

In view of the starting point that a holomorphic function is a collection of elements n n of Oc (c ∈ C ), it is natural to extend the class of domains in C to the domains over Cn. Definition 1.2 A domain over a topological space X is a connected topological space X˜ with a local homeomorphism p : X˜ → X. X˜ is said to be finitely sheeted if the cardinality of p−1(c) is bounded from above by some m ∈ N independent of c. A domain D in Cn is naturally identified with a domain over Cn with respect to the inclusion map. A domain over X will be referred to also as a Riemann domain over X. For two domains (Dk,pk)(k= 1, 2) over n C , (D1,p1) is called a subdomain of (D2,p2) if there exists an injective local n homeomorphism ι : D1 → D2 such that p2 ◦ ι = p1.LetD be a domain over C . A C-valued function f on D is said to be holomorphic if every point x ∈ D has a −1 neighborhood U such that p|U is a homeomorphism and f ◦ (p|U ) ∈ O(p(U)). The set of holomorphic functions on D will be denoted by O(D). For any c ∈ Cn n and for any f ∈ Oc, a pair of a domain (D,p)over C and fD ∈ O(D) is called an extension of f if there exist c˜ ∈ D with p(c)˜ = c and a neighborhood U ˜c −1 such that f is the germ of fD ◦ (p|U ) at c. Extensions of f are ordered by the inclusion relation defined as above. 8 1 Basic Notions and Classical Results

Definition 1.3 A domain (D,p)over Cn is called a domain of holomorphy if it is the domain of definition of the maximal extension of an element of Op(c) for some c ∈ D. n Example 1.1 (C , idCn ) is a domain of holomorphy. In contrast to this trivial example, the following is highly nontrivial. Theorem 1.9 Every domain over C is a domain of holomorphy. For the proof, the idea of Oka for the characterization of domains of holomorphy for any n is essential. (See [B-S] or Chap. 2.) Theorem 1.1 shows that not every domain in Cn is a domain of holomorphy if n ≥ 2. Accordingly, the classification theory of holomorphic functions can be geometric, with respect to the Euclidean distance

n 1 2 2 n dist(z, w) := z − w= |zj − wj | ,z,w∈ C j=1 for instance. (z will be simpliﬁed as |z| in some places.) The convexity notion is important in this context as one can see from: Theorem 1.10 For a domain Ω ⊂ Rn, the domain {z ∈ Cn; Re z ∈ Ω} is a domain of holomorphy if and only if Ω is convex.

Sketch of proof If Ω is convex and x0 ∈ ∂Ω, there exists an afﬁne linear function n n on C such that (R ) = R, (x0) = 0 and (x) > 0ifx ∈ Ω. Then

1 ∈ O({Re z ∈ Ω}) (z + iy0)

n for any y0 ∈ R . Hence, considering an inﬁnite sum of such functions, it is easy to see that {Re z ∈ Ω} is a domain of holmorphy. For the converse, see [Hö-2, Theorem 2.5.10]. Convexity is naturally attached to any σ ∈ C{z} as follows. Let

◦ Rσ := {(z1,...,zn); σ is convergent at (z1,...,zn)} .

For any domain Ω ⊂ Cn satisfying

Ω ={(ζ1z1,...,ζnzn); (z1,...,zn) ∈ Ω and ζj ∈ D},

n we put log |Ω|={(log r1,...,log rn) ∈[−∞, ∞) ; (r1,...,rn) ∈ Ω}. Then it is easy to see that the set log |Rσ | is convex for any σ . Conversely, Ω = Rσ for some σ ∈ C{z} if log |Ω| is convex (cf. [Oh-21, Corollary 1.19]). Rσ is called the Reinhardt domain of σ. On the other hand, the observation after Theorem 1.1 can be stated as follows. 1.2 Complex Manifolds and Convexity Notions 9

Theorem 1.11 Let D ⊂ C be a domain, let ϕ be an upper semicontinuous function ˜ 2 −ϕ(z ) ˜ on D, and let D ={z = (z1,z2) ∈ C ;|z1|

n δD (x) = sup {r; p maps a neighborhood of x bijectively to B (p(x), r)}, where Bn(c, r) := {z ∈ Cn;z − c

1.2 Complex Manifolds and Convexity Notions

Plurisubharmonic functions play an important role in the study of basic existence problems. This is also the case on complex manifolds, objects on which the theory of holomorphic mappings can be discussed in full generality. In the study of global coordinates on complex manifolds, analytic tools available on the domains over Cn work as well on those manifolds that satisfy certain convexity properties. Basic convexity notions needed in this theory are recalled and important existence theorems due to Oka and Grauert will be reviewed.

1Bn will stand for Bn(0, 1), for simplicity. 10 1 Basic Notions and Classical Results

1.2.1 Complex Manifolds, Stein Manifolds and Holomorphic Convexity

By a complex manifold, we shall mean a Hausdorff space M with an open covering {Uj }j∈I, for some index set I, such that a homeomorphism ϕj from a domain Dj Cn = −1 ◦ in (n n(j)) to Uj is attached for each j, in such a way that ϕj ϕk is −1 ∩ ∩ = ∅ holomorphic on ϕk (Uj Uk) whenever Uj Uk . Unless otherwise stated, every connected component of M is assumed to be paracompact (In most cases M is −1 tacitly assumed to be connected.) (Uj ,ϕj ) is called a chart of M and the collection { −1 } C (Uj ,ϕj ) j∈I of charts is called an atlas of M.A -valued function f on M is said to be holomorphic if f ◦ ϕj are holomorphic on Dj . The set of holomorphic functions on M is denoted by O(M). The set of germs of holomorphic functions at x will be denoted by OM,x. For any local coordinate z around x, one has an ∼ isomorphism OM,x = C{z}. Unless stated otherwise, atlases are taken to be maximal −1 ∈ with respect to the inclusion relation. ϕj are called local coordinates around x −1 ◦ Uj and ϕj ϕk are called coordinate transformations. By an abuse of language, for a local coordinate ψ around a ﬁxed point x ∈ M, the condition ψ(x) = 0 will be assumed in many cases. A complex manifold M is said to be of dimension n if maxj dim Dj = n and of pure dimension n if dim Dj = n for all j. Unless stated otherwise, complex manifolds will be assumed to be ﬁnite dimensional and of pure dimension. By an abuse of language, topological spaces with discrete topology are regarded as 0-dimensional complex manifolds. It is conventional to call connected 1-dimensional complex manifolds Riemann surfaces. Cn and the domains over Cn are regarded as a complex manifold in an obvious way. n Example 1.2 Let Dj = C (j = 0, 1,...,n)and let

n n CP = Dj / ∼, j=0 where ∼ is an equivalence relation deﬁned by

Dj (z1,...,zn) ∼ (w1,...,wn) ∈ Dk

⇐⇒ (z1,...,zj , 1,zj+1,...,zn)//(w1,...,wk, 1,wk+1,...,wn).

Here v//w means that there exists ζ ∈ C \{0} such that ζv = w. n Then, with respect to the quotient topology and the natural maps ϕj : Dj → CP induced from the inclusion, CPn is (or rather becomes, more precisely speaking) a compact complex manifold. CPn is called the complex projective space of dimension n. 1.2 Complex Manifolds and Convexity Notions 11

= { −1 } Let Mμ (μ 1, 2) be two complex manifolds with atlases (Uμ,j ,φμ,j ) j∈Iμ , respectively. Then the product space M1 ×M2 is a complex manifold with respect to { × −1 } a (non-maximal) atlas (U1,j U2,k,(φ1,j ,φ2,k) ) (j,k)∈I1×I2 . A continuous map −1 ◦ ◦ F from M1 to M2 is called a holomorphic map if ϕ2,k F ϕ1,j are all holomorphic. The set of holomorphic maps from M1 to M2 will be denoted by O(M1,M2). Given a surjective holomorphic map f : M1 → M2,amaps from an open set U in M2 to M1 will be called a section if f ◦ s = idU holds. (s need not be holomorphic.) A holomorphic map F is said to be biholomorphic if it has a holomorphic inverse. If O(M1,M2) contains a biholomorphic map, M1 and M2 are ∼ said to be isomorphic to each other (denoted by M1 = M2). Aut M will stand for the group of biholomorphic automorphisms of M. A proper holomorphic map F : M1 → M2 is called a modification if there exists a nowhere-dense subset A of | M1 such that F M1\A is a biholomorphic map onto its image. We shall say that M1 and M2 are modifications of each other. Example 1.3 Let π : Cn+1 \{0}→CPn be defined by

π(ξ0,ξ1,...,ξn) = φj (z1,...,zn) for ξj = 0 and

(ξ0,ξ1,...,ξn)//(z1,...,zj−1, 1,zj ,...,zn).

n+1 n Then π ∈ O(C \{0}, CP ). ξ = (ξ0,ξ1,...,ξn) is called the homogeneous n coordinate of CP . π(ξ0,ξ1,...,ξn) will be denoted by [(ξ0,...,ξn)] (the equivalence class) or (ξ0 : ξ1 : ··· : ξn) (continued ratio). For any (C-vector) subspace V ⊂ Cn+1 of codimension one, π(V \{0}) is called a complex hyperplane.

A holomorphic map F : M1 → M2 is called an embedding if the following are satisﬁed: (1) F is injective. (2) For any point p ∈ M1 there exist a neighborhood U p and a chart (V, ψ) of M2 such that F(p) ∈ V , ψ(F(p)) = 0 and

F(U) ={q ∈ V ; ψ(q)1 =···=ψ(q)k = 0} for some k = k(p).

The image F(M1), equipped with the topology of M1, of a holomorphic embedding F will be called a complex submanifold of M2. The integer minp∈M1 k(p) is called the codimension of F(M1). By an abuse of language, we shall call F(M1) a closed complex submanifold of M if F is a proper holomorphic embedding. A closed complex submanifold of an open subset of M is called a locally closed complex submanifold of M. Closed submanifolds of codimension one are called complex hypersurfaces. A holomorphic embedding from D to M is called a holomorphic disc in M.An upper semicontinuous function Φ : M →[−∞, ∞) is called a plurisubharmonic 12 1 Basic Notions and Classical Results function on M if Φ ◦ ι is subharmonic for any holomorphic disc ι : D → M.Itis easy to see that Φ is plurisubharmonic if and only if Φ ◦ ψ−1 is plurisubharmonic for every chart (U, ψ) in the sense of Deﬁnition 1.5. The set of plurisubharmonic functions on M will be denoted by PSH(M). We put

PH(M) ={u;±u ∈ PSH(M)}.

Elements of PH(M) are called pluriharmonic. Pluriharmonic functions are locally characterized as real parts of holomorphic functions. PH(M) is a real vector space and PSH(M) is a convex cone containing PH(M) as an edge. Given any subgroup G ⊂ Aut M such that

(1) γ · x := γ(x) = x if G γ = idM (2) {γ ∈ G; γ(K)∩ K = ∅} < ∞ for any compact set K ⊂ M, where A := the cardinality of A, the projection π : M → M/G := {G · x ; x ∈ M} naturally induces on M/G a complex manifold structure. n Example 1.4 (complex semitori) Let Γ be an additive subgroup of C of the form m Z · ∈ Cn R j=1 vj (vj ) such that v1,v2,...,vm are linearly independent over . Then Γ is naturally identiﬁed with a subgroup of Aut Cn by

Γ v −→ { z −→ z + v}∈Aut Cn.

Since (1) and (2) are obviously satisﬁed by Γ , one has a complex manifold Cn/Γ which is called a complex semitorus. Cn/Γ is called a complex torus if it is compact, or equivalently m = 2n. A well-known theorem of Riemann says that a complex torus Cn/Γ can be embedded holomorphically into CP2n+1 if

(v1,v2,...,v2n) = (I, Z) holds for the n × n identity matrix I and an n × n symmetric matrix Z whose imaginary part is positive definite. Here vj are identified with the corresponding column vectors. Complex semitori are typical examples of pseudoconvex manifolds. (For the definition of pseudoconvex manifolds, see Sect. 1.2.3.) From this viewpoint, a generalization of Riemann’s theorem by Kodaira and its recent refinements will be discussed in Chap. 2 as an application of the L2 method. Example 1.5 The map

+ + ι : Cn 1 \{0}ξ −→ (ξ, [ξ]) ∈ Cn 1 × CPn is a holomorphic embedding. The closure of ι(Cn+1 \{0}) is a closed complex submanifold. The restriction of the projection Cn+1 × CPn → Cn+1 to ι(Cn+1 \{0}), 1.2 Complex Manifolds and Convexity Notions 13 say , is a modification. Note that −1(0) =∼ CPn. is called the blow-up centered at 0 ∈ Cn+1. Blow-ups centered at (or along) closed complex submanifolds are defined similarly. Compact complex manifolds which are isomorphic to closed complex submanifolds of CPn are called projective algebraic manifolds (over C). A theorem of Chow [Ch] says that every projective manifold is the set of zeros of some homogeneous polynomial in ξ. It may be worthwhile to mention that Chow’s theorem is a corollary of a continuation theorem of Hartogs type (cf. [R-S]). Example 1.6 (Hopf manifolds) Let H = (Cn \{0})/ ∼, where ∼ is an equivalence relation defined by

m (z1,...,zn) ∼ (w1,...,wn) ⇐⇒ wk = e · zk (1 ≤ k ≤ n) for some m ∈ Z.

Then, with respect to the quotient topology and the restrictions of the canonical n projection p : C \{0}→H to the domains D such that p|D is injective, H becomes a compact complex manifold. By applying a continuation theorem of Hartogs type, or appealing to the fact that the p-th Betti numbers of projective algebraic manifolds are even integers if p is odd, one knows that H is not projective algebraic. Definition 1.6 A Stein manifold is a complex manifold M such that any closed discrete subset of M is mapped bijectively to some closed discrete subset of C by some element of O(M). Theorem 1.13 (cf. [Bi, R-1, N]) A complex manifold M of dimension n is Stein if and only if there exists a proper holomorphic embedding from M to C2n+1. Remark 1.1 It is known that Stein manifolds of dimension n are properly and [ 3n ]+1 holomorphically embeddable into C 2 if n ≥ 2(cf.[E-Grm, Sm, F’17]). Definition 1.7 A complex manifold M is said to be holomorphically convex if any closed discrete subset of M is properly mapped onto some closed discrete subset of C by some element of O(M). Theorem 1.14 (cf. [Gra-1]) An n-dimensional complex manifold M is Stein if and only if the following are satisfied: (1) M is holomorphically convex. (2) For any two distinct points p, q ∈ M, there exists f ∈ O(M) such that f(p) = f(q). (3) For any p ∈ M there exist a neighborhood U p and f1,...,fn ∈ O(M) such that (U, (f1,...,fn)) is a chart of M. Remark 1.2 The class of Stein manifolds was first introduced by K. Stein in [St] by the properties (1) to (3) as above. So, Definition 1.6 was originally one of the characterizations of Stein manifolds. 14 1 Basic Notions and Classical Results

Grauert also established another characterization of Stein manifolds by generalizing Oka’s theory on pseudoconvex domains over Cn, which will be reviewed in Sect. 1.2.3 after a preliminary in Sect. 1.2.2.

1.2.2 Complex Exterior Derivatives and Levi Form

Let us recall that differentiable manifolds of class Cr ,for0≤ r ≤∞or r = ω, n m are deﬁned by replacing the domains Dj in C by domains in R and requiring −1 ◦ r ϕj ϕk to be of class C . Basic terminology on differentiable manifolds such as Cr maps, tangent bundles, differential forms, exterior derivatives, etc. will be used freely (cf. [W]). By an abuse of notation, Cr (M) will stand for the set of C-valued Cr functions on M. The set of germs of C-valued Cr functions at x will be denoted C r by M,x. C Let M be a complex manifold of dimension n.ByTM we shall denote the complex tangent bundle of M, i.e. the complexiﬁcation of the tangent bundle TM of M as a differentiable manifold. Recall that C = C TM TM,x x∈M as a set, where

C ={ ∈ C ∞ C ; = + } TM,x v Hom( M,x, ) v(fg) f(x)v(g) g(x)v(f ) .

Here Hom(A, B) denotes the set of C linear maps from A to B. Let C ∗ = C ∗ ∗ = C (TM ) (TM,x) (V Hom(V, ) for any complex vector space V) x∈M

C ∈ be the complex cotangent bundle of M, i.e. the dual bundle of TM . For any x M we put

0,1 ={ ∈ C ; = ∈ O } TM,x v TM,x v(f ) 0iff M,x , 1,0 = 0,1 TM,x TM,x (complex conjugate) and 0,1 = 0,1 1,0 = 1,0 TM TM,x,TM TM,x. x∈M x∈M 1.2 Complex Manifolds and Convexity Notions 15

1,0 TM is called the holomorphic tangent bundle of M. We put 1,0 ∗ = 1,0 ∗ (TM ) (TM,x) x∈M and p q p,q ∗ = 1,0 ∗ ⊗ 0,1 ∗ (TM ) (TM ) (TM ) .

Then

r C ∗ =∼ p,q ∗ (TM ) (TM ) . p+q=r

∞ According to this decomposition, the exterior derivative d actingonthesetofC r C ∗ sections of (TM ) decomposes naturally into the sum of the complex exterior derivative of type (1,0), denoted by ∂, and its conjugate ∂¯, the complex exterior derivative of type (0,1). In terms of a local coordinate z, = ∧ ¯ = + ¯ du d uIJ¯ dzI dzJ ∂u ∂u, where ∂uIJ¯ ∂ u ¯ dz ∧ dz¯ = dz ∧ dz ∧ dz IJ I J ∂z j I J I,J j,I,J j and ¯ ∂uIJ¯ ∂ u ¯ dz ∧ dz = dz ∧ dz ∧ dz . IJ I J ∂z j I J I,J j,I,J j

For any chart (U, ψ) of M,sayψ = (z1,...,zn), there is a natural identiﬁcation

Cn −→ 0,1 ∈ TM,x x U 1 n ξ = (ξ ,...,ξ ) −→ vx(ξ) by j ∂f ∞ vx(ξ)(f ◦ ψ) = ξ (ψ(x)), f ∈ C (ψ(U)). ∂zj 16 1 Basic Notions and Classical Results

→ → 0,1 1,0 The sectionx νx(ξ) (resp. x νx(ξ)) of TM (resp.TM ) over U will be denoted by ξ j ∂ (resp. ξ j ∂ ). ∂zj ∂zj Given a real-valued C2 function ϕ on M,theLevi form of ϕ at x ∈ M is deﬁned as a Hermitian form

n ∂2ϕ ξ j ξ k ∂zj ∂z¯k j,k=1 on ∂ 1,0 =∼ j ; ∈ Cn TM,x ξ ξ . ∂zj z=ψ(x)

Although the deﬁnition uses ψ = z, it is easy to see that the above Hermitian form 1,0 on TM,x is independent of the choice of local coordinates. The Levi form of ϕ is ¯ denoted simply by Lϕ, or more explicitly by ∂∂ϕ, but by an abuse of notation. ϕ is said to be q-convex (resp. weakly q-convex)atx if Lϕ has at most q − 1 nonpositive (resp. negative) eigenvalues at x. Note that the set of q-convex functions is not a convex cone unless q = 1. It is easy to verify and a fact of basic importance that a C2 function ϕ is plurisubharmonic on M if and only if ϕ is everywhere weakly 1-convex. If ϕ is 1-convex at x, we shall also say that ϕ is strictly plurisubharmonic at x. Given a complex manifold M and an upper semicontinuous function ϕ : M → [−∞, ∞), the domain {(ζ, x) ∈ C × M;|ζ |

1.2.3 Pseudoconvex Manifolds and Oka–Grauert Theory

Loosely speaking, the Levi problem asks to characterize holomorphically convex manifolds by geometric properties such as pseudoconvexity, or more weakly to ﬁnd nonconstant holomorphic functions on pseudoconvex manifolds. A complex manifold M is said to be Cr -pseudoconvex if M admits a Cr plurisubharmonic exhaustion function. Here a real-valued function, say Ψ on a topological space X, is called an exhaustion function on X if its sublevel sets

Xc ={x ∈ X; Ψ(x)

D ∩ U ={z ∈ U; ρ(z) < 0} and dρ vanishes nowhere on ∂D. We shall call ρ a deﬁning function of ∂D,or sometimes that of D if ρ is deﬁned on D ∪ U. We put

1,0 = 1,0 ∩ ⊗ C T∂D TM (T∂D ).

2 If D is C -smooth and Lρ| 1,0 is everywhere semipositive on ∂D for some T∂D defining function ρ of D, ∂D is said to be pseudoconvex. ∂D is called strongly pseudoconvex at x ∈ ∂D if Lρ| 1,0 is positive definite at x. T∂D Definition 1.8 A strongly pseudoconvex domain in M is a relatively compact domain in M whose boundary is everywhere strongly pseudoconvex. Strongly pseudoconvex domains admit strictly plurisubharmonic defining functions. In fact, for any defining function ρ of D, eAρ − 1 becomes strictly plurisubharmonic on a neighborhood of ∂D for sufficiently large A. Strongly pseudoconvex domains are 1-convex because − log (−ρ) is an exhaustion function on D which is plurisubharmonic outside a compact subset of D. Remark 1.5 A smoothly bounded pseudoconvex domain is called weakly pseudoconvex if it is not strongly pseudoconvex. There exist Cω-smooth weakly pseudoconvex domains which do not admit plurisubharmonic defining functions (cf. [B]). For a further extensive account of the Levi problem, see [Siu’78]. 1.3 Oka–Cartan Theory 19

1.3 Oka–Cartan Theory

In order to discuss the questions on the rings and modules of holomorphic functions, it is often necessary to approximate locally deﬁned functions by globally deﬁned ones. The language of sheaf cohomology is useful to describe such a procedure.

Once these notions are transplanted from the ﬁeld of algebraic functions to that of general analytic functions, various new questions naturally arise, because analytic functions show up (to us) not as global objects, but only as local ones. (Kiyoshi Oka—in a letter to Teiji Takagi)

1.3.1 Sheaves and Cohomology

Let {Fx}x∈X be a family of Abelian groups with the identity elements 0x ∈ Fx parametrized by a topological space X.Let F = Fx x∈X and let p : F → X be deﬁned by p(Fx) ={x}. For any open set U ⊂ X,let

F [U]={s : U −→ F ; p ◦ s = idU }.

By an abuse of language, elements of F [U] will be called possibly discontinuous sections of F .Ifs ∈ F [U] and s(x) = 0 (= 0x) for all x ∈ U, s will be called the zero section of F over U and denoted simply by 0.

Deﬁnition 1.9 A family {F (U)}U of subsets F (U) of F [U] is called a presheaf if the following are satisﬁed:

(1) s ∈ F (U), U ⊃ V ⇒ s|V ∈ F (V ). (2) f ∈ Fx ⇒ there exists a neighborhood U x and s ∈ F (U) satisfying s(x) = f . (3) s ∈ F (U), x ∈ U,s(x) = 0x ⇒ s = 0 on a neighborhood of x. (4) s ∈ F (U), t ∈ F (V ) ⇒ (s − t)|U∩V ∈ F (U ∩ V). {F } F A presheaf (U) U induces a topology on the set in such a way that { ; ∈ F } F F U⊂X s(U) s (U) is a basis of open sets of . Elements of are continuous with respect to this topology. Deﬁnition 1.10 A presheaf {F (U)} is called a sheaf if F (U) ={s ∈ F [U]; For any x ∈ U there exists a neighborhoood V x such that s|V ∈ F (V )}. 20 1 Basic Notions and Classical Results

Clearly, for any presheaf {F (U)} (an abbreviation for {F (U)}U ), one can find a sheaf {F (U)} such that F (U) ⊂ F (U) ⊂ F [U] uniquely. {F (U)} will be called the sheafification of {F (U)}. For simplicity, the topological space F will also stand for the sheaf {F (U)}.To be explicit, F is called a sheaf over X.Themapp : F → X will be referred to as a sheaf projection. Fx is called a stalk of F at x, and the elements of Fx the germs at x. Elements of F (U) will be called the sections of F over U. By (3) above, the germs at x of sections in F (U) are naturally identified with elements of Fx if U x, i.e.

Fx = ind.limUxF (U) with respect to the inductive system induced from the natural restriction maps F [U]→F [V ] for U ⊃ V x. For any s ∈ F [U] the germ of s at x will be denoted by sx. In short, s(x) = sx if s ∈ F (U).LetG be another sheaf over X. G is called a subsheaf of F if G (U) ⊂ F (U) for any open set U ⊂ X.The constant sheaf CX → X is defined as the sheaf whose stalks are C. CX will be simply denoted by C. Note that the family {F [U]} itself is not necessarily a sheaf because the condition (3) may not be satisfied. However, if we put ˆ Fx = ind.limUxF [U], ˆ ˆ F = Fx xX and ˆ ˆ F (U) ={ˆs : U −→ F ;ˆs(x) = sx for some s ∈ F [U]}, ˆ ˆ then {F (U)}U is a sheaf over X. F has a property that any section over any open set extends to X as a section. Sheaves having this property are called flabby sheaves. Since F is a subsheaf of Fˆ, we shall call the sheaf Fˆ the canonical flabby extension of F . For any two sheaves Fj (j = 1, 2) over X, the direct sum F1 ⊕ F2 is a sheaf defined by {F1(U) ⊕ F2(U)}U . For any continuous map β : X → Y ,thedirect image sheaf of F by β, denoted by β∗F , is defined over Y by

−1 (β∗F )x = ind.limUxF1(β (U))

−1 and (β∗F )(U) = F (β (U)). ⊂ F F | F | := If A X, the sheaf x∈A x is denoted by A.Here A(U) ind.limV ⊃U F (V ). F |A is called the restriction of F to A. A is called the support of F if “Fx ={0x}⇔x/∈ A ”. The support of F is denoted by supp F . Sheaves of rings and sheaves of modules are deﬁned similarly. 1.3 Oka–Cartan Theory 21

Definition 1.11 A ringed space is a topological space equipped with a sheaf of rings. For any complex manifold M, the family {O(U);U is open in M} is naturally regarded as a sheaf by identifying an element of O(U) as the collection of its germs. This sheaf is called the structure sheaf of M and denoted by OM ,orsimplybyO. (M, O) is the most important example of ringed space for our purpose. We note that the domains of holomorphy are nothing but the connected components of OCn .For meromorphic functions, domains of meromorphy can be characterized similarly. Namely, in the sheaf theoretic terms, meromorphic functions are identified as the sections of a sheaf in the following way: Let Mx be the quotient field of Ox,let M = Mx, x∈M and

M (U) ={h ∈ M [U]; for every x ∈ U there exist a neighborhood V x and

fy f, g ∈ O(V ) such that h(y) = for all y ∈ V } gy for any open set U ⊂ M. Sections of the sheaf {M (U)}U are called meromorphic functions. Connected components of the sheaf M as the topological space are called the domains of meromorphy. A sheaf p1 : F1 → X1 is said to be isomorphic to a sheaf p2 : F2 → X2 if there exists a homeomorphism ψ : X1 → X2 and a bijection β : F1 → F2 such that

| ∈ F F β F1,x Hom( 1,x, 2,ψ(x)) for all x ∈ X1. Complex manifolds are naturally identiﬁed with ringed spaces which are locally n isomorphic to (D, OD) for some domain D in C . Deﬁnition 1.12 An ideal sheaf of a ringed space (X, R) is a sheaf of R-modules (X, I ) such that Ix is an ideal of Rx for each x ∈ X.

Let R → X be a sheaf of commutative rings with units and let Ej → X(j = 1, 2) be sheaves of R-modules (i.e. Ej,x are Rx-modules, etc.). A collection of Rx-homomorphisms

αx : E1,x −→ E2,x,x∈ X, denoted by α : E1 → E2 is called a homomorphism between R-modules if

s ∈ E1(U) ⇒ α ◦ s ∈ E2(U) 22 1 Basic Notions and Classical Results holds for any open set U ⊂ X. α◦s will also be denoted by α(s) (for a typographical reason). For the sheaves of Abelian groups and those of rings, homomorphisms are deﬁned similarly. Sheaves of O-modules are called analytic sheaves. The stalkwise direct sum R⊕m is called a free R-module of rank m. A sheaf of R-modules is called locally free if it is locally isomorphic to a free sheaf. Locally free sheaves of rank one are said to be invertible. A sheaf E of R-modules is said to be torsion free if Ex are torsion free Rx-modules. Locally free R-modules are torsion free. A holomorphic map ψ between two complex manifolds (Mj , Oj )(j = 1, 2) induces a homomorphism

: O | −→ O | ψ∗ 2 ψ(M1) ψ∗ 1 ψ(M1) by ψ∗(fψ(x)) = (f ◦ ψ)ψ(x). Conversely, a continuous map ψ : M1 → M2 is : O | → O | holomorphic if there exists a homomorphism β 2 ψ(M1) ψ∗ 1 ψ(M1) which induces at every point x ∈ M1 a homomorphism from O2,ψ(x) to O1,x which maps the invertible elements of O2,ψ(x) to those of O1,x. For any homomorphism α : E1 → E2, the collection of preimages of 0, which is called the kernel of α, is naturally equipped with a sheaf structure whose sections over U are precisely the elements of {s ∈ E1(U); α ◦ s = 0}, the kernel of the homomorphism

αU : E1(U) s −→ α ◦ s ∈ E2(U).

The kernel of α will be denoted by Ker α. Deﬁnition of the cokernel of α is more delicate: Let coker α = coker αx, x∈X : E → let π 2 x∈X coker αx be the canonical projection, and let

coker α(U) ={s ∈ coker α[U]; s = π ◦ˆs for some sˆ ∈ E2(U)}.

Then {coker α(U)} is clearly a presheaf. The sheafification of {coker α(U)} will be called the cokernel sheaf of α and denoted by Coker α. When α is an inclusion, Coker α will be denoted by E1/E2.Theimage sheaf Im α of α is defined similarly. Given an ideal sheaf I of R, the cokernel R/I of the inclusion morphism ι : I → R carries naturally the induced structure of a sheaf of commutative rings. A sequence

+ + ···−→E k −→ E k 1 −→ E k 2 −→··· (1.4) 1.3 Oka–Cartan Theory 23 of sheaves of Abelian groups or R-modules is called an exact sequence if, for any two successive morphisms αk : E k → E k+1 and αk+1 : E k+1 → E k+2,Imαk = Ker αk+1 holds. The family E ∗ ={(E k,αk)} is called a complex of sheaves if Im αk ⊂ Ker αk+1 holds for all k.Aresolution of a sheaf F is by deﬁnition an exact sequence of the form

0 −→ F −→ E 0 −→ E 1 −→··· .

Definition 1.13 The canonical flabby resolution of a sheaf F → X is a complex ∗ k k F ={(F ,j )}k∈Z defined by k F = 0 (= {0x}) for k ≤−1, − F 0 = Fˆ,j1 = 0, + ∧ F k 1 = (Coker j k) (the canonical flabby extension) and j k+1 = the composite of the canonical projection F k → Coker j k and the inclusion Coker j k → (Coker j k)∧,fork ≥ 0, inductively. Clearly, {(F k,jk)} is a complex of sheaves and the sequence

0 −→ F −→ F 0 −→ F 1 −→ F 2 −→··· (1.5) is exact. Definition 1.14 The p-th cohomology group of X with values in the sheaf F → X, denoted by H p(X, F ), is by definition the p-th cohomology group of { F k k } the complex ( (X), jX) . The elements of H p(X, F ) will be referred to as the F -valued p-th cohomology classes. The restriction homomorphism F (U) → F (V ) naturally induces a homomorphism H p(U, F ) → H p(V, F ). Note that H 0(X, F ) = F (X).AsforH p(V, F ), p ≥ 1, let us briefly recall a 1 F ∈ 1 description of the cohomology classes in H (X, ). Given any v Ker jX, there 1 1 exists an open covering U ={U } of X and u ∈ F (U ) such that j (u ) = v 1 holds on U . Hence u − u ∈ Ker j ∩ = F (U ∩ U ). If the cohomology U U class represented by v is zero, there exists u ∈ F 0 such that j 0(u) = v. As a result one has u − u ∈ F (U ). Therefore the collection of u − u , as an element of 1 ⊕ , Ker j ∩ , is in the image of the map U U

0 δ : F (U ) {u } −→ { u − u } , ∈ F (U ∩ U ). , 24 1 Basic Notions and Classical Results

Consequently, letting

p U F = { ∈ F ∩···∩ ; C ( , ) u 0... p (U 0 U p ) u 0... p 0,..., p

is alternating in 0,..., p} and deﬁning

p p p+1 δU : C (U , F ) −→ C (U , F ) and H p(U , F ) respectively by p { } = − j δU ( u ... p ,..., p ) ( 1) u ...... 0 0 0 j−1 j+1 p+1 ,..., 0≤j≤p 0 p+1

p p−1 and Ker δU /Im δU , one has a homomorphism

1 1 1 γ : H (X, F ) −→ ind.limU H (U , F )

[ ]→[{ − } ] p : defined by the correspondence v u u , , and similarly γ H p(X, F ) → ind.limU H p(U , F ) for all p. Here the inductive system {H p(U , F )} is with respect to the restriction homomorphisms H p(U , F ) → H p(V , F ) for the refinements V of U . See [G-R] (for instance) for the detail of the construction of γ p for p ≥ 1 and for the proof of the following extremely important fact. Theorem 1.18 γ p are isomorphisms if X is a paracompact Hausdorff space. For any paracompact Hausdorff space X, a sheaf G → X of Abelian groups is said to be fine if, given any open covering U of X, there exists a locally finite refinement V ={Vj } of U and homomorphisms hj : G → G such that

supp hj := {x ; hj | Gx = 0}⊂Vj = and j hj 1. Corollary of Theorem 1.18. If G → X is a ﬁne sheaf,

H p(X, G ) = 0 for any p ≥ 1. A resolution 0 → F → E 0 → E 1 → ··· is said to be fine if E k are fine sheaves. Another basic fact is the existence of a canonically defined exact sequences of the cohomology groups: Let 1.3 Oka–Cartan Theory 25

0 −→ E −→ F −→ G −→ 0 (1.6) be an exact sequence of sheaves over X. Then it is easy to see that the induced sequence

0 −→ E (X) −→ F (X) −→ G (X) is exact. By the exactness of (1.6), this sequence can be prolonged canonically as

E (X) → F (X) → G (X) → H 1(X, E ) → H 1(X, F ) → H 1(X, G ) → H 2(X, E ) →··· , which is called the long exact sequence associated to (1.6)(cf.[G-R]).

1.3.2 Coherent Sheaves, Complex Spaces, and Theorems A and B

In the study of ideals of holomorphic functions, Oka and Cartan were led to introduce a notion characterizing a class of ideal sheaves of O,thecoherence (cf. [O-2] and [C]). Deﬁnition 1.15 An R-module E over a topological space X is called coherent if:

(1) E is locally finitely generated, i.e. for any x0 ∈ X there exist a neighborhood U x0 and finitely many sections of M over U whose values at x ∈ U generate Ex over Rx for any x ∈ U. (2) For any m ∈ N and for any morphism α from the direct sum R⊕m to E ,Kerα is locally finitely generated. A penetrating insight (definitely shared by Oka and Cartan) was that a principal basic question of several complex variables is to establish a criterion for the analytic sheaves to be globally generated. For any complex manifold (M, O), Oka established the following basic result by exploiting the Weierstrass division theorem to run an induction argument on the dimension. Theorem 1.19 (Oka’s coherence theorem) O is coherent. For the proof, the reader is referred to [G-R, Hö-2], or [Nog]. By this theorem, for any coherent O-module F and for any x ∈ M, one can find a neighborhood U x and an exact sequence over U of the form

⊕mk ⊕m2 ⊕m1 ···−→O |U −→ · · · −→ O |U −→ O |U −→ F |U −→ 0, 26 1 Basic Notions and Classical Results which is called a free resolution of F over U. Since C{z1,...,zn} is a regular local ⊕m ⊕m − ring of dimension n, the kernel of O n |U → O n 1 |U is locally free by Hilbert’s syzygy theorem (cf. [G-R]). For any A ⊂ M, IA will stand for the ideal sheaf of O consisting of the germs of holomorphic functions vanishing along A, i.e.

IA(U) ={f ∈ O(U); f |U∩A = 0}.

IA is called the ideal sheaf of A, for short. Definition 1.16 AclosedsetA ⊂ M is called an analytic set if for every point x ∈ A there exist a neighborhood U, m ∈ N, and f1,...,fm ∈ O(U) such that U ∩ A ={w ∈ U; f1(w) =···=fm(w) = 0}. From the definition, it is clear that supp(O/I ) is analytic if I is a coherent ideal sheaf. The vector-valued holomorphic function (f1,...,fm) is called a local defining function of A around x.Bythedimension of an analytic set A at x ∈ A, we shall mean the minimal number of holomorphic functions f1,...,fk defined on ∩ k −1 a neighborhood of U such that x is isolated in A ( j=1 fj (0)).

Theorem 1.20 (Rückert’s Nullstellensatz) Let (f1,...,fm) be a local deﬁning function of an analytic set A around x and let f ∈ IA,x. Then there exists p ∈ N such that

m p f ∈ fj · OM,x. j=1

For the proof, the reader is referred to [G-R, Chapter 3, A].

Theorem 1.21 (Cartan’s coherence theorem) IA is coherent if A is analytic.

Sketch of proof Let x ∈ A,let(f1,...,fm) be a local deﬁning function of A around x, and let IA be the ideal sheaf generated by fj (1 ≤ j ≤ m) over a neighborhood U x. Then one has an exact sequence of O-modules ˆ ˆ 0 −→ IA −→ IA|U −→ IA|U /IA −→ 0. ˆ Since IA is coherent by Theorem 1.19, the coherence of IA follows by a descending induction on the codimension of A. Remark 1.6 The ideal sheaf J of the form 2 −ϕ Jx = fx ∈ Ox; |f | e dλ < ∞ for some neighborhood U of x U turns out to be coherent if ϕ is plurisubharmonic (see Chap. 3). 1.3 Oka–Cartan Theory 27

Roughly speaking, complex spaces are complex manifolds with singularities. In function theory, such things arise as ringed spaces which are locally isomorphic to those whose underlying spaces are the sets of common zeros of holomorphic functions. Definition 1.17 A ringed space (X, O) is called a complex space if every point x ∈ X has a neighborhood U such that (U, O|U ) is isomorphic to N (supp(OD/I ), OD/I ) for some domain D in C (N = N(x)) and for some coherent ideal sheaf I of OD. Example 1.7 Let D be a domain in Cn,letF ∈ O(D),letX = F −1(0) and let O = OD/F · OD. Then (X, O) is a complex space, since F · OD is coherent by Theorem 1.19. X will be referred to as the underlying space of the complex space (X, O). (X, O) is said to be compact if so is the underlying space X. A point x ∈ X is called ∼ a regular point of (X, O) if one can find U and D such that O|U = OD.Thesetof regular points of X is denoted by Xreg. X \ Xreg is denoted by Sing X. O is called the structure sheaf of X. The structure sheaf is denoted also by OX. Coherent O- sheaves will be called coherent analytic sheaves. A closed set A ⊂ X is called an analytic set if it is the support of some coherent analytic sheaf over X. An analytic set of X is naturally equipped with the structure of a reduced complex space induced n from OX (see Definition 1.18 below). Analytic sets of CP are called projective algebraic sets. The implicit function theorem naturally implies that Sing X is an analytic set of X. X is called nonsingular if Sing X = ∅. (X, O) is said to be irreducible if every proper analytic set is nowhere dense. An irreducible complex space (Y, OY ) is called an irreducible component of (X, OX) if Y ⊂ X and the inclusion map ι : Y → X is accompanied with a surjective sheaf homomorphism OX|Y → ι∗OY |Y whose kernel has a nowhere dense support in Y . Irreducible complex spaces are called varieties. By a routine argument one can infer the following from Theorem 1.19. Theorem 1.22 The structure sheaf of a complex space is coherent. For any complex space (X, O), the elements of O(X) will be called holomorphic functions on X. Holomorphic functions on X naturally induce genuine C-valued functions on X. If a holomorphic function f on X is zero as a function, then, for each x ∈ Xfx is nilpotent, i.e. some power of fx is zero, by Rückert’s Nullstellensatz. By an abuse of notation, the values of f in C will be denoted by f(x). Given two complex spaces (X, OX) and (Y, OY ),aholomorphic map from (X, OX) to (Y, OY ) is by definition a pair of continuous map ψ : X → Y and a homomorphism

β : OY |ψ(X) −→ ψ∗OX|ψ(X) between the sheaves of rings which maps invertible elements to invertible elements. 28 1 Basic Notions and Classical Results

For any holomorphic map (ψ, β) from (X, OX) to (Y, OY ), a homomorphism p p from H (Y, OY ) to H (X, OX) is induced canonically. For simplicity, (ψ, β) will p p be referred to as ψ and the induced homomorphism H (Y, OY ) → H (X, OX) by ψ∗. Deﬁnition 1.18 A complex space (X, O) is said to be reduced if no stalk of O contains a nilpotent element. For any reduced complex space (X, O) and f, g ∈ O(X), f = g if and only if f(x) = g(x)(∈ C) for all x ∈ X. For any complex space (X, O), the collection of the nilpotent elements in the stalks of O is an ideal sheaf, say J . Then (X, O/J ) is a reduced complex space. We shall call it the reduction of (X, O). If (X, O) is reduced, Xreg is an everywhere dense subset of X. We then deﬁne the dimension of X by dim X := dim Xreg and put

dimx X = sup {dim Ureg; U is a neighborhood of x} for any x ∈ X. dim X and dimx X will stand for those for the reduction of (X, O).Itiseasyto verify that this definition of the dimension agrees with that for analytic sets when (X, O) = (A, O/IA). The codimension of A ⊂ X is defined as dim X − dim A. It will be denoted by codimX A. X is called a complex curve if dim X = 1. Definition 1.19 A complex space (X, O) is called a Stein space (resp. a holomorphically convex space) if any discrete closed subset of X is mapped injectively (resp. properly) into a discrete closed subset of C by a holomorphic function on X. Theorem 1.23 (Cartan’s theorem A) Let (X, O) be a Stein space. Then, for any coherent analytic sheaf F over X and for any point x ∈ X, the image of the natural restriction map

F (X) −→ Fx generates Fx over Ox. Combining Theorem 1.23 with Cartan’s coherence theorem, we obtain for instance the following. Proposition 1.1 Let (X, O) be a Stein space, let A ⊂ M be an analytic set, and let x ∈ A be any point. Then there exist a neighborhood U x and f1,...,fm ∈ O(X) such that U ∩ A ={y ∈ U; f1(y) =···=fm(y) = 0}. Theorem 1.23 was ﬁrst established by Oka when X is a domain of holomorphy in Cn and F is a coherent ideal sheaf (idéal de domaine indéterminé) over X.For the proof, Oka and Cartan solved a problem which P. Cousin had solved in 1895 in a very special case to construct meromorphic functions with given poles on the products of plane domains. This argument was extended eventually to show that Theorem 1.23 is a consequence of the assertion that H 1(X, F ) = 0 holds for any 1.3 Oka–Cartan Theory 29 coherent analytic sheaf over a Stein space (X, O). An ultimately strengthend form of such a cohomology vanishing theorem on Stein spaces is the following. Theorem 1.24 (Cartan’s theorem B) H p(X, F ) = 0 for any p ≥ 1 if F is a coherent analytic sheaf over a Stein space (X, O). There are a lot of implications of the vanishing of cohomology groups. It may be worthwhile to recall that there is a characterization of Stein spaces among them. Theorem 1.25 (X, O) is a Stein space if and only if H 1(X, I ) = 0 for any coherent ideal sheaf of O.

Proof For any discrete closed set Γ ⊂ X, the ideal sheaf IΓ of Γ is coherent by 1 Theorem 1.21. Hence H (X, IΓ ) = 0, so that from the exact sequence

0 −→ IΓ −→ OX −→ OX/IΓ −→ 0 one has the surjectivity of the natural restriction homomorphism O(X) → CΓ . Remark 1.7 The surjectivity of O(X) → CΓ is equivalent to the injectivity of H 1(X, I ) → H 1(X, O) by the long exact sequence. This point has a significance in the development of the application of the L2 technique. If (X, O) is a domain over Cn, it is easy to see that X is Stein if and only if H q (X, O) = 0 for any 1 ≤ q ≤ n − 1(cf.[Oh-21, 2.3]). Actually, H n(X, O) = 0 holds whenever X is a complex space of dimension n which does not contain any compact n-dimensional analytic subset (cf. [Siu-1]). Once it is recognized that Theorem 1.23 is a consequence of a vanishing theorem as above, it is not difficult to show the following for instance. Theorem 1.26 (cf. [Gra-2]) Every analytic subset of a Stein space of dimension n is the set of common zeros of at most n + 1 holomorphic functions. Remark 1.8 Forster and Ramspott [F-R] proved that actually n functions suffice. It was known by Kronecker [K] that every algebraic set in Cn is the set of common zeros of n+1 polynomials. Eisenbud and Evans proved in [Eb-E] that n polynomials suffice. This theory is closely related to the Oka principle asserting the categorical equivalence between topology and analysis on Stein manifolds. (See [F’17]forthe detail.) The L2 method provides another effective way to analyze the sheaf cohomology groups. As a result, Theorem 1.24 can be extended to pseudoconvex spaces and certain ideal sheaves arising naturally in basic questions of complex geometry. (See Chap. 3.) 30 1 Basic Notions and Classical Results

1.3.3 Coherence of Direct Images and a Theorem of Andreotti and Grauert

A complex space (X, O) is said to be normal if, for any open set U ⊂ X and for any nowhere-dense analytic set A ⊂ U, bounded holomorphic functions on U \ A extend holomorphically to U. It is easy to see that normal complex curves are nonsingular. In [O-3], Oka proved that every reduced complex space has a “normal model”, i.e. for any reduced complex space (X, O) there exist a reduced complex ˆ O : ˆ → space (X, Xˆ ) and a proper surjective holomorphic map p X X such that ˆ ˆ | −1 : → p p (Xreg) is a biholomorphic map onto Xreg and the inclusion map ι Xreg X O =∼ O induces an isomorphism ι∗ Xˆ . It can be stated as another coherence theorem as follows. Theorem 1.27 (Oka’s normalization theorem) Let (X, O) be a reduced complex space and let ι : Xreg → X be the inclusion map. Then the direct image sheaf O + ≤ ι∗ Xreg is a coherent sheaf of rings if dimx Sing X 2 lim infy→x dimy X for all x ∈ Sing X. Substantially, this is a primitive form of Hironaka’s desingularization theorem. Recall that Hironaka’s desingularization theorem asserts that every reduced complex space has a nonsingular model, i.e. for any reduced complex space (X, OX) one can find a complex manifold (M, OM ) and a proper holomorphic map π˜ : M → X such ˜ | that π M\˜π−1(Sing X) is a biholomorphic map onto Xreg. Let ψ : X → Y be a holomorphic map. For any analytic sheaf F → X,the p-th direct image sheaf of F , denoted by Rpψ∗F , is defined as the sheafification of the presheaf p −1 U −→ { s : U → (ind.limV yH (ψ (V ), F )); yU p −1 there exists u ∈ H (ψ (U), F ) such that s(y) = uy for all y}.

Note that R0ψ∗F = ψ∗F . The following is a very profound result. Theorem 1.28 (cf. [Gra-4, Gra-R-2]) Let ψ : X → Y be a proper holomorphic map between complex spaces and let F → X be a coherent analytic sheaf. Then Rpψ∗F (p ≥ 0) are coherent analytic sheaves over Y . Corollary 1.2 (Remmert’s proper mapping theorem) For any proper holomorphic map ψ : X → Y , ψ(X) is an analytic set of Y . Theorem 1.28 is a generalization of the following. Theorem 1.29 (Cartan–Serre ﬁniteness theorem) For any compact complex space (X, O) and for any coherent analytic sheaf F over X, dim H p(X, F )<∞ for all p. 1.4 ∂¯-Equations on Manifolds 31

Andreotti and Grauert studied intermediate results between Theorems 1.24 and 1.29. For that they introduced a class of q-convex spaces. Deﬁnition 1.20 A continuous function φ : X → R is called q-convex (resp. plurisubharmonic) around x ∈ X if one can ﬁnd a neighborhood U x, a domain N D ⊂ C , a coherent ideal sheaf I ⊂ OD such that ∼ (U, OU ) = (supp(OD/I ), OD/I ) (1.7)

˜ and a q-convex function φD on D whose restriction to supp(OD/I ) coincides with φ|U by the isomorphism (1.7). Definition 1.21 A complex space (X, O) is said to be q-complete (resp. q- convex, resp. pseudoconvex)ifX admits a continuous exhaustion function which is q-convex everywhere (resp. q-convex outside a compact subset of X, resp. plurisubharmonic everywhere). In view of Theorem 1.15, the following is a generalization of Theorem 1.24. Theorem 1.30 (cf. Andreotti and Grauert [A-G]) Let X be a q-complete (resp. q-convex) space and let F → X be a coherent analytic sheaf. Then H p(X, F ) are 0 (resp. finite dimensional) for p ≥ q. It is known that n-dimensional noncompact complex spaces without n- dimensional compact irreducible components are n-complete (cf. [Oh-9]). However, it is not known, except for the cases q = 1 and q ≥ dim X, whether or not (X, O) is q-complete (resp. q-convex) if H p(X, F ) are all zero (resp. finite dimensional) for any p ≥ q and for any analytic sheaf F . The so-called Grauert conjecture is “It is the case”. A reason to expect it is in the theory of cycle spaces. It is known that the set of certain equivalence classes of holomorphic maps from compact complex manifolds to a complex space X is canonically equipped with a structure of a complex space, say B(X) (cf. [B-1]), and the connected components of B(X) for the maps from purely (q − 1)-dimensional manifolds are Stein if X is q-complete (cf. [B-2]). (See [G-W-1, Dm-1] and [Oh-9] for the case q = dim X.) For q-convex manifolds M and locally free analytic sheaves F over M, one can analyze H p(M, F ) by the L2 method. For instance, one has an extension of the Hodge theory on compact complex manifolds to certain q-convex manifolds (cf. Chap. 2).

1.4 ∂¯-Equations on Manifolds

By extending Theorem 1.18, one has a very basic result on the cohomology groups of complex manifolds with values in the locally free analytic sheaves, the Dolbeault isomorphism theorem. Based on this, one can represent the cohomology classes by differential forms. The cohomology classes in H p(X, F ) can be studied in a way 32 1 Basic Notions and Classical Results closely related to the geometry of X and F by exploiting such an expression. This is the method of L2 estimates to be discussed in subsequent chapters. Basic notions needed for that are holomorphic vector bundles, Dolbeault cohomology groups, Chern connections, curvature forms, etc., which belong to differential geometry and recalled below together with classical results as Dolbeault’s isomorphism theorem and Serre’s duality theorem, which are natural counterparts of the de Rham isomorphism and the Poincaré duality in the classical global differential geometry.

1.4.1 Holomorphic Vector Bundles and ∂¯-Cohomology

A holomorphic vector bundle over a complex manifold is by deﬁnition a C∞ complex vector bundle whose transition functions are holomorphic. In other words, a complex manifold E is called a holomorphic vector bundle over a complex manifold M if E is of the form x∈M Ex, as a set for some vector spaces Ex parametrized by M, satisfying the following requirements: (1) The map

π : E ⊃ Ex −→ { x}⊂M

: 1,0 → 1,0 is holomorphic and dπ TE TM is everywhere of maximal rank. (2) For any open set U ⊂ M, f, g ∈ O(U) and holomorphic sections s,t : U → E, fs+ gt is also holomorphic. The rank of E (= rank E := dim Ker dπ) is a locally constant function on M, which will be assumed to be constant unless stated otherwise. From the definition, it is clear that the direct sums, the duals and the tensor products of holomorphic vector bundles are naturally defined by fiberwise construction. They will be denoted ⊕ ∗ ⊗ ∗ ⊗ by E1 E2, E and E1 E2, respectively. Hom(E1,E2) will stand for E1 E2. r E will be denoted by det E if r = rank E. Holomorphic vector bundles over complex spaces are defined similarly. For any subset A ⊂ M, E|A will stand for the vector bundle π −1(A) → A.Alocal (holomorphic) frame of E over an open set U is an r-tuple of holomorphic sections, say s1,...,sr of E|U → U such that s1(x),...,sr (x) are linearly independent for all x ∈ U. For two holomorphic vector bundles Ej (j = 1, 2) over M, a holomorphic map α : E1 → E2 is called a bundle homomorphism, or simply a homomorphism, if α|E ∈ Hom(E1,x,E2,x) for all x ∈ M and dim α(E1,x) is a locally constant func- 1,x ∼ tion on M. The isomorphism E1 = E2 will mean that there exists a biholomorphic bundle homomorphism from E1 to E2. Given any bundle homomorphism, its kernel and cokernel are naturally defined as holomorphic vector bundles. Subbundles and quotient bundles are defined similarly. A sequence of bundle homomorphisms is said to be exact if it is exact fiberwise. Given a holomorphic map f : M → N and a holomorphic vector bundle π : E → N, f ∗E will stand for the fiber product of f and π, naturally equipped with the structure of a holomorphic vector bundle over 1.4 ∂¯-Equations on Manifolds 33

M. Holomorphic vector bundles of rank one are called holomorphic line bundles. For any holomorphic line bundle L → M and m ∈ N ∪{0}, L⊗m will be denoted simply by Lm, since there will be no fear of confusing the abbreviated L⊗2 with “square integrable”. If −m ∈ N, Lm := (L∗)(−m). p,0 p,0 ∗ Important examples of holomorphic vector bundles are TM and (TM ) .If M is of pure dimension n, (T n,0)∗ is called the canonical bundle and denoted by KM .LetS be a (not necessarily closed) complex submanifold of M. Then 1,0| TM S is a holomorphic vector bundle over S, so that one has a natural inclusion 1,0 → 1,0| 1,0| 1,0 homomorphism TS TM S. The quotient bundle TM S/TS is by deﬁnition the holomorphic normal bundle of S in M, and denoted by NS/M, or more explicitly 1,0 by NS/M in some context. There is a neat relation between the canonical bundles of M and S:

∼ −1 KM |S = KS ⊗ det NS/M (adjunction formula).

Let us mention a more speciﬁc example. Example 1.8 By extending the natural projection

+ π : Cn 1 \{0}−→CPn, one has a holomorphic line bundle π˜ : π −1(x) −→ CPn. x∈CPn Here π −1(x) denotes the closure of π −1(x) in Cn+1. The bundle π −1(x) is called ∼ the tautological line bundle and denoted by τCPn . It is easy to verify that KCPn = n+1 −1 τCPn . τCPn is called the hyperplane section bundle. A local trivialization of E around a point x ∈ M is a chart of the form (π −1(U), ψ) such that x ∈ U and ψ maps π −1(U) to U × Cr (r = rank E on U) r ◦ | ∈ Cr ∈ r in such a way that prC ψ Ey Hom(Ey, ) for all y U, where prC r r −1 denotes the projection U ×C → C . Given two local trivializations (π (U), ψU ) −1 ◦ −1 = · ∈ and (π (V ), ψV ), ψU ψV (x, ζ ) (x, eUV (x) ζ) holds for any (x, ζ ) r (U ∩ V)× C for some eUV ∈ O(U ∩ V,GL(r, C)). eUV is called a transition −1 function of E. A system of local trivializations {π (U), ψU }U∈U associated to an open covering U of M yields a system of transition functions

{eUV ; U,V ∈ U ,U ∩ V = ∅}.

Obviously {eUV } satisﬁes

eUV eVW = eUW on U ∩ V ∩ W = ∅. (1.8) 34 1 Basic Notions and Classical Results

Conversely, given any open covering U of M and a system of GL(r, C)-valued holomorphic functions eUV (U,V ∈ U ) satisfying the transition relations (1.8), a holomorphic vector bundle E → M is deﬁned by E = U × Cr / ∼ . U∈U

Here the equivalence relation ∼ is deﬁned by

r r U × C (x, ζ ) ∼ (y, ξ) ∈ V × C ⇐⇒ x = y and ζ = eUV (x)ξ.

Given a holomorphic vector bundle E → M and an open set U ⊂ M, we put

Γ(U,E)={s ∈ O(U, E); π ◦ s = idU }, OE = { ; ∈ } ∈ x sx s Γ(U,E) (x M), Ux where sx denotes the germ of s at x, and

E E O (U) ={σ ∈ O [U]; there exists s ∈ Γ(U,E)such that sx = σ(x) for all x ∈ U}. OE = OE {OE } OE Namely, we consider the sheaf x∈M x (or (U) U ). Clearly, is a locally free analytic sheaf. OE will be identiﬁed with E in many contexts. For ⊗ p,0 ∗ p OE (TM ) simplicity, we shall denote by ΩM,E. Further, by an abuse of notation, p p 0 O O ΩM,E will be denoted by Ω (E) and ΩM,E by (E). (E) is an invertible sheaf if and only if rank E = 1. Let A ⊂ M be an analytic set whose ideal sheaf is invertible. ∼ Then, there exists a holomorphic line bundle L → M such that IA = O(L).It’s −1 ∗ a convention that L (= L )is denoted by [A].Adivisor on M is by deﬁnition a formal linear combination k m A (m ∈ Z) such that I are invertible. j=1 j j Aj The bundle k [A ]mj is often denoted additively as [ k m A ]. A divisor j=1 j j=1 j j k m A is called an effective divisor if m ≥ 0 for all j. Given an effective j=1 j j j = k divisor δ j=1 mj Aj with mj > 0, we put |δ|= Aj . j

|δ| is called the support of δ.IfM = CPn and A is a complex hyperplane, [mA] is denoted by O(m) in many places. O(1) is called the hyperplane section bundle. ∞ Similarly to the case of holomorphic vector bundles, for any C complex vector → C ∞ = C ∞ bundle E1 M, we deﬁne the sheaf x∈M of the germs ∞ ∞ ∞ M,E1 M,E1,x ∞ C of C sections of E1. C (U) will be denoted simply by C (U, E1). M,E1,x M,E1 = ⊗ p,q ∗ = ⊗ r C ∗ ∞ If E1 E0 (TM ) (resp. E1 E0 (TM ) )forsomeC vector bundle E0, 1.4 ∂¯-Equations on Manifolds 35

C ∞ C p,q C r C p,q we shall denote M,E simply by (E0) (resp. (E0)) and (E0)(U) (resp. r p,q1 r p,q C (E0)(U))byC (U, E0) (resp. C (U, E0)). Elements of C (U, E0) (resp. r C (U, E0)) will be referred to as E0-valued (p, q)-forms (resp. r-forms) on U. For any holomorphic vector bundle E → M and for any open covering U = ∼ r p,q {Uj } of M such that E|U = Uj × C , the elements of C (M, E) are naturally j ∞ identiﬁed with the systems of vector-valued C (p, q)-forms {uj }, uj being deﬁned = := ∩ = ∅ on Uj , such that uj ejkuk (ejk eUj Uk ) holds whenever Uj Uk . In particular, the complex exterior derivative ∂¯ of type (0,1) maps Cp,q(U, E) to p,q+1 ¯ p,q ¯ C (U, E) (U ⊂ M) so that ∂ induces a complex {C (E), ∂}q≥0. The associated sequence

0 −→ Ωp(E) −→ C p,0(E) −→ C p,1(E) −→··· (1.9) is exact (Dolbeault’s lemma). The proof of Dolbeault’s lemma is based on the characterization of holomorphic functions as C1 solutions of the Cauchy–Riemann equation and Pompeiu’s formula 1 u(ζ ) ∂u/∂ζ¯ u(z) = dζ + dζ ∧ dζ¯ ,z∈ D, (1.10) 2πi ∂D ζ − z D ζ − z which holds for any C1 function u on the closure of a bounded domain D ⊂ C with smooth boundary (cf. [G-R, Hö-2]). Here ¯ · dζ ∧ dζ := −2i · dλ1.

In fact, (1.10) implies in particular that, for any compactly supported C∞ function ϕ on C, the function −1 ϕ(z − ζ) u(z) = dζ ∧ dζ¯ 2πi C ζ satisﬁes the equation

∂u = ϕ, ∂z¯ so that an induction argument works to prove the exactness of (1.9) (for the detail, see [G-R]or[W] for instance). The sequence (1.9) is called the Dolbeault complex. Deﬁnition 1.22

H p,q(M, E) := Ker ∂¯ ∩ Cp,q(M, E)/Im ∂¯ ∩ Cp,q(M, E). (1.11)

H p,q(M, E) is called the E-valued Dolbeault cohomology group of M of type (p, q), or simply the ∂¯-cohomology of E of type (p, q). 36 1 Basic Notions and Classical Results

E will not be referred to if E =∼ M × C, i.e. if E is isomorphic to the trivial line bundle. Accordingly H p,q(M, E) will be denoted by H p,q(M) in such a case. Since we have assumed that any connected component of M admits a countable open basis, for any open covering U of M one can find a C∞ partition of unity on M,say{ρα}, such that {supp ρα} is a refinement of U . Therefore the sequence (1.9) is a fine resolution, i.e. a resolution by fine sheaves, of Ωp(E). Since

H k(U, C p,q(E)) = 0,k≥ 1,p,q≥ 0 for any open set U ⊂ M, by the corollary of Theorem 1.18, similarly to Theorem 1.18 one has: Theorem 1.31 (Dolbeault’s isomorphism theorem)

H p,q(M, E) =∼ H q (M, Ωp(E)).

For the detail of the proof, the reader is referred to [G-R]or[W]. Example 1.9 If M is a Stein manifold, Theorems 1.31 and 1.24 imply that

H p,q(M, E) = 0,q≥ 1 holds for any E.

1.4.2 Cohomology with Compact Support

Let X be a topological space. By deﬁnition, a family of supports on X is a collection of closed subsets of X,sayΦ, satisfying the following two requirements: (1) “A ∈ Φ and K ⊂ A” implies K ∈ Φ. (2) A, B ∈ Φ ⇒ A ∪ B ∈ Φ. Let F → X be a sheaf. We put

FΦ (U) ={s ∈ F (U); supp s ∈ Φ}.

Since any homomorphism α : F → G satisﬁes

αU (FΦ (U)) ⊂ GΦ (U),

F ∗ := {F k k } one has a complex Φ (X) Φ (X), jX k≥0 associated to the canonical ﬂabby resolution of F . 1.4 ∂¯-Equations on Manifolds 37

Deﬁnition 1.23 The p-th F -valued cohomology group of X supported in Φ, p F denoted by HΦ (X, ), is deﬁned as the p-th cohomology group of the complex F ∗ Φ (X). Let

Φ0 ={K ⊂ X; K is compact}.

Then Φ is obviously a family of supports. F (X)(resp. H p (X, F )) will 0 Φ0 Φ0 F p F be simply denoted by 0(X)(resp. H0 (X, )). The following exact sequence is useful:

−→ 0 F −→ 0 F −→ 0 \ F −→ 0 H0 (X, ) H (X, ) ind.limKXH (X K, ) −→ 1 F −→ 1 F −→ 1 \ F −→··· H0 (X, ) H (X, ) ind.limKXH (X K, ) .

Here K X means that K is relatively compact in X. We put

p,q ={ ∈ p,q ; } C0 (M, E) u C (M, E) supp u M and

+ Ker(∂¯ : Cp,q(M, E) −→ Cp,q 1(M, E)) H p,q(M, E) = 0 0 . 0 ¯ : p,q−1 −→ p,q Im(∂ C0 (M, E) C0 (M, E))

Then, similarly to Theorem 1.31, given any holomorphic vector bundle E → M one has: p,q =∼ q p Theorem 1.32 H0 (M, E) H0 (M, Ω (E)). We note that, combining the vanishing of H p,q(Cn) for q ≥ 1 with Theorem 1.1, p,1 Cn = ≥ ∞ ¯ Cn one has H0 ( ) 0ifn 2. In fact, for any C ∂-closed (p, 1)-form v on with compact support, there exists a C∞ (p, 0)-form u satisfying ∂u¯ = v because H p,1(Cn) = 0, but there exists f ∈ O(Cn) such that f = u holds outside a compact subset of Cn by Theorem 1.1. Therefore v is the ∂¯-image of a compactly supported function u − f . This argument can be generalized immediately to show p,1 = ⊂ Cn ≥ that H0 (D) 0 for any domain D (n 2) with unbounded and p,q Cn = ≤ − connected complement. That H0 ( ) 0forq n 1 can be shown similarly, but much more general and straighforward reasoning is given by Serre’s duality theorem explained below. 38 1 Basic Notions and Classical Results

1.4.3 Serre’s Duality Theorem

The duality between the space of compactly supported C∞ functions and the space of distributions is carried over to the spaces of ∂¯-cohomology groups. Such a duality theorem holds on complex manifolds and can be extended on complex spaces after an appropriate modification. We shall restrict ourselves to the duality on complex manifolds here. For the duality theorem, an object of basic importance is the space of currents. By definition, a current of type (p, q) on M,a(p, q)-current for short, is n−p,n−q K p,q an element of the (topological) dual space of C0 (M),say (M), where n−p,n−q the topology of C0 (M) is that of the uniform convergence of all derivatives with uniformly bounded supports. The topology of K p,q(M) is defined as that of the uniform convergence on bounded sets (the strong dual topology). Similarly, the K p,q space 0 (M) of compactly supported (p, q)-currents is defined as K p,q ={ ∈ K p,q ; } 0 (M) u (M) supp u M .

For any holomorphic vector bundle E over M, the space K p,q(M, E) of E-valued n−p,n−q ∗ (p, q)-currents is similarly defined as the dual of the space of C0 (M, E ). K p,q p,q 0 (M, E) is defined as well. C (M, E) is naturally identified with a subset of K p,q(M, E). Since the Dolbeault complex with Dolbeault’s lemma is naturally extended to the complex of sheaves of the germs of currents, which are obviously fine, one has canonical isomorphisms

Ker(∂¯ : K p,q(M, E) −→ K p,q+1(M, E)) H p,q(M, E) =∼ Im(∂¯ : K p,q−1(M, E) −→ K p,q(M, E)) and

+ Ker(∂¯ : K p,q(M, E) −→ K p,q 1(M, E)) H p,q(M, E) =∼ 0 0 . 0 ¯ : K p,q−1 −→ K p,q Im(∂ 0 (M, E) 0 (M, E))

The pairing

K p,q × n−p,n−q ∗ −→ C (M, E) C0 (M, E ) is compatible with the exterior derivatives so that from the complexes

· K p, (M, E) : 0 −→ K p,0(M, E) −→ K p,1(M, E) −→··· and

n−p,· ∗ : −→ n−p,0 ∗ −→ p,1 ∗ −→··· C0 (M, E ) 0 C0 (M, E ) C0 (M, E ) 1.4 ∂¯-Equations on Manifolds 39 a pairing

p,q × n−p,n−q ∗ −→ C H (M, E) H0 (M, E ) is induced. Therefore one has a canonically deﬁned continuous linear map

p,q : n−p.n−q ∗ −→ p,q ι H0 (M, E ) (H (M, E)) , where V denotes for any locally convex space V the dual equipped with the strong topology. The map ιp,q is surjective. To see this, ﬁrst observe that any η ∈ (H p,q(M, E)) lifts to a continuous linear map from Cp,q(M, E) ∩ Ker ∂¯ to C, ˜ p,q = K n−p,n−q ∗ so that it also lifts to an element η of (C (M, E)) 0 (M, E ). Since η˜ vanishes on the image of ∂¯ : Cp,q−1(M, E) → Cp,q(M, E), one has ∂¯η˜ = 0. Hence ιp,q(η)˜ = η. Similarly, we have natural surjective linear maps

p,q : p,q −→ n−p.n−q ∗ ι0 H (M, E) (H0 (M, E )) induced by the pairing

K n−p,n−q × p,q ∗ −→ C 0 (M, E) C (M, E ) .

Serre’s duality theorem describes a necessary and sufﬁcient condition for p,q p,q the maps ι and ι0 to be topological isomorphisms. Here the dual spaces p,q n−p,n−q ∗ (H (M, E)) and (H0 (M, E )) are equipped with the topology of uniform convergence on bounded sets. Theorem 1.33 The following are equivalent: (1) ιp,q is a topological isomorphism. p,q+1 (2) ι0 is a topological isomorphism. (3) H p,q+1(M, E) is a Hausdorff space. n−p,n−q ∗ (4) H0 (M, E ) is a Hausdorff space. (5) Im(∂¯ : K p,q(M, E) → K p,q+1(M, E)) ={f ∈ K p,q+1(M, E);!f, g"= ∈ n−p,n−q−1 ∗ ∩ ¯} 0 for any g C0 (M, E ) Ker ∂ . ¯ : n−p,n−q−1 ∗ → n−p,n−q ∗ ={ ∈ n−p,n−q ∗ ; (6) Im(∂ C0 (M, E ) C0 (M, E )) g C0 (M, E ) ! "= ∈ K p,q ∩ ¯} f, g 0 for any f 0 (M, E) Ker ∂ . Proof It is standard that (5) and (6) are equivalent. Indeed, given two reﬂexive locally convex vector spaces say A and B, a continuous linear map α : A → B and its transpose α ; B → A , we have an equivalence

Im α = Im α ⇐⇒ Im α = Im α .

Equivalences (3) ⇔ (5) and (4) ⇔ (6) are obvious. 40 1 Basic Notions and Classical Results

(5) ⇒ (1): By Banach’s open mapping theorem, it sufﬁces to show that ιp,q is bijective. Since the proof of surjectivity is over, it remains to show the injectivity. Suppose that ιp,q([v]) = 0forsomev ∈ Cn−p,n−q (M, E∗)∩Ker ∂¯. Then !u, v"=0 for any u ∈ K p,q(M, E) ∩ Ker ∂¯. Since Im(∂¯ : K p,q(M, E) → K p,q+1(M, E)) is closed by (5), by Banach’s open mapping theorem one can ﬁnd a continuous linear map

+ w : Im(∂¯ : K p,q(M, E) → K p,q 1(M, E)) −→ C such that

w(∂u)¯ =!u, v" for any u ∈ K p,q(M, E).

Therefore, by the Hahn–Banach theorem there exists a w˜ ∈ K p,q+1(M, E) = n−p,n−q−1 ∗ !¯ ˜ "=! " ∈ K p,q C0 (M, E ) such that ∂u,w u, v holds for all u (M, E), which means (−1)deg u+1∂¯w˜ = v, so that [v]=0. (6) ⇒ (2): Similar to the above. ⇒ p,q n−p,n−q ∗ (1) (4): Since H (M, E) is Hausdorff, (1) implies that H0 (M, E ) is Hausdorff. The proof of (2) ⇒ (5) is similar. Thus we have shown

(5) ⇐⇒ (3) and

(5) ⇒ (1) ⇒ (4) ⇐⇒ (6) ⇒ (2) ⇒ (3) ⇐⇒ (5).

Hence (1) ∼ (6) are all equivalent. By the unique continuation theorem for analytic functions, obviously p,0 = ≥ H0 (M, E) 0 holds for any p 0ifM is connected and noncompact. Hence Serre’s duality theorem implies that (H p,n(M, E)) = 0(p ≥ 0) holds for any connected noncompact complex manifold M. But actually, H p,n(M, E) = 0in such a situation (cf. [Siu-1]). Exploiting this fact and the Serre duality, let us note some examples of non-Hausdorff cohomology. = C2 \ R ×{ } 0,1 0,2 Example 1.10 If M ( 0 ), H (M) and H0 (M) are not Hausdorff. In fact, if H 0,1(M) were Hausdorff, since H 0,2(M) = 0 as above, Serre’s duality theorem and the remark after Theorem 1.32 would imply that H 0,1(M) = 0. But any domain D ⊂ C2 with H 0,1(D) = 0 is a domain of holomorphy, because every holomorphic function on D ∩{z1 = const} can be holomorphically extended to D in this situation. But C2 \ (R ×{0}) is not a domain of holomorphy, as is easily seen from Theorem 1.1. Therefore H 0,1(M) = 0, which is a contradiction. Further, since 0,1 2,2 = ∼ 0,0 H (M) is not Hausdorff, it follows that H0 (M) (H (M)) . 1.4 ∂¯-Equations on Manifolds 41

For any complex space X and a coherent analytic sheaf F → X, H 0(X, F ) is naturally equipped with the Hausdorff topology induced from that of the local uniform convergence in the space of holomorphic functions on complex manifolds.

1.4.4 Fiber Metric and L2 Spaces

Let M be a complex manifold of dimension n and let E → M be a holomorphic vector bundle of constant rank r.Byafiber metric of E we shall mean a collection of positive definite Hermitian forms on the fibers Ex (x ∈ M) which is of class C∞ as a section of Hom(E, E∗).AHermitian metric on M is by definition a fiber 1,0 metric of the holomorphic tangent bundle TM . Since M has a countable basis of open sets, fiber metrics of E can be constructed by patching locally defined fiber ∞ metrics of E|U (U ⊂ M)byaC partition of unity. For any fiber metric h ∈ C∞(M, Hom(E, E∗)),atwist

p,q p+1,q ∂h : C (M, E) −→ C (M, E) of ∂ : Cp,q(M) → Cp+1,q (M) is deﬁned by

−1 ∂h = h ◦ ∂ ◦ h. ¯ The operator Dh = ∂h + ∂ is called the Chern connection of (E, h).Itis 2 easy to see that Dh is naturally identified with the exterior multiplication by a Hom(E, E)-valued (1, 1) form, say Θh from the left-hand side. The cohomology i = i 2 Z class represented by 2π Θdet h 2π Tr Θh in H (M, ) is called the first Chern class of E and denoted by c1(E). If, moreover, M is compact and dim M = 1, we put i deg E = Tr Θh. 2π M and call it the degree of E.degτCPn =−1. The degree is a topological invariant of E. This notion is generalized to the bundles over complex curves and further to higher-dimensional cases by fixing a set of divisors. However, we shall not go into this aspect of the theory of vector bundles in subsequent chapters. (See [Kb-2]for these materials.) For any trivialization

r E|U ξ −→ (π(ξ), ψ(ξ)) ∈ U × C

1 r with ψ(ξ) = (ξ ,...,ξ ), the length |ξ|h of ξ with respect to h is expressed as

| |2 = t ξ h ψ(ξ)hU ψ(ξ) 42 1 Basic Notions and Classical Results

∞ for some matrix-valued C function hU on U. Hence a fiber metric of E is naturally ∞ identified with a system of matrix-valued C functions hj on Uj (M = Uj and =∼ × Cr = EUj Uj ) such that hj (x) are positive definite Hermitian matrices and hj t ekj hke¯kj is satisfied on Uj ∩Uk for the system of transition functions ejk associated | to the local trivializations of E Uj . A holomorphic vector bundle equipped with a fiber metrics is called a Hermitian holomorphic vector bundle. For a Hermitian holomorphic vector bundle (E, h), a local frame s = (s1,...,sr ) of E defined on a neighborhood Uof x ∈ M is said to be normal at x if the matrix representation hs of the fiber metric h ∈ C∞(M, Hom(E, E¯ ∗)) with respect to the local trivialization

r j r E|U ξ sj (y) −→ (y, ξ1,...,ξr ) ∈ U × C j=1 satisﬁes ⎛ ⎞ 10··· 0 ⎜ ⎟ ⎜ 01··· 0 ⎟ = = hs(x) ⎜ . . . . ⎟ and dhs(x) 0. ⎝ . . .. . ⎠ 0 ··· 01

It is easy to see that normal local frames exist for any (E, h) and x ∈ M.Theyare useful to check the validity of local formulas on the differential geometric quantities. Anyway, once we have a Hermitian metric on M and a ﬁber metric on E, the vector p,q space C0 (M, E) is naturally equipped with a topology of pre-Hilbert space which is much closer to the space we live in than those used in the proof of Serre’s duality theorem. The purpose of the remaining four chapters is to make use of this advantage as far as possible.

1.5 Notes and Remarks

For the reader’s convenience, some of the basic properties of plurisubharmonic functions on D ⊂ Cn are listed below. (1) PSH(D) is convex in [−∞, ∞)D. (2) For any decreasing sequence ϕμ in PSH(D), limμ→∞ ϕμ ∈ PSH(D). (3) For any locally bounded sequence ϕμ ∈ PSH(D), the upper regularization of sup ϕμ is plurisubharmonic on D. (4) Let ϕ ∈ PSH(D) and let −2n ζ ϕ(z) = ϕ(z + ζ)χ dλ ( > 0), ζ<1 1.5 Notes and Remarks 43

where χ is a C∞ nonnegative function on R with supp χ ⊂ (−∞, 1] and χ(ζ )dλ= 1. Cn

n Then ϕ ∈ PSH(D), where D ={z ∈ D; dist(x, C \ D) > }, and ϕ ϕ as → 0. n (5) Let D ⊂ C be a domain of holomorphy, let D ∩{zn = 0}=D ×{0} and let δ : D → (0, ∞) be deﬁned by

1 δ(z ) = sup {r;{z }×B (0,r)⊂ D} (z = (z1,...,zn−1)).

Then − log δ(z ) ∈ PSH(D ) (Oka’s lemma, substantially). The notion of plurisubharmonic function was introduced independently by Lelong [Ll’42] and Oka [O-1]. For any hypersurface S ⊂ Cn, one can ﬁnd ∈ O Cn i ¯ | | F ( ) such that 2π ∂∂ log F is identical as a (1,1)-current deﬁned by −→ ∈ n−1,n−1 Cn u u(uC0 ( )). Sreg

[ ]= i ¯ | | This correspondence is often written as S 2π ∂∂ log F and referred to as Poincaré–Lelong formula. F satisfies the formula whenever F is locally a minimal local defining function of S. Holomorphic convexity of a complex space X is equivalent to saying that every compact subset K ⊂ X has a compact holomorphic hull Kˆ defined by

Kˆ ={x ∈ X;|f(x)|≤sup |f | for all f ∈ O(X)}. K

Cartan and Thullen [C-T] introduced the notion of holomorphic convexity in this form. By this alternate definition, it is easy to see that X is holomorphically convex if and only if one can find for any unbounded sequence xμ(μ = 1, 2,...) some f ∈ O(X) satisfying limμ→∞ sup |f(xμ)|=∞. Theorem 1.8 was first due to Skoda [Sk-2]. It was generalized by Yoshioka [Y’81] as follows. Theorem 1.34 Let A be an analytic set of dimension ≤ n−1 in an open set D ⊂ Cn ∈ p,q ¯ = and let u L(2),loc(D) (see Sect. 2.2.5 for the notation). If ∂u 0 is satisfied on D \ A in distribution sense (cf. Theorem 1.2), then ∂u¯ = 0 holds on D. The proof is straightforward from the definition of ∂u¯ as a distribution. Skoda’s original proof was based on the Laurent expansion. Let D be a domain in a complex manifold. If ∂D is strongly pseudoconvex at x, then one can find a local coordinate z = (z1,...,zn) around x such that ∂D is defined by an equation of the form 44 1 Basic Notions and Classical Results

2 2 2 Re z1 = (Im z1) +|z2| +···+|zn| + O(2) on a neighborhood of x. In particular, D is strictly convex near x in this coordinate system. On the other hand, weakly pseudoconvex domains are not always convexi- ﬁable in this sense. For instance, it was shown in [K-N] that the domain

15 Ω = z ∈ C2; Re z +|z |8 + |z |2Re z6 < 0 2 1 7 1 1 is pseudoconvex and (0,0) has no neighborhood in which there exists a hypersurface that contains (0,0) and does not intersect with Ω. Theorem 1.16 on the embedding of real analytic manifolds was preceded by Morrey’s work [M’58], where the compact case had been settled by the L2 method. A real analytic manifold S embedded into its complexiﬁcation S(C) has an obvious property that TS contains no nontrivial complex subspaces of TS(C). For any complex manifold M of dimension n, a submanifold N ⊂ M of real dimension n is said to be totally real if

⊗ C ∩ 1,0 ={ } (TN,x ) TM 0 holds for all x ∈ N. Weierstrass-type approximation theorems are known to hold on certain totally real submanifolds (cf. [H-W’68, Nm’76, W’09]). Furthermore, a connection between complex analysis and Riemannian geometry is known in this context. Morimoto and Nagano [M-N’63] studied the complexification of compact and simply connected Riemannian symmetric spaces of rank 1, generalizing the 2 +···+ 2 = Rn indentification between the tangent bundle of the sphere x1 xn 1in 2 +···+ 2 = Cn and the hypersurface z1 zn 1in .[M-N’63] was followed by a geometric theory of complex Monge–Ampère equations [G-S’92, L-S’91, P-W’91]. In a letter to Takagi, Oka wrote in 1943 that a new question on the locally pseudoconvex ramified domains arose after the solution of the Levi problem. The manuscript for the full solution of the Levi problem had already been written in Japanese. At this point, Cartan had started to study the ideals of holomorphic functions in [C’40] to which Oka showed a great concern. In [C’44] Cartan asked several basic questions. Although Oka had no chance to read it because of WWII, he solved some of them and sent the manuscript in 1948 to Cartan via USA with help of Yasuo Akizuki, Hideki Yukawa, Shizuo Kakutani and André Weil. It appeared in Bull. Soc. Math. France with [C]. Grauert’s direct image theorem (Theorem 1.28) was preceded by a theorem of Grothendieck in algebraic geometry as well as a theorem of Kodaira and Spencer in the deformation theory of complex structures. Basically it asserts the upper p,q −1 semicontinuity of dim H (Mt ,Et ), where Mt = f (t) for a proper surjective holomorphic map f → D such that df is everywhere of maximal rank and = | → Et E Mt for a holomorphic vector bundle E M. As a consequence of Remmert’s proper mapping theorem, the following factorization theorem is obtained. References 45

Theorem 1.35 (cf. [C’60]) For any holomorphically convex space X, there exists a Stein space Y and a proper surjective holomorphic map ϕ : X → Y satisfying the following properties: (1) Fibers of ϕ are connected. (2) ϕ∗OX = OY . (3) For any Stein space Z and any holomorphic map σ : X → Z, there exists a unique holomorphic map τ : Y → Z satisfying σ = τ ◦ ϕ. ϕ is called the Stein factorization of X and Y is called the Remmert reduction of X. Theorem 1.35 is closely related to the quotients of group actions on complex spaces. A complex space X is said to be (p, q)-convex-concave if there exists a C2 proper map ϕ : X → (−1, ∞) and a compact set K ⊂ X such that on every connected component of X \ K, either ϕ is positive and p-convex or ϕ is negative and q-convex. There is a generalization of Theorem 1.31 to (p, q)-convex-concave spaces. As an application of a finiteness theorem on (1,1)-convex-concave space, it is known that every irreducible (1,1)-convex-concave space of dimension ≥3is embeddable as an open subset of a 1-convex space (cf. [R’65]). Grauert’s method of the proof of Theorem 1.28 entails generalizations to nonproper maps whose fibers are (p, q)-convex-concave. Some of them play important roles in the theory of modifications and deformations (cf. [F-K’72, K-S’71, L’73, Siu’70, Siu’72, F’82]). See Theorem 2.84 in Chap. 2, for instance. Since coherent analytic sheaves over X are locally free on a dense open subset of X, questions on the sheaves can be reduced to those on holomorphic vector bundles in certain circumstances. For instance, by virtue of Hironaka’s desingularization theorem, any torsion free coherent analytic sheaf F → X can be pulled back by a modification f : X˜ → X to a sheaf which becomes locally free after taking the quotient by the torsion subsheaf (cf. [R’68]). For any holomorphic map f : X → Y and a coherent analytic sheaf F → X, there exists a sequence of doubly indexed vector spaces starting j k q from {H (X, R f∗F )}j,k which naturally approximates {H (X, F )}q≥0 (Leray’s spectral sequence). By virtue of Remmert’s reduction, combining Leray’s spectral sequence with Theorem 1.28 and Cartan’s theorem B, one can deduce that H q (X, F ) are Hausdorff if X is holomorphically convex. On the other hand, it is known that some complex Lie groups have non-Hausdorff H 0,1 (cf. [Kz-2, Vo]).

References

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Abstract For the bundle-valued differential forms on complex manifolds, a method of solving ∂¯-equations with a control of L2 norm is discussed. Basic results are existence theorems for such solutions under curvature conditions. They are variants of Kodaira’s cohomology vanishing theorem on compact Kähler manifolds, and formulated as vanishing theorems with L2 conditions. Some of these L2 vanishing theorems are generalized to ﬁnite-dimensionality theorems under the assumptions on the bundle-convexity. Besides applications to holomorphic functions, extensions of the Hodge theory to noncompact manifolds will also be discussed.

2.1 Orthogonal Decompositions in Hilbert Spaces

The method of orthogonal projection introduced by H. Weyl [Wy-1] was an inno- vation in potential theory in the sense that it provided a general method of solving the Laplace equations without appealing to the fundamental solutions. This method has developed into a basic existence theory which is useful in complex analysis on complex manifolds. Its basic part can be stated in an abstract form that certain inequality implies the solvability of an equation with an estimate for the norms.

2.1.1 Basics on Closed Operators

Let Hj (j = 1, 2) be two Hilbert spaces. Unless stated otherwise, we shall only consider complex Hilbert spaces. We shall denote by (·, ·)j and j respectively the · · inner products and the norms of Hj . Later we shall use also the notations ( , )Hj and C Hj . By a closed operator from H1 to H2, we mean a -linear map T from a dense linear subspace Ω ⊂ H1 to H2 whose graph GT ={(u, T u) ∈ H1 ×H2 ; u ∈ Ω} is closed in H1 × H2. Ω is called the domain of T and denoted by Dom T . The image T(Ω)of T will be denoted by Im T unless it is confused with the “imaginary part”

© Springer Japan KK, part of Springer Nature 2018 47 T. Ohsawa, L2 Approaches in Several Complex Variables, Springer Monographs in Mathematics, https://doi.org/10.1007/978-4-431-56852-0_2 48 2 Analyzing the L2 ∂¯-Cohomology of T . Accordingly, Im T stands for the closure of the image of T , and not for the conjugate of the imaginary part of T . The kernel {u; Tu= 0} of T will be denoted by Ker T . Note that Ker T is closed since so is GT . The adjoint of a closed operator T , denoted by T ∗, is by deﬁnition a closed operator from H2 to H1 satisfying

GT ∗ ={(v, w) ∈ H2 × H1; (v, T u)2 = (w, u)1 for all u ∈ Dom T }. (2.1)

⊥ Note that GT ∗ = GT ∗ because the right-hand side of (2.1)is(G−T ) ,the orthogonal complement of G−T in H1 × H2, up to the exchange of components.

That (v, w), (v, w ) ∈ GT ∗ implies w = w follows from Dom T = H1. Vice versa, ∗ ∗∗ that Dom T = H2 is because T is single-valued. Obviously T = T . Proposition 2.1 Im T ⊥ = Ker T ∗. ∗ Corollary 2.1 H2 = Im T ⊕ Ker T . ∗ ∗∗ Similarly, H1 = Im T ⊕ Ker T ,forT = T .

2.1.2 Kodaira’s Decomposition Theorem and Hörmander’s Lemma

Let Hj (j = 1, 2, 3) be three Hilbert spaces with norms ∗j .LetT be a closed operator from H1 to H2, and let S be a closed operator from H2 to H3 satisfying Dom S ⊃ Im T and ST = 0. Then, by Proposition 2.1 one has

∗ ∗ H2 = Im T ⊕ Ker T = Im S ⊕ Ker S. (2.2)

Since ST = 0, Im T ⊂ Ker S so that Im T ⊂ Ker S. Similarly Im S∗ ⊂ Ker T ∗, since T ∗S∗ = 0 follows immediately from ST = 0. Hence Im T and Im S∗ are othorgonal to each other. Combining these one has the following decomposition theorem ﬁrst due to K. Kodaira [K-1, Theorem 5]. ∗ ∗ Theorem 2.1 H2 = Im T ⊕ Im S ⊕ (Ker S ∩ Ker T ). In order to analyze this decomposition more in detail, the following is of basic importance.

Lemma 2.1 (cf. [Hö-2, Proof of Lemma 4.1.1]) Let v ∈ H2. Then v ∈ Im T if and only if there exists a nonnegative number C such that

∗ |(u, v)2|≤CT u1 (2.3)

∗ holds for any u ∈ Dom T . Moreover, the inﬁmum of such C is min {w1; Tw= v}. 2.1 Orthogonal Decompositions in Hilbert Spaces 49

∗ Proof If v = Tw for some w ∈ H1, |(u, v)2|=|(u, T w)2|=|(T u, w)1|≤ ∗ T u1 ·w1. Hence one may put C =w1. Conversely, suppose that (2.3) holds ∗ for any u ∈ Dom T . Then the correspondence u → (u, v)2 induces a continuous C ∗ linear map from Im T to C, and further, one from H1 by composing the orthogonal ∗ projection H1 → Im T . Therefore, there exists a w ∈ H1 such that w ∈ H1 such ∗ ∗ that w1 ≤ C and (u, v)2 = (T u, w)1 holds for any u ∈ Dom T . Corollary 2.2 Ker S = Im T if and only if there exists a function C : Ker S → ∗ ∗ [0, ∞) such that |(u, v)2|≤C(v)T u1 holds for any u ∈ Dom T and v ∈ Ker S. Corollary 2.3 Ker S = Im T if there exists a constant C>0 such that

∗ u2 ≤ C(T u1 +Su3) (2.4) holds for any u ∈ Dom T ∗ ∩ Dom S. Proof Suppose that (2.4) holds for any u ∈ Dom T ∗ ∩ Dom S and take any v ∈ ∗ ∗ Ker S.Letu ∈ Dom T and let u = u1 + u2,u1 ∈ Ker S,u2 ∈ Im S be the ∗ ∗ orthogonal decomposition. Then T u = T u1, and (u, v)2 = (u1,v)2 since u2⊥v. ∗ ∗ Hence |(u, v)2|=|(u1,v)2|≤CT u1·v=CT u·v, so that v ∈ Im T by Corollary 2.1. Extension of Corollary 2.3 to the following is immediate. ∗ ∗ Theorem 2.2 H2 = Im T ⊕ Im S ⊕ (Ker S ∩ Ker T ) if there exists a constant C>0 such that (2.4) holds for any u ∈ Dom T ∗ ∩ Dom S ∩ (Ker S ∩ Ker T ∗)⊥. The hypothesis of Theorem 2.2 is fulfilled in a situation naturally arising in certain existence questions in Sect. 1.5 above. To see this, the following will be applied later. ∗ Proposition 2.2 (cf. [Hö-1]) Assume that from every sequence uk ∈ Dom T ∩ ∗ Dom S with uk2 bounded and T uk → 0 in H1, Suk → 0 in H3, one can select a strongly convergent subsequence. Then (2.4) holds for some C>0 and any u ∈ Dom T ∗ ∩ Dom T ∗ ∩ Dom S ∩ (Ker S ∩ Ker T ∗)⊥, and Ker S ∩ Ker T ∗ is finite dimensional. Proof By hypothesis the unit sphere in Ker S∩Ker T ∗ is compact, so Ker S∩Ker T ∗ has to be finite dimensional. If (2.4) were not true for any C, one could choose a ∗ ∗ sequence uk⊥Ker S ∩ Ker T such that uk2 = 1, T uk → 0inH1 and Suk → 0 in H3.Letu be a strong limit of the sequence uk, which exists by hypothesis. Then ∗ ∗ ∗ u2 = 1 and u is orthogonal to Ker S ∩ Ker T ∩ Ker T although T u = Su = 0, so that u ∈ Ker S ∩ Ker T ∗. This contradiction proves (2.4). The following is also applied later:

Theorem 2.3 (cf. Theorem 1.1.4 in [Hö-1]) Let F be a closed subspace of H2 ∗ containing Im T . Assume that u2 ≤ C(T u1 +Su3) holds for any u ∈ Dom T ∗ ∩ Dom S ∩ F . Then: 50 2 Analyzing the L2 ∂¯-Cohomology

(i) For any v ∈ Ker S ∩ F one can ﬁnd w ∈ Dom T such that Tw = v and w1 ≤ Cv2. ∗ ∗ ∗ (ii) For any w ∈ Im T one can ﬁnd v ∈ Dom T such that T v = w and v2 ≤ Cw1. Proof ∗ (i) Let v ∈ Ker S ∩ F ,letu ∈ Dom T , and let u = u1 + u2 + u3, where u1 ∈ Ker S ∩ F , u2 ∈ Ker S and u2⊥F , and u3⊥Ker S. Since Im T ⊂ F and u2⊥F , ∗ ∗ ∗ u2⊥Im T so that u2 ∈ Ker T . Moreover, u3 ∈ Im S ⊂ Ker T . Therefore ∗ ∗ u2 + u3 ∈ Ker T , so that u1 ∈ Dom T , and

∗ |(u, v)2|=|(u1,v)2|≤CT u11 ·v2 (2.5)

∗ Hence we have |(u, v)2|≤CT u1 ·v2 from (2.5). Therefore the linear ∗ ∗ functional u → (u, v)2 on Dom T is continuous in T u, so that there exists w ∈ H1 such that w1 ≤ Cv2 and

∗ (u, v)2 = (T u, w)1

holds for every u ∈ Dom T ∗ so that v = Tw. ∗ ∗ ∗ (ii) Let w = T v0 and v0 = v1 + v2, where v1⊥Ker T and v2 ∈ Ker T . Then ∗ v1 ∈ Im T so that v1 ∈ F . Hence v1 ∈ F ∩ Dom T ∩ Ker S so that

∗ ∗ v12 ≤ CT v1=CT v0=Cw.

Thus it sufﬁces to put v = v1.

2.1.3 Remarks on the Closedness

Let the situation be as above. A basic observation of meta-theoretical importance is | that Im T is closed if and only if T (Ker T)⊥ is invertible. In other words the following holds. Proposition 2.3 The following are equivalent: (i) Im T = Im T . (ii) There exists a constant C such that u≤CTu holds for any u ∈ Dom T ∩ (Ker T)⊥. Proof (ii) ⇒ (i) is obvious. (i) ⇒ (ii) follows from Banach’s open mapping theorem, or closed graph theorem, or uniform boundedness theorem. Combining Proposition 2.3 with Corollary 2.1 one has: 2.2 Vanishing Theorems 51

Theorem 2.4 The following are equivalent: (i) Im T = Im T . (ii) Im T ∗ = Im T ∗. Accordingly, Corollary 2.2 is also strengthened to the following. ∗ Theorem 2.5 H2 = Im T ⊕ Im S if and only if there exists a constant C such that (2.4) holds for any u ∈ Dom T ∗ ∩ Dom S. Similarly, the converse of Theorem 2.2 also holds. Let us add one more remark which is not so often mentioned but seems to be useful. For an application see Example 2.2 below. (See also [Oh-5].) The proof is left to the reader as an exercise. Proposition 2.4 Im T = Im T if dim Ker S/Im T<∞.

2.2 Vanishing Theorems

Solvability criteria for ∂¯-equations on complex manifolds are often described as cohomology vanishing theorems. In order to apply the abstract theory presented in the previous section, it is necessary to know that certain inequality holds for the bundle-valued differential forms under some curvature condition. The first vanishing theorem of this type was established by Kodaira [K-2] on compact Kähler manifolds and substantially by Oka [O-1, O-4] on pseudoconvex Riemann domains over Cn. A vanishing theorem for L2 ∂¯-cohomology groups on complete Kähler manifolds unifies Kodaira’s vanishing theorem and Cartan’s Theorem B on Stein spaces. This viewpoint was first presented in a paper of Andreotti and Vesentini [A-V-1] and later effectively developed in [A-V-2]. Independently and more thoroughly, Hörmander [Hö-1] established the method of L2 estimates for the ∂¯-operator, extending also a preceding work of Morrey [Mry]. The advantage of this method is its flexibility in the limiting procedures as in Theorems 2.14 and 2.16. The argument below is based also on the method of Andreotti and Vesentini. It will be refined in the next section to recover a finiteness theorem of Hörmander.

2.2.1 Metrics and L2 ∂¯-Cohomology

Let (M, ω) be a (not necessarily connected but pure dimensional) Hermitian manifold of dimension n and let (E, h) be a Hermitian holomorphic vector bundle over M. In order to analyze the ∂¯-cohomology groups of (M, E), the metric structure (ω, h) is useful. As before, we denote by Cp,q(M, E) the set of E-valued ∞ p,q p,q C (p, q)-forms on M and by C0 (M, E) the subset of C (M, E) consisting of compactly supported forms. 52 2 Analyzing the L2 ∂¯-Cohomology

∈ p,q The pointwise length of u C (M, E) with respect to the ﬁber metric induced p 1,0 ∗ ⊗ q 0,1 ∗ ⊗ by ω and h, measured by regarding u as a section of (TM ) (TM ) E, is denoted by |u|(=|u|ω,h). The pointwise inner product of u and v is denoted by 2 !u, v"(=!u, v"ω,h). Then the L norm of u denoted by uh,orsimplybyu,is deﬁned as the square root of the integral ωn |u|2 , (2.6) M n!

∈ p,q which is ﬁnite if u C0 (M, E). The inner product of u and v associated to the norm is denoted by (u, v)h. (u, v)h is ωn !u, v"ω,h M n! or 1 i (u + v2 −u2 −v2) − (iu + v2 −u2 −v2) (2.7) 2 2 by deﬁnition, but has an expression more convenient for computation. Namely, (u, v)h = u ∧ h ∗ v = h(u) ∧ ∗v . (2.8) M M

Here h is identiﬁed with a section of E∗ ⊗ E¯ ∗(=∼ Hom(E, E¯ ∗) and ∗ is a map from Cp,q(M, E) to Cn−q,n−p(M, E) induced from the unique isometric bundle − morphism ∗ between r (T CM)∗ and 2n r (T CM)∗ (r = p + q) that satisﬁes

2 n e1 ∧ e2 ∧···∧er ∧ ∗(e1 ∧ e2 ∧···∧er ) =|e1 ∧ e2 ∧···∧er | ω /n! (2.9)

≤ ≤ C ∗ ∗ for all ej (1 j r) in a ﬁber of (TM ) .Themap is called Hodge’s star operator. For simplicity we put ∗v = ∗v. Then ∗ is a map from Cp,q(M, E) to Cn−p,n−q (M, E). = Cn = i n ∧ Example 2.1 For M and ω 2 j=1 dzj dzj ,

∗(dzI ∧ dzJ ) = cIJ dzI ∧ dzJ , (2.10)

(n−p)q n2 p+q−n where I and J complement I and J , respectively, and cIJ = (−1) i 2 , where p =|I| and q =|J |. p,q p,q Let L(2) (M, E) be the completion of the pre-Hilbert space C0 (M, E) with 2 p,q respect to the L norm. By Lebesgue’s theory of integration, L(2) (M, E) is naturally identiﬁed with a subset of E-valued (p, q)-forms with locally square 2.2 Vanishing Theorems 53

2 p,q integrable (=Lloc) coefficients. Then every element f of L(2) (M, E) is naturally identified with a C-linear function on Cp,q(M, E) by the inner product ¯ 0 ·→(·,f)h. For simplicity, ∂ will also stand for a densely defined map from p,q p,q+1 ¯ L(2) (M, E) to L(2) (M, E) whose domain of definition, denoted by Dom ∂,is { ∈ p,q ; ¯ ∈ p,q+1 } ¯ f L(2) (M, E) ∂f L(2) (M, E) , where ∂f is defined in the sense of ∈ p,q ¯ distribution for any f L(2) (M, E). In other words, ∂f is regarded as an element n−p,n−q−1 ∗ ∗ of C0 (M, E ) by the equality + + ∂f¯ ∧ v(= ∂f¯ (v)) = (−1)p q 1 f ∧ ∂v¯ M M

∈ n−p,n−q−1 ∗ for all v C0 (M, E ), (2.11)

+ and “∂f¯ ∈ Lp,q 1(M, E)” means that there exists a unique element w ∈ (2) p,q+1 ∧ = − p+q+1 ∧ ¯ ∈ L(2) (M, E) such that M w v ( 1) M f ∂v holds for any v n−p,n−q−1 ∗ C0 (M, E ). 2 ¯ p,q We deﬁne the L ∂-cohomology groups H(2) (M, E) by

p,q := ¯ ∩ p,q ¯ ∩ p,q H(2) (M, E) Ker ∂ L(2) (M, E)/Im ∂ L(2) (M, E). (2.12)

p,q p,q p,q L(2) (M, E) and H(2) (M, E) will be denoted by L(2) (M,E,ω,h) and p,q 2 H(2) (M,E,ω,h), respectively, whenever (ω, h) must be visible. L de Rham r cohomology groups H(2)(M) are defined similarly with respect to the exterior ¯ ¯ p,q derivative instead of ∂. ∂ is obviously a closed operator from L(2) (M, E) to p,q+1 ¯ ∗ L(2) (M, E) so that it has its adjoint. It will be denoted by ∂h , or more simply ¯ ∗ p,q =∼ ¯ ∩ ¯ ∗ ∩ p,q by ∂ . A basic fact is that H(2) (M, E) Ker ∂ Ker ∂ L(2) (M, E) if ¯ ∩ p,q Im ∂ L(2) (M, E) is closed (cf. Theorem 2.1). Example 2.2 With respect to the Euclidean metric, 0 if q = 0 or q > n, dim H p,q(Cn) = (2) ∞ otherwise, for any n ∈ N. p,0 Cn = p,q Cn = Indeed, H(2) ( ) 0 follows from Cauchy’s estimate. That H(2) ( ) 0for p,1 Cn =∞ q>nis trivial. To see that dim H(2) ( ) , it suffices to apply Propositions 2.3 p,0 Cn = and 2.4, combining H(2) ( ) 0 with an obvious fact that one can find a sequence ∈ p,0 Cn = ¯ → →∞ uk L(2) ( ) such that uk 1 and ∂uk 0ask . The infinite dimensionality for general q follows similarly. Namely, the non-Hausdorff property 54 2 Analyzing the L2 ∂¯-Cohomology

p,q Cn ≤ ≤ ∈ p,q Cn of H(2) ( ) for 2 q n follows from that there exists a sequence uk C0 ( ) ¯ ∗ ¯ ¯ ¯ ∗ such that ∂ uk(⊥Ker ∂) are of norm 1 but ∂∂ uk→0ask →∞, which is also obvious as in the case q = 1. To obtain more advanced results, one needs to ﬁnd natural conditions on (ω, h) in order to apply abstract existence theorems in Sect. 2.1.2. An effective condition acceptable in most cases is the completeness of ω which guarantees in particular the p,q ¯ ∗ ∩ p,q density of C0 (M, E) in Dom ∂ L(2) (M, E) with respect to the graph norm of ∂¯ ∗, which will be explained below.

2.2.2 Complete Metrics and Gaffney’s Theorem

A Hermitian manifold (M, ω) is said to be complete if M is complete as a metric space with respect to the distance associated to ω. Recall that √ the distance between ∈ 1 ∗ x,y M with respect to ω is defined as the infimum of 0 γ g where g is the 1,0 1,0 ∗ ⊗ 0,1 ∗ fiber metric of TM associated to ω regarded as a section of (TM ) (TM ) and γ runs through C∞ maps from [0,1] to M satisfying γ(0) = x and γ(1) = y.This distance will be denoted by distω(x, y),orsimplybyd(x,y). Cn i ∧ Example 2.3 ( , 2 dzj dzj ) is complete. Proposition 2.5 (M, ω) is complete if and only if {y; d(x,y) < R} is relatively compact for any x ∈ M and R>0. Proof The “if” part is obvious. The converse is easy to see from the Bolzano– Weierstrass theorem. Since d(x,y) is Lipschitz continuous on M × M, it can be approximated uniformly by a C∞ function, say d(x,y)˜ with bounded gradient. Let us fix a point ˜ ∞ x0 ∈ M and put ρ(x) = d(x0,x).Letχ : R →[0, ∞) be a C function such that (i) χ|(−∞, 1) ≡ 1 and (ii) supp χ ⊂ (−∞, 2]. = ρ(x) Then we put χR(x) χ( R ) for R>1. An important property of χR is that |dχR|≤C/R holds for some C>0. Proposition 2.6 Let (M, ω) be a Hermitian manifold and let (E, h) be a Hermitian ∈ ¯ ∩ p,q ∈ holomorphic vector bundle over M. Then, for any u Dom ∂ L(2) (M, E), χRu ¯ ∩ p,q − +¯ − ¯ → →∞ Dom ∂ L(2) (M, E) and χRu u ∂(χRu) ∂u 0 as R . 2.2 Vanishing Theorems 55

¯ = ¯ ∧ + ¯ ∈ p,q+1 − = Proof That ∂(χRu) ∂χR u χR∂u L(2) (M, E) and limR→∞ χRu u ¯ ¯ 0 is obvious. In order to see that limR→∞ ∂(χRu)−∂u=0, it sufﬁces to combine ¯ − ¯ = |¯ | = limR→∞ χR∂u ∂u 0 and limR→∞(supM ∂χR ) 0.

If (M, ω) is complete, supp(χRu) M for all R. Hence there exists for each R ∈ p,q − +¯ − ¯ → a sequence uk C0 (M, E) satisfying uk χRu ∂uk ∂(χRu) 0as k →∞. Combining this observation with Proposition 2.6,wehave: Proposition 2.7 Let (M, ω) be a complete Hermitian manifold and let (E, h) be a p,q ¯ Hermitian holomorphic vector bundle over M. Then C0 (M, E) is dense in Dom ∂ with respect to the norm u+∂u¯ . ¯ ∗ p,q+1 −∗ −1 ¯ ∗ Similarly, since ∂ acts on C0 (M, E) as a differential operator h ∂h , which is easy to see from the Stokes’ formula and ∗∗u = (−1)deg uu, Propo- sition 2.7 can be extended to the following important result which is ﬁrst due to Gaffney [Ga] for the exterior derivative d and formulated for ∂¯ by Andreotti and Vesentini in [A-V-1, A-V-2]. p,q ¯ ∩ Theorem 2.6 In the situation of Proposition 2.7, C0 (M, E) is dense in Dom ∂ Dom ∂¯ ∗ with respect to the norm u+∂u¯ +∂¯ ∗u. The importance of Theorem 2.6 for our purpose lies in that integration by parts is available without worrying about the boundary terms to obtain the estimates imply- ∞ ing the existence theorems. To proceed in this way, formulas in the C (M) algebra 2n p,q of operators on p+q=0 C (M, E) are useful. They will be described next.

2.2.3 Some Commutator Relations

Before presenting formulas involving ∂¯, let us prepare some abstract formalism. Let R be a commutative ring and let M be a graded R module, i.e. M is a direct sum of submodules say Mj (j ∈ Z).Ifu ∈ Mj −{0}, j is called the degree of u and denoted by deg u.Let

M Πk(M ) ={T ∈ M ; T(Mj ) ⊂ Mj+k for all j}. ∈ M −{ } = M For any T Πk( ) 0 we put deg T k. Then k∈Z Πk( ) is a graded left R M algebra whose product is deﬁned by composition. Elements of k∈Z Πk( ) are said to be homogeneous. Given S ∈ Πk(M ) and T ∈ Π (M ), we deﬁne the graded commutator of S and T by

deg S deg T [S,T ]gr = S ◦ T − (−1) T ◦ S, (2.13) where we put deg 0 = 0. The following straightforward consequence of the deﬁnition is very important. 56 2 Analyzing the L2 ∂¯-Cohomology

Lemma 2.2 (Jacobi’s identity) For any homogeneous S,T,U ∈ Π(M ),

deg S deg T +1 [[S,T ]gr,U]gr −[S,[T,U]gr]gr = (−1) [T,[S,U]gr]gr. (2.14)

∞ Now let (M, ω)(dim M ≥ 1) and (E, h) be as before and set R = C (M), M = 2n p,q M = p,q p+q=0 C (M, E) and j p+q=j C (M, E). Then, with respect ¯ p,q to this natural grading, ∂ ∈ Π1(M ). We shall identify the elements of C (M, E) with those in Πp+q (M ) by letting them act on M by exterior multiplication from p,q ∗ the left-hand side. Given θ ∈ C (M), we deﬁne θ ∈ Π−p−q (M ) by requiring ∗ the equality !θ ∧ u, v"=!u, θ v" for the pointwise inner product !·, ·" = !·, ·"h to hold for any u ∈ Cr,s (M, E) and v ∈ Cr+p,s+q (M, E). Since ω ∈ C1,1(M), ω∗ ∈ ∗ Π−2(M ). We shall use the notation Λ for ω following [W-1, W-2] and [N-1]. Some formulas involving Λ are of special importance. They will be recalled below. = Cn i ∧ For the special case (M, ω) ( , 2 dzj dzj ), it is easy to see that 1 ∗ ∗ Λ = (dz ) (dz ) 2i j j and

∗ [dzj ,Λ]gr =−i(dzj ) (1 ≤ j ≤ n).

Since this formula can be applied pointwise, one has the following in general. ∗ 0,1 Lemma 2.3 For any θ ∈ C (M), [θ,Λ]gr =−iθ . 0,1 ∗ ∗ Proposition 2.8 For any σ, τ ∈ C (M), [σ, τ ]gr +[σ , τ]gr = 0. ∗ ∗ ∗ ∗ Proof Since τ = i[τ,Λ]gr and σ =−i[σ, Λ]gr, one has [σ, τ ]gr +[σ , τ]gr = i[σ, [τ,Λ]gr]gr + i[[σ, Λ]gr, τ]gr =[i[σ, τ]gr,Λ]gr = 0. ¯ ¯ Similarly, replacing θ by ∂ one has a useful expression for [∂,Λ]gr. To describe it, let us put

∂$ =−∗∂∗. (2.15) ∈ p,q ∈ p+1,q ∧ ∗ = ∧ Proposition 2.9 For any u C0 (M) and v C (M), M ∂u v M u ∗∂$v. ∧ ∗ = ∧ ∗ − − p+q ∧ ∗ =−− deg v−1 ∧ ∗ Proof M ∂u v M d(u v) ( 1) M u d v ( 1) M u ∂ v =− ∧ ∗∗ ∗ = ∧ ∗ −∗ ∗ M u ∂ v M u ( ∂ v). ¯ $ Lemma 2.4 [∂,Λ]gr = i∂ if dω = 0. Proof Since ∂¯ and ∂$ are differential operators of the first order, it suffices to show the assertion for the Euclidean case. In this situation, first note that for any u ∈ C∞(Cn), 2.2 Vanishing Theorems 57

1 ∂u ∗ [∂,Λ¯ ] (u dz ∧ dz ) = dz (dz ∧ dz ). (2.16) gr I J i ∂z j I J j∈I j

∈ ∞ Cn { }= Hence, if v C0 ( ) and j,K I,

n [¯ ] ∧ = 1 ∂u ∗ ∧ ∧ ∗ ∧ ( ∂,Λ gr(u dzI dzJ ), v) dzj (dzI dzJ ) (v dzK dzJ ) i Cn ∂zj j=1 1 ∂u n2 = v · i (dz1 ∧···∧dzn ∧ dz1 ∧···∧dzn) i Cn ∂zj 1 ∂v n2 =− u i (dz1 ∧···∧dzn ∧ dz1 ∧···∧dzn) i Cn ∂zj ∂v = i udzI ∧ dzJ ∧ ∗ dzj ∧ dzK ∧ dzJ Cn ∂zj

= i(u,∂v)(= i(u,∂v)ω).

¯ $ Hence [∂,Λ]gr = i∂ . ¯ ∗ ¯ ∗ ¯ $ ¯ $ For the sake of consistency, we shall denote ∂ (resp. ∂h )by∂ (resp. ∂h) when p,q p,q it operates on C (M) (resp. C (M, E)) as a differential operator. Let ∂h = −1 ◦ ◦ [¯ ] + $ [ ] − ¯ $ h ∂ h. Then ∂,Λ gr i∂ and ∂h,Λ gr i∂h are operators of order zero. (For the explicit expressions of them, see [Dm-3]or[Oh-7].) ¯ ¯ $ $ Proposition 2.10 [∂,∂ ]gr −[∂ ,∂]gr = 0 if dω = 0. $ ¯ ¯ $ ¯ ¯ $ $ Proof Since ∂ =−i[∂,Λ]gr and ∂ = i[∂,Λ]gr, one has [∂,∂ ]gr −[∂ ,∂]gr = ¯ ¯ ¯ i[∂,[∂,Λ]gr]gr + i[[∂,Λ]gr,∂]gr = i[[∂,∂]gr,Λ]gr = 0. [¯ ¯ $] − We note that the above computation works to show also that ∂,∂h gr [ $ ] = [[¯ ] ] = ¯ $ ¯ $ ∂ ,∂h gr i ∂,∂h gr,Λ gr if dω 0, because ∂h and ∂h coincide with ∂ and ∂ respectively at a point x ∈ M with respect to a normal frame of E around x.The ﬁrst-order terms in the Taylor expansion of the coefﬁcients of the zero order terms ¯ 1,1 of ∂h appear in [∂,∂h]gr = Θh ∈ C (M, Hom(E, E)). Identifying Θh naturally with an element of Π2(M ),wehave [¯ ¯ $] −[ $ ] =[ ] = Theorem 2.7 (Nakano’s identity) ∂,∂h gr ∂ ,∂h gr iΘh,Λ gr if dω 0. Similarly, combining Lemmas 2.3 and 2.4 we obtain: 0,1 ¯ ∗ $ ¯ Theorem 2.8 For any θ ∈ C (M), [∂,θ ]gr +[∂ , θ]gr =[i∂θ,Λ]gr holds if dω = 0. As a remark, we note that Lemma 2.3, Proposition 2.8 and Theorem 2.8 can be generalized to commutator relations for θ ∈ C0,1(M, Hom(E, E)) by letting ∗ −1 t θ = ∗θ∗ and θ h = h ◦ θ ◦ h as follows. 58 2 Analyzing the L2 ∂¯-Cohomology

∗ Lemma 2.5 [θ,Λ]=iθ for any θ ∈ C0,1(M, Hom(E, E)). 0,1 ∗ ∗ Proposition 2.11 For any σ, τ ∈ C (M, Hom(E, E)), [σ, τ ]gr +[σ , τ h]gr = [[σ, τ h]gr,Λ]gr. 0,1 ¯ ∗ $ Theorem 2.9 For any θ ∈ C (M, Hom(E, E)), [∂,θ ]gr +[∂ , θ h]gr = ¯ [i∂θ h,Λ]gr holds if dω = 0. (M, ω) is called a Kähler manifold if dω = 0.

2.2.4 Positivity and L2 Estimates

Let (M, ω) be a Kähler manifold of dimension n and let (E, h) be a Hermitian holomorphic vector bundle of rank r over M. Then, Nakano’s identity implies, by integration by parts, that

¯ 2 ¯ ∗ 2 ∗ 2 2 ∂u +∂ u −∂ u −∂hu = (i(ΘhΛ − ΛΘh)u, u)

∈ p,q holds for any u C0 (M, E). In particular, one has

¯ 2 ¯ ∗ 2 ∂u +∂ u ≥ (i(ΘhΛ − ΛΘh)u, u) (basic inequality), which simpliﬁes to

¯ 2 ¯ ∗ 2 ∂u +∂ u ≥ (iΘhΛu, u) if p = n. (2.17)

From this inequality, we shall derive a useful estimate under some positivity assumption on Θh which turns out to be satisfied in many situations arising in complex geometry. ∈ Let x M and let (z1,z2,...,zn) be a local coordinate around x such that = i ∧ ω 2 dzj dzj at x, and let (e1,e2,...,er ) be a local frame of E . Then ∈ C ≤ ≤ = ∗ ⊗ ∗ we define hμν (1 μ, ν r) by requiring h μν hμνeμ eν to hold ∗ ∗ at x ∈ M, identifying Hom(E, E ) with E∗ ⊗ E . Similarly, we define Θμ ∈ αβν C (1 ≤ α, β ≤ n, 1 ≤ μ, ν ≤ r) by ∗ Θ = Θμ (e ⊗ e )dz ∧ dz (2.18) h αβν ν μ α β α,β μ,ν at x, by identifying Hom(E, E) with E∗ ⊗ E. Then take any u ∈ Cn,q (M, E) and let = μ ∧···∧ ∧ u uJ eμ dz1 dzn dzJ (2.19) J μ 2.2 Vanishing Theorems 59 | |2 = n+q μ ν hold at x. Then u 2 J μ,ν hμνuJ uJ and + !iΘ Λu, u"=2n q Θμ h uν uκ (2.20) h αβν μκ {K,α} {K,β} K α,β,ν,κ μ at x. Similarly, + !iΛΘ u, u"=2n q Θμ h uν uκ (2.21) h αβν μκ L\{α} L\{β} L α,β,ν,κ μ at x. Note that ( Θμ h )ξ ανξ βκ ((ξ αν) ∈ Cnr ) gives a quadratic form on μ αβν μκ the fibers of T 1,0M ⊗ E as x varies.

Definition 2.1 (E, h) is said to be Nakano positive (resp. Nakano semipositive) if the quadratic form ( Θμ h )ξ ανξ βκ is positive (resp. semipositive) μ αβν μκ at every point of M. Nakano negativity and Nakano seminegativity are defined similarly. In other words, a Hermitian holomorphic vector bundle E is said to be Nakano (semi-) positive if, for any point x0 ∈ M, there is a neighborhood U = U(x0) with local coordinate z = (z1,...,zn) around x0 and a coordinate w = (w1,...,wr ) on the fibers of V |U coming from a holomorphic trivialization such that: (1) over U we have the representation of the fiber metric μν¯ hμν¯ (z)wμwν, (2) the matrix (hμν¯ (x0)) is the unit matrix, (3) the total derivative dhμν¯ (x0) = 0, and 2 (4) the Hermitian form − (∂ hμν¯ (x0)/∂zα∂zβ )γμαγνβ is positive (semi-) definite. In particular, for any Nakano semipositive bundle (E, h) and for any holomor- ∗ ∗ phic section s of E , it is easy to see that log sh∗ ∈ PSH(M).Hereh denotes the dual of h. We shall say that a holomorphic vector bundle E is Nakano (semi-)positive if it admits a fiber metric whose curvature form is (semi-)positive in the sense of Definition 2.1. In accordance with the positivity of the Kähler form ω, Nakano positivity (resp. semipositivity) of the curvature form Θh in the above sense will be denoted by iΘh > 0 (resp. ≥ 0). By an abuse of language we shall call the eigenvalues of Θh also those of iΘh. Nakano positive (resp. semipositive) line bundles are simply called positive (resp. semipositive) line bundles. The curvature form of a positive line bundle is naturally identified with a Kähler metric. 60 2 Analyzing the L2 ∂¯-Cohomology

By Corollary 2.3 and the inequality (2.17), the equality (2.20) eventually implies the following. Theorem 2.10 Let (M, ω) be a complete Kähler manifold and let (E, h) be a Hermitian holomorphic vector bundle over M such that iΘh − c IdE ⊗ ω ≥ 0 for some c>0. Then

n,q = H(2) (M, E) 0 for all q>0. (2.22)

Corollary 2.4 (Kodaira–Nakano vanishing theorem) If (M, ω) is a compact Kähler manifold and (E, h) is a Nakano positive vector bundle over M,

n,q q H (M, E) = 0 (or equivalently H (M, O(KM ⊗ E)) = 0 ) for all q>0. (2.23)

Corollary 2.5 Positive line bundles over compact complex manifolds are ample. Here, a holomorphic line bundle L → M is said to be ample if there exists m m ∈ N such that L is very ample in the sense that there exist s0,s1,...,sN ∈ 0,0 m N−1 H (M, L ) such that the ratio (s0 : s1 : ··· : sN ) maps M injectively to CP as a (not necessarily locally closed for noncompact M) complex submanifold. ¯ 2 +¯ ∗ 2 ≥ − ∈ 0,q Similarly, the inequality ∂u ∂ u ( iΛΘhu, u) for u C0 (M, E) implies: Theorem 2.11 Suppose that (M, ω) is a complete Kähler manifold and there exists + ⊗ ≤ 0,q = a c>0 such that iΘh c IdE ω 0. Then H(2) (M, E) 0 for all q

2.2.5 L2 Vanishing Theorems on Complete Kähler Manifolds

Let (M, ω) be a complete Kähler manifold of dimension n and let (B, a) be a Hermitian holomorphic line bundle over M. 2.2 Vanishing Theorems 61

= p,q = Theorem 2.12 (cf. [A-N] and [A-V-1]) If ω iΘa, then H(2) (M, B) 0 for + p,q ∗ = + p q>nand H(2) (M, B ) 0 for p q

!(ωΛ − Λω)u, u"=(p + q − n)|u|2 (2.24) for any u ∈ Cp,q(M, B), whence the conclusion follows similarly to Theorems 2.10 and 2.11. Corollary 2.6 (Akizuki–Nakano vanishing theorem) Let M be a compact complex manifold of dimension n and B → M a holomorphic line bundle which admits a ﬁber metric whose curvature form is positive. Then H p,q(M, B) = 0 for p+q>n and H p,q(M, B∗) = 0 for p + q

Theorem 2.13 If the eigenvalues λ1 ≤···≤λn of Θa with respect to ω everywhere satisfy

λ1 +···+λp − λq+1 −···−λn (= λ1 +···+λq − λp+1 −···−λn) ≥ c

p,q = n−p,n−q ∗ = for some positive constant c, then H(2) (M, B) 0 and H(2) (M, B ) 0. A reﬁnement of Theorem 2.10 in another direction is: Theorem 2.14 (cf. [Dm-2] and [Oh-2, Oh-8]) Let M be a complex manifold of dimension n which admits a complete Kähler metric, and let (B, a) be a positive line bundle over M. Then, for any Kähler metric ω on M satisfying ω ≤ iΘa, n,q = ∈ n,q ∩ ¯ H(2) (M,B,ω,a) 0 for q>0. Moreover, for any v L(2) (M,B,ω,a) Ker ∂, ∈ n,q−1 ∩ ¯ ¯ = 2 ≤ one can ﬁnd w L(2) (M,B,ω,a) Dom ∂ satisfying ∂w v and w qv2. ∈ n,q ∩ ¯ Proof Let v L(2) (M,B,ω,a) Ker ∂. Taking any complete Kähler metric ω∞ on M,letω = ω + ω∞ for any ≥ 0, let !, " denote the pointwise inner product with respect to ω,letΛ denote the adjoint of ω(= ω∧) with respect to ω,let n = ! " ω ·2 = · · ∈ n,q (, ) M , n! and let ( , ). Then, for any u C0 (M, B), n 2 ω −1 !u, v" ≤(iΘaΛu, u)((iΘaΛ) v,v) (Cauchy–Schwarz inequality). M n! (2.25) ∈ = ∧···∧ ∧ = i ∧ Let x M be any point. Let v vJ dz1 dzn dzJ , ω 2 dzj dzj = i ∧ and ω∞ 2 θj dzj dzj (θj > 0) at x. 62 2 Analyzing the L2 ∂¯-Cohomology

Then

n n − ω − ω !(iΘ Λ ) 1v,v" ≤!(ωΛ ) 1v,v" a n! n! n + − − ω = 2n q (1 + θ ) 1 |v |2 (1 + θ ) 1 j J j n! J j∈J j∈J ωn ≤ q|v|2 (2.26) n! at x (for almost all x). Hence

2 2 |(u, v)| ≤ qv (iΘaΛu, u). (2.27)

But

≤¯ 2 +¯ ∗ 2 (iΘaΛu, u) ∂u ∂ u , (2.28)

¯ ∗ ¯ where ∂ denotes the adjoint of ∂ with respect to (ω,a). ¯ Therefore, by Theorem 2.3, there exists for each a w ∈ Dom ∂ ∩ p,q−1 ¯ = L(2) (M,B,ω,a)such that ∂w v and

2 ≤ 2 w q v . (2.29)

From (2.29) one sees that there exists a locally weakly convergent subsequence ¯ 2 2 of w 1 (k ∈ N). The limit w satisﬁes ∂w = v and w ≤ qv . k n,q = n,q Since H(2) (M,B,iΘa,a) H(2) (M,B,ciΘa,a)for any c>0, one has: n,q = Corollary 2.7 In the above situation, H(2) (M,B,ciΘa,a) 0 for q>0 holds for any c>0. Corollary 2.8 Let M be as above. Then, for any C∞ strictly plurisubharmonic function Φ : M → R,

n,q ¯ −αΦ = H(2) (M, i∂∂Φ,e ) 0 for q>0 (2.30) holds for any α>0. Example 2.5 M = Cn,Φ =z2. (Compare with Example 2.2.) Proposition 2.12 If there exist a Kähler metric and a plurisubharmonic function Φ on M such that Φ−1([−R,R]) are compact for all R ∈ R, then M admits a complete Kähler metric. 2.2 Vanishing Theorems 63

Proof Let λ be a convex increasing function on R such that − log (−t) if t ≤−e2, λ(t) = t2 if t ≥ 1.

Then i∂∂(λ¯ ◦ Φ) is a complete Kähler metric on M. If (E, h) is a Nakano positive vector bundle over a complete Kähler manifold (M, ω∞), the above proof of Theorem 2.14 works word for word to show that n,q = = − ⊗ H(2) (M,E,ω,h) 0forq>0ifdω 0 and iΘh IdE ω is Nakano semipositive. However, if rank E ≥ 2, the existence of such ω becomes a delicate question. Therefore, in view of applications, the following generalization of Theorem 2.14 is more appropriate. Theorem 2.15 Let M be a complex manifold of dimension n which admits a complete Kähler metric, and let (B, a) be a Nakano positive line bundle over ∈ n,q ∩ M. Then, for any Kähler metric ω on M and for any v L(2),loc(M, E) ¯ −1 Ker ∂ satisfying ((iΘaΛ) v,v) < ∞ with respect to ω, one can ﬁnd w ∈ n,q−1 ∩ ¯ ¯ = 2 ≤ −1 L(2) (M,B,ω,a) Dom ∂ satisfying ∂w v and w ((iΘaΛ) v,v). 0,q ∗ ∗ Warning. In contrast to Theorems 2.10 and 2.11, H(2) (M, B ,ω,a ) may not 0,0 ¯| |2 |z|2 vanish for q

Ia,x = {fx ∈ Ox; f ∈ O(U) and − |f |2e φ dV < ∞ for some neighborhood U x , (2.31) U 64 2 Analyzing the L2 ∂¯-Cohomology where φ is a plurisubharmonic function on U such that aeφ is a smooth fiber metric, and dV is any smooth volume form on M. Ia is called the multiplier ideal sheaf of the singular Hermitian line bundle (B, a). ⊂ p,q For any Hermitian metric ω on M and for any open set U M, L(2),loc(U,B,a) will denote the set of locally square integrable B-valued (p, q)-forms s on U with respect to a. By an abuse of notation, for any d-closed (1, 1)-form θ on M, we shall mean by ≥ + ¯ − ¯ iΘa iθ that Θa0 ∂∂Φ θ is locally of the form ∂∂ψ for some plurisubharmonic function ψ. By an abuse of language, we shall say that the singular fiber metric a has semipositive curvature current if iΘa ≥ 0 holds in the above sense. It is remarkable that by taking a “limit” of Theorem 2.14, as Demailly did in [Dm-4], one can strengthen the assertion very much as follows. Theorem 2.16 (cf. [Dm-3, Dm-4]) Let M be as in Theorem 2.14,letω be a Kähler metric on M, and let B be a holomorphic line bundle over M with a singular fiber ≥ ∈ n,q ∩ ¯ metric a satisfying iΘa ω. Then for any v L(2) (M,B,ω,a) Ker ∂ , one can ∈ n,q−1 ∩ ¯ ¯ = 2 ≤ 2 find w L(2) (M,B,ω,a) Dom ∂ satisfying ∂w v and w q v .

Proof For any sequence of positive numbers k converging to 0, let ak be smooth ≥ − fiber metrics of B converging to a from below and iΘak (1 k)ω. To find such ak, first do it locally by the convolution with respect to the Kähler metric, and then patch these approximating functions together by a partition of unity. Then, by applying Theorem 2.14 for each (ω, ak) and letting k →∞, one has the conclusion.

We shall say that a singular ﬁber metric a has strictly positive curvature current if there exists a Hermitian metric ω on M satisfying iΘh ≥ ω. A sheaf theoretic interpretation of Theorem 2.16 has important applications (cf. Sects. 2.2.7 and 3.3.2). Remark 2.3 The multiplier ideal sheaf was named after Kohn’s work on the ideals arising in the complex boundary value problem (cf. [Kn]). Besides this, it may be worthwhile to note that Ia had appeared implicitly in Bombieri’s work [Bb-1] which solved a question of analytic number theory by the L2 method. In fact, Bombieri applied a theorem of Hörmander in [Hö-1] which is a prototype of Theorem 2.16. It may be worthwhile to note that there is another limiting procedure which leads n,q to a result of different nature. To state it, we introduce a subset L(2) (M,E,σ,h)of n,q L(2),loc(M, E) (= the set of locally square integrable E-valued (n, q)-forms on M) for any smooth semipositive (1,1)-form σ on M as follows:

n,q L (M,E,σ,h)={u; lim u exists for any Hermitian metric ω0 on M}. (2) →0 (2.32)

Here u denotes the norm of u with respect to (σ + ω0,h). 2.2 Vanishing Theorems 65

n,q · = · Note that L(2) (M,E,σ,h)is a Hilbert space with norm σ lim→0 because of the monotonicity property “u ≤u if ≥ ”, so that n,q n,q H(2) (M,E,σ,h)and H(2),loc(M,E,σ)are deﬁned similarly. Theorem 2.17 (cf. [Oh-8, Theorem 2.8]) Let M be a complex manifold of dimension n admitting a complete Kähler metric, and let (E, h) be a Nakano semipositive vector bundle over M. Assume that σ is a smooth semipositive (1,1)-form on M such = − ⊗ n,q = that dσ 0 and iΘh IdE σ is Nakano semipositive. Then H(2) (M,E,σ,h) 0 for q>0. n,q ¯ Proof Let v ∈ L (M,E,σ,h)∩Ker ∂ (q>0), let ω0 be a complete Kähler metric (2) ∗ on M and let Λ = (σ + ω0) . Then

⊗ + −1 ≤ 2 ((IdE ω0 iΘh)Λ) v,v) v σ .

∈ n,q−1 + ¯ = ≤ Hence one can ﬁnd u L(2) (M,E,σ ω0,h)such that ∂u v and u vσ . Choosing a subsequence of u as → 0 which is locally weakly convergent, we are done. Remark 2.4 The prototype of Theorem 2.17 is a vanishing theorem of Grauert and Riemenschneider in [Gra-Ri-1, Gra-Ri-2] which ﬁrst generalized Kodaira–Nakano’s vanishing theorem for semipositive bundles. In Sects. 2.2.7 and 2.3, more on the results in this direction will be discussed. n,q ¯ As we have remarked in Sect. 2.2.1, H(2) (M) may not vanish even if i∂∂Φ is a complete metric (e.g. (M, ω) = (Cn,i∂∂¯z2) and 0

∗ ∗ C(∂u¯ +∂¯ u+∂ u+∂u)u≥|p + q − n|u2 (2.33)

∈ p,q = | | for any u C0 (M).HereC supM ∂Φ . Since

∗ ∂u¯ 2 +∂¯ u2 =∂u2 +∂u2,

(2.33) implies an estimate equivalent to the assertion. Example 2.6 M = Bn,Φ =−log (1 −z2) (cf. Chap. 4). 66 2 Analyzing the L2 ∂¯-Cohomology

Example 2.7 M = Bn \{0},Φ =−log (− log z2) (i∂∂Φ¯ is the Poincaré metric of D \{0} if n = 1). We shall say that a C∞ plurisubharmonic function Φ on M is of self bounded gradient,orof SBG for short, if

i(∂∂Φ¯ − ∂Φ ∧ ∂Φ)¯ ≥ 0forsome>0. (2.34)

Note that, if Φ is of SBG, then arctan (Φ) is a bounded plurisubharmonic function for some >0, which is strictly plurisubharmonic if so is Φ. With this condition of SBG, a limiting process works similarly to the proof of Theorem 2.14. For instance, one can deduce the following from Theorem 2.18. Corollary 2.9 Let (M, ω) be a (not necessarily complete) Kähler manifold of dimension n admitting a potential Φ of SBG. Assume that there exist an interval I ⊂ R and a C∞ function λ : I → R such that Φ(M) ⊂ I, λ ◦ Φ is of SBG and + 2 ∞ p,q = + sup ((λ λ )/(λ ) )< . Then H(2) (M, ω) 0 for p q>n. ={ ; 1 } = −1 = =− Example 2.8 M z 0 < z < e , Φ log z , I (0, 1), λ(t) log t.

2.2.6 Pseudoconvex Cases

From the above-mentioned L2 vanishing theorems, we shall deduce here vanishing theorems for the ordinary cohomology groups on pseudoconvex manifolds. First, vanishing theorems for positive bundles will be obtained from Theorems 2.12 and 2.14. As before, let (M, ω) be a Kähler manifold of dimension n. Recall that M is said to be pseudoconvex (resp. 1-complete)ifM is equipped with a C∞ plurisubharmonic (resp. strictly plurisubharmonic) exhaustion function. Pseudo- convex manifolds are also called weakly 1-complete manifolds (cf. [N-2, N-3]). Since every pseudoconvex Kähler manifold admits a complete Kähler metric by Proposition 2.12, an immediate consequence of Theorem 2.12 is: Theorem 2.19 (cf. [N-3]) Let M be a pseudoconvex manifold of dimension n and let (B, a) be a positive line bundle over M. Then H p,q(M, B) = 0 for p + q>n. + ∈ p,q ∩ ¯ Proof Let p q>nand v L(2),loc(M, B) Ker ∂. Then, it is easy to see that for any smooth plurisubharmonic exhaustion function Φ on M, one can ﬁnd a p,q − ◦ ∈ − ◦ λ Φ convex increasing function λ such that v L(2) (M,B,iΘae λ Φ ,ae ). Hence, ∈ ¯ p,q−1 v ∂(L(2),loc(M, B)) by Theorem 2.12. Similarly, from Theorem 2.14 we obtain: Theorem 2.20 (cf. [Kz-1]) Let M be a pseudoconvex manifold of dimension n and let (E, h) be a Nakano positive vector bundle over M. Then H n,q (M, E) = 0 for q>0. 2.2 Vanishing Theorems 67

∈ n,q ∩ ¯ Proof Let q>0 and v L(2),loc(M, E) Ker ∂. It is clear that for the Kähler metric iΘdet h there exists a smooth plurisubharmonic exhaustion function Φ on M −1 ∞ −Φ such that ((iΘhe−Φ Λ) v,v) < holds with respect to (iΘdet h,he ). Hence v ∈ Im ∂¯ by Theorem 2.14. Similarly, but using the L2 estimates for a sequence of solutions of the ∂¯- equation, we have: Theorem 2.21 Let M be a pseudoconvex manifold of dimension n and let (E, h) 0,q = be a Nakano negative vector bundle over M. Then H0 (M, E) 0 for q1(cf.[Siu-5]). Combining Theorems 2.20 and 2.21 (or 2.22), we obtain: Theorem 2.23 Let M be a 1-complete manifold of dimension n. Then, for any 0,q = 0,q = holomorphic vector bundle E over M, H (M, E) 0 (resp.H0 (M, E) 0) holds for any q>0 (resp.q0 follows from Theorem 2.20. The rest is similar. Similarly to the above, one can remove the L2 condition at inﬁnity from Theorem 2.16. Since Theorem 2.16 can be applied for all Stein domains in M,the result can be formulated in terms of the sheaf cohomology groups as in the spirit of the theorems of de Rham and Dolbeault. We shall summarize such interpretations in the next subsection.

2.2.7 Sheaf Theoretic Interpretation

Let M be a complex manifold of dimension n and let (B, a) be a singular Hermitian line bundle over M.ByW p,q(B, a) we shall denote a sheaf over M whose sections ⊂ p,q ⊂ p,q over an open set U M are those elements of L(2),loc(U,B,a)( L(2),loc(U, B)) ¯ p,q+1 whose images by ∂ (in the distribution sense) belong to L(2),loc(U,B,a). Then, n,q ¯ Theorem 2.16 implies that the complex (W (B, a)q≥0, ∂) is a ﬁne resolution of the sheaf O(KM ) ⊗ Ia. Hence, as an immediate consequence of Theorem 2.16 we obtain: 68 2 Analyzing the L2 ∂¯-Cohomology

Theorem 2.24 (Nadel’s vanishing theorem, cf. [Nd]) In the situation of Theo- rem 2.16, assume moreover that M is pseudoconvex. Then

q H (M, O(KM ⊗ B) ⊗ Ia) = 0 for q>0. Theorem 2.24 was proved for the compact case by Nadel [Nd] who applied it to prove the existence of Kähler–Einstein metrics on certain projective algebraic manifolds. Nadel proved also the coherence of Ia (cf. Chap. 3). As well as the generalization to noncompact manifolds, Nadel’s vanishing theorem can be extended easily to complex spaces with singularities.

Deﬁnition 2.3 Given a reduced complex space X of pure dimension n, a sheaf ωX over X is called the L2-dualizing sheaf of X if

={ ∈ ∩ O K ; ∈ n,0 ∩ } Γ(U,ωX) u Γ(U Xreg, ( Xreg )) u L(2) (V Xreg) for every V U for any open subset U ⊂ X. Similarly, for any singular Hermitian line bundle (B, a) over X,themultiplier dualizing sheaf ωX,a is deﬁned as the collection of square integrable germs of B- valued holomorphic n-forms with respect to a. : ˜ → = O K For any desingularization π X X, one has ωX π∗ ( X˜ ), since the support of effective divisors are negligible as the singularities of L2 holomorphic functions (cf. Theorem 1.8). In particular, ωX is a coherent analytic sheaf over X. Theorem 2.25 (Nadel’s vanishing theorem on complex spaces) Let X be a reduced and pseudoconvex complex space of pure dimension n with a Kähler metric, and let (B, a) be a singular Hermitian line bundle over X with strictly positive curvature current. Then

q H (X, O(B) ⊗ ωX,a) = 0 for q>0. Generally speaking, precise vanishing theorems are important in complex geometry not only because they yield effective results, but also because they give a wider perspective in the theory of symmetry and invariants. In fact, their generalizations and variants have been found in the literature of geometry, analysis and algebra. Let us review an example of such a development motivated by more algebraic ideas. Soon after the appearance of [Gra-Ri-1, Gra-Ri-2], Ramanujam [Rm-1] came up with a similar generalization of Kodaira’s vanishing theorem. Ramanujam’s background was Grothendieck’s theory [Grt-1, Grt-2], which is a foundation of algebraic geometry over the fields of arbitrary characteristic. In this situation, he could prove a vanishing theorem only for surfaces. No one could have done better because it is false for higher dimensions in positive characteristic. Note that a counterexample to Kodaira’s vanishing theorem in positive characteristic was found only after the publication of [Rm-2](cf.[Rn]). Anyway, from this new perspective, 2.2 Vanishing Theorems 69 inspired also by Bombieri’s work [Bb-2] on pluricanonical surfaces, he could strengthen the vanishing theorem in the following way. Theorem 2.26 (cf. [Rm-2]) Let X be a nonsingular projective algebraic variety of dimension n ≥ 2 over C and let L → X be a holomorphic line bundle whose first 2 · C := | ≥ Chern class c1(L) satisfies c1(L) > 0 and c1(L) ( deg L Cred ) 0 for any compact complex curve C in X. Then H 1(X, O(L∗)) = 0. In the case where L =[D] for some effective divisor D on X, H 1(X, O(L∗)) = 0 implies in particular that the support |D| of D is connected. The formulation of Ramanujam’s theorem is in the same spirit as in Nakai’s numerical criterion for ampleness of line bundles (cf. [Na-1, Na-2, Na-3]). Mumford [Mm] gave an alternate proof of Theorem 2.26 but did not proceed to the higher- dimensional cases. Kawamata [Km-1] and Viehweg [V] overcame this shortcoming independently by establishing the following: Theorem 2.27 Let X be a nonsingular projective algebraic variety of dimension n n and let L → X be a holomorphic line bundle with c1(L) > 0 such that c1(L)·C ≥ 0 for any compact complex curve C in X. Then

k H (X, O(KX ⊗ L)) = 0 holds for all k>0. By the Serre duality, this contains Theorem 2.26 as a special case. Being a numerical criterion for the cohomology vanishing, Theorem 2.27 is of basic importance in birational geometry. (See [Km-M-M] for instance.) A simple analytic proof of Theorem 2.27 was later given by Demailly [Dm-4]. According to the modern terminology, L is said to be nef (= numerically effective or numerically eventually free) if c1(L) · C ≥ 0 for any complex curve C on X. The notion of nef line bundles naturally extends to Kähler manifolds. Namely, a holomorphic line bundle L over a compact Kähler manifold is called nef if c1(L) is in the closure of the cone of Kähler classes. Nef bundles over nonsingular projective varieties are nef in the latter sense because any Kähler class is in the closure of the cone generated over R+ by the ﬁrst Chern classes of positive line bundles. Nef line bundles also make sense over the proper images of compact Kähler manifolds by almost biholomorphic maps. Furthermore, in virtue of a theorem of Varouchas [Va], they can be deﬁned similarly over proper holomorphic images of compact Kähler manifolds. Demailly and Peternell [Dm-P] proved: Theorem 2.28 Let X be a compact and normal complex space of dimension n 2 admitting a Kähler metric, and let L be a nef line bundle over X with c1(L) = 0. q Then H (X, O(KX ⊗ L)) = 0 for q ≥ n − 1. Similarly to Theorem 2.26,ifn ≥ 2 and L =[D] for some effective divisor D, it follows from the assumption of Theorem 2.28 that |D| is connected. In [Oh-27] 1 \| | O = 1 it is proved that H0 (X D , ) 0 under the same hypothesis. Recall that H0 denotes the cohomology with compact support. A question was raised in [Dm-P-S] 70 2 Analyzing the L2 ∂¯-Cohomology

q n−q+1 whether or not H (X, O(KX ⊗ L)) = 0 holds if L is nef and c1(L) = 0. It was recently settled by J.-Y. Cao [CJ]. There exists another sheaf theoretic interpretation of Theorem 2.14, slightly different from that of Theorem 2.16. It is based on an observation that, n,q n,q although L(2) (M,E,σ,h) are subsets of L(2),loc(M, E) for semipositive σ , n,q H(2),loc(M,E,σ) can be naturally isomorphic to certain sheaf cohomology group on a quotient space of M. Assume that there exists a proper holomorphic map π from M to a complex space X admitting a Kähler metric ω. Then, Theorem 2.14 n,q ∗ =∼ q O K ⊗ ≥ implies that H(2),loc(M,E,π ω) H (X, π∗ ( M E)) for q 0ifE is Nakano semipositive on some neighborhood of π −1(x) for any x ∈ X. In particular, one has the following. Theorem 2.29 (cf. [Oh-8, Theorem 3.1]) Let M be a pseudoconvex Kähler manifold, let X be a complex space with a Kähler metric ω, and let π : M → X be a proper holomorphic map. Then, for any Nakano semipositive vector bundle (E, h) ∗ q over M satisfying iΘh − IdE ⊗ π ω ≥ 0, H (X, π∗O(KM ⊗ E)) = 0 for q>0. Remark 2.6 In [Oh-8], this was stated only for compact M. As in the case of Theorem 2.25, Theorem 2.29 has several predecessors besides the Kodaira–Nakano vanishing theorem. These are semipositivity theorems for the direct image sheaves K ⊗ ∗K−1 of the relative canonical bundles M π X for Kählerian M and nonsingular X (cf. [Gri-1, Gri-3, F-1]). Quite recently, there was an unexpected development in the theory of L2 holomorphic functions closely related to the semipositivity theorem of this type (see Chaps. 3 and 4). See [Ko’86-1, Ko’86-2] and [Ko’87] for the higher K ⊗ ∗K−1 direct images of M π X . Roughly speaking, the choice of singular ﬁber metrics and degenerate base metrics (=pseudometrics) amounts to the choice of boundary conditions in the problems of partial differential equations. There are inductive arguments to produce “good” singular ﬁber metrics (cf. Lemma 3.2 in Chap. 3). Concerning the L2 ∂¯-cohomology groups of type (p, q), there are results which relate analytic invariants and topological invariants on complex spaces with singularities. Let us review some of these results in the next subsection.

2.2.8 Application to the Cohomology of Complex Spaces

¯ ¯ $ $ The symmetry [∂,∂ ]gr =[∂ ,∂]gr of the complex Laplacian on a compact Kähler manifold M yields Hodge’s decomposition theorem

H r (M) =∼ H p,q(M) p+q=r H p,q(M) =∼ H q,p(M)

(cf. [W]). Hence it is natural to expect that the corresponding decomposition for the spaces of L2 harmonic forms carries similar geometric information also on 2.2 Vanishing Theorems 71 some noncompact Kähler manifolds. Grauert [Gra-2] has shown that, for every compact Kähler space X, Xreg carries a complete Kähler metric. Based on this, it will be shown below after [Sap] and [Oh-10, Oh-11, Oh-14] that the L2 cohomology groups of Xreg with respect to some class of (not necessarily complete) metrics are canonically isomorphic to the ordinary ones in certain degrees. Accordingly, the Hodge decomposition remains true for M = Xreg there. The method also gives a partial solution to a conjecture of Cheeger, Goresky and MacPherson in [C-G-M]. Related results are also reviewed. Let X be a (reduced) complex space. A Hermitian metric on X is deﬁned as a Hermitian metric or a positive (1,1)-form say ω on Xreg, such that, for any point x0 ∈ Sing X there exists a neighborhood U x0 in X, a proper holomorphic embedding ι of U into a polydisc DN for some N and a C∞ positive (1, 1) form Ω on DN such ∗ that ω = ι Ω holds on U ∩ Xreg. A complex space X equipped with a Hermitian metric is called a Hermitian complex space. A Hermitian complex space (X, ω) is called a Kähler space if dω = 0. Lemma 2.6 Let (X, ω) be a compact Kähler space. Then there exists a continuous ¯ function ϕ : X →[0, 1] such that ω + i∂∂ϕ is a complete Kähler metric on Xreg for which the length of ∂ϕ is bounded.

Proof Let x0 ∈ Sing X be any point and let f1,...,fm be holomorphic functions I on a neighborhood U of x0 which generate the ideal sheaf Sing X on U. Then, by m | |2 −e shrinking U if necessary so that j=1 fj

1 ϕ = . U − m | |2 log ( log j=1 fj )

By an abuse of notation, we put ϕU = 1ifU ∩ Sing X = ∅. In order to see that a desired function ϕ can be obtained by patching ϕU by a partition of unity, let us take another generator (g1,...,g ) of ISing X over U. Then it is easy + ¯ to verify that one can find a neighborhood V x0 such that ω i∂∂ϕU and + ¯ − | |2 −1 ω i∂∂(log ( log k=1 gk )) are positive and quasi-isometrically equivalent to each other on Xreg ∩V . Moreover, it is also immediate that, for any >0 one can find a neighborhood W x0 such that the length of ∂ϕU on W ∩ Xregwith respect ¯ ¯ to ω + i∂∂ϕU (or even with respect to i∂∂ϕU ) is less than . Now let U ={Uα}α∈A be a finite open cover of X by such U, and let ρα be a C∞ partition of unity associated to U . We put = ϕ ραϕUα . (2.35) α∈A

−1 Then ϕ is a continuous function on X with values in [0, 1] such that ϕ (0) = | ∞ = ¯ = Sing X and ϕ Xreg is C . Furthermore, since ∂( ρα) 0 and ∂∂( ρα) 0, ¯ = ¯ − + ¯ − ¯ ∂∂ϕ ∂∂ρα(ϕUα ϕUβ ) ∂ρα(∂ϕUα ∂ϕUβ ) + − ¯ + ¯ (∂ϕUα ∂ϕUβ )∂ρα ρα∂∂ϕUα . (2.36) 72 2 Analyzing the L2 ∂¯-Cohomology

¯ Combining (2.34) with the above remarks on ω + i∂∂ϕU and ∂ϕU , it is clear that ¯ ω + i∂∂ϕ is a complete Kähler metric on Xreg for 0 < 1. Any function ϕ of the form (2.35) will be called a Grauert potential on X.A metric of the form ω+i∂∂ϕ¯ will then be called a Grauert metric. The boundedness condition for ∂ϕ is important when one wants to extend the Hodge theory to complex spaces with singularities. A basic fact for that is the following: Proposition 2.13 Let (M, ω) be a complete Kähler manifold, let (E, h) be a Hermitian holomorphic vector bundle over M and let ϕ be a real-valued bounded C∞ function on M such that ∂ϕ is bounded with respect to ω. Then, for any ∈ p,q ∈ ¯ ∩ nonnegative integers p and q, and for any u L(2) (M.E) satisfying u Dom ∂ Dom ∂¯ ∗, u belongs to the domain of the adjoint of ∂¯ with respect to the modified −ϕ ¯ fiber metric he . If moreover Θh and ∂∂ϕ are also bounded, u belongs also to the $ domains of ∂h,∂he−ϕ and ∂ . Proof The proof of the first assertion follows immediately from the definition of the adjoint of ∂¯. The second assertion follows from Nakano’s identity. ⊂ 2 p,q ∩ r ∩ For any open set U X,theL cohomology groups H(2) (U Xreg), H(2)(U p,q r Xreg) with respect to ω will be denoted by H(2) (U), H(2)(U), for simplicity. The L2 cohomology groups with supports restricted to relatively compact subsets of p,q r 2 U will be denoted by H(2),0(U), H(2),0(U). Similarly, the L cohomology groups with respect to a Grauert metric ω + i∂∂ϕ¯ and those with “compact support in p,q r p,q r U” will be denoted by H(2),ϕ(U), H(2),ϕ(U), H(2),ϕ,0(U) and H(2),ϕ,0(U). Then the vanishing of the L2 cohomology of Akizuki–Nakano type on complete Kähler manifolds implies that these L2 cohomology groups do not see the singularities in higher degrees. For instance the following holds. Theorem 2.30 Let (X, ω) be a compact Kähler space of pure dimension n and let ϕ be a Grauert potential on X.Ifdim Sing X = 0, then

p,q =∼ p,q =∼ p,q + + H(2) (X) H(2),ϕ(X) H0 (Xreg) for p q>n 1 and

r =∼ r =∼ r + H(2)(X) H(2),ϕ(X) H0 (Xreg) for r>n 1.

p,q p,q p,q Moreover, the natural homomorphisms from H0 (Xreg) to H(2) (X) and H(2),ϕ(X) + = + n+1 n+1 are surjective for p q n 1, and so are those from H0 (Xreg) to H(2) (X) n+1 and H(2),ϕ(X).

Proof Let x0 ∈ Sing X and let W {x0} be a neighborhood such that W ∩Sing X = {x0}.Letδ>0 be sufﬁciently small so that Wδ := {x ∈ W; ϕ(x) < δ} W. ¯ −1 Then, with respect to the complete Kähler metric ω,δ := ω + i∂∂(log (δ − ϕ)) \{ } 2 p,q \{ } (0 < δ<1) on Wδ x0 ,theL cohomology groups H(2) (Wδ x0 ,ω,δ) 2.2 Vanishing Theorems 73 vanish for p + q>nby Theorem 2.12. On the other hand, it is easy to see that, for 2 + | 2 any L (p, q) form f on Wreg with p q>n, f Wreg\{x0} is L with respect to ω,δ. Therefore the natural homomorphism

p,q −→ p,q H0 (Xreg) H(2),ϕ(X) is surjective if p + q>nand injective if p + q>n+ 1. Concerning the L2 de r r \{ } Rham cohomology groups H(2), that H(2)(Wδ x0 ,ω,δ) vanish for r>ncan be ∈ r \{ } = shown as follows: Let u L(2)(Wδ x0 ,ω,δ), du 0 and r>n. Then u is = r,0 + r−1,1 +···+ 0,r p,q ∈ p,q \{ } decomposed as u u u u with u L(2) (Wδ x0 ,ω,δ). = ¯ 0,r = ∈ 0,r−1 \{ } Since du 0, ∂u 0, so that there exists v L(2) (Wδ x0 ,ω,δ) such ¯ = 0,r ¯ ∗ = that ∂v u and ∂ v 0. By Proposition 2.13, it follows in particular that ∈ 1,r−1 \{ } − ∈ r j,r−j \{ } ∂v L(2) (Wδ x0 ,ω,δ). Hence u dv j=1 L(2) (Wδ x0 ,ω,δ). Proceeding similarly, we obtain that the L2 de Rham cohomology class of u is 0. p,q r 2 Thus we obtain the assertion for H(2),ϕ(X) and H(2),ϕ(X). As for the ordinary L p,q r cohomology groups H(2) (X) and H(2)(X), they are considered respectively as the p,q r limits of H(2),ϕ(X) and H(2),ϕ(X). For that, we make a special choice of ϕ as in the proof of Lemma 2.6. Then, after ﬁxing δ, we consider ω as the limit of ω,δ → p,q \{ } ⊂ p,q \{ } ( 0). Then, as is easily checked, L(2) (Wδ x0 ,ω) L(2) (Wδ x0 ,ω,δ) ¯ ¯ 2 if p + q>n, so that by solving the ∂ equation ∂v = u with L norm estimates ∈ p,q−1 \{ } for v L(2) (Wδ x0 ,ω,δ) uniformly in , and by taking the weak limit of p,q \ a subsequence of v, we obtain the required the vanishing results for H(2) (Wδ { } r \{ } x0 ,ω)as well as those for H(2)(Wδ x0 ,ω). In view of the long exact sequences

···−→ p,q −→ p,q H0 (Xreg) H(2) (X) −→ p,q \ −→ p,q+1 −→··· −lim→ H(2) (X K) H0 (Xreg) ···−→ p,q −→ p,q −→ p,q \ (resp. H0 (Xreg) H(2),ϕ(X) −lim→ H(2),ϕ(X K) −→ p,q+1 −→··· H0 (Xreg) ), where− lim→ denotes the inductive limit of the system

p,q \ −→ p,q \ H(2) (X K1) H(2) (X K2) p,q \ −→ p,q \ ⊂ (resp. H(2),ϕ(X K1) H(2),ϕ(X K2)) (K1 K2 Xreg),

Theorem 2.30 says

p,q \ = p,q \ = + −lim→ H(2) (X K) −lim→ H(2),ϕ(X K) 0ifp q>n. (2.37) 74 2 Analyzing the L2 ∂¯-Cohomology

The proof shows that

r \ = r \ = −lim→ H(2)(X K) −lim→ H(2),ϕ(X K) 0forr>n is a consequence of (2.37). Existence of the natural homomorphisms

p,q \ = p,q \ + −lim→ H(2),ϕ(X K) −lim→ H(2) (X K), p q>n is crucial to deduce p,q \ = −lim→ H(2) (X K) 0 from p,q \ = −lim→ H(2),ϕ(X K) 0.

The Kähler condition is superﬂuous here. Obviously Theorem 2.30 holds for compact Hermitian complex spaces. Thus, an essential part of Theorem 2.30 can be stated as a Dolbeaut-type lemma: N Lemma 2.7 Let V be an analytic set of pure dimension n in D containing z0 as an p,q = isolated singularity. Then there exists a neighborhood U z0 such that H(2) (U) p,q = + r = r = 0, H(2),ϕ(U) 0 for p q>nand H(2)(U) 0, H(2),ϕ(U) 0 for r>n. Because of the presence of singularities, it is not allowed immediately to apply the ordinary duality theorems due to Poincaré and Serre to obtain the results for p + q, r < n, simply reversing the direction of the arrows. Nevertheless there is a method to prove the following (see [Oh-10, Supplement]).

Lemma 2.8 Let V and z0 be as above. Then there exists a neighborhood U z0 such that

p,q = p,q = + H(2),0(U) 0,H(2),ϕ,0(U) 0 for p q

r = r = H(2),0(U) 0,H(2),ϕ,0(U) 0 for r

As a result, the dual of Theorem 2.30 is stated as follows. Theorem 2.31 Let (X, ω) and ϕ be as in Theorem 2.30. Then

p,q =∼ p,q =∼ p,q + − H(2) (X) H(2),ϕ(X) H (Xreg) for p q

r =∼ r =∼ r − H(2)(X) H(2),ϕ(X) H0 (Xreg) for r

p,q p,q p,q Moreover, the natural homomorphisms from H(2) (X) and H(2),ϕ(X) to H (Xreg) + = − n−1 n−1 are injective for p q n 1, and so are those from H(2) (X) and H(2),ϕ(X) to n−1 H (Xreg). Proof First we shall show the surjectivity of

p,q −→ p,q + − H(2),ϕ(X) H (Xreg) for p q

p,q ¯ For that, it sufﬁces to prove that, for any u ∈ C (Xreg) ∩ Ker ∂ (p + q< − ∈ p,q−1 − ¯ ∈ p,q n 1), there exists w L(2),loc(Xreg) such that u ∂w L(2),ϕ(X).Letρ be a C∞ function on X such that ρ = 1 on a neighborhood of Sing X and supp ρ ⊂{ϕ<δ} for sufﬁciently small δ. We put Vδ ={ϕ<δ}\Sing X. ∈ p,q ¯ 1 ¯ = ¯ Then take any v L(2) (Vδ,i∂∂(log (δ−ϕ))) with ∂v ∂(ρu) on Vδ. (Note that + ¯ + ¯ 1 ω i∂∂ϕ i∂∂(log (δ−ϕ)) is a complete Kähler metric on Vδ.) We put v on V , v˜ = δ 0 on Xreg \ Vδ.

˜ ∈ p,q ¯ ˜ − = ˜ − ⊂ Then v L(2),ϕ(Xreg), ∂(v ρu) 0 and supp(v ρu) Vδ. Hence, similarly to the above, by applying Theorem 2.13 for a = e−μ(ϕ) for a family of convex ∈ p,q−1 ⊂ increasing functions μ, one can ﬁnd w L(2),loc(Xreg) such that supp w Vδ and ˜ − = ¯ − ¯ ∈ p,q v ρu ∂w. Hence u ∂w L(2),ϕ(X). Considering a long exact sequence, we conclude that the natural homomorphisms

p,q −→ p,q H(2),ϕ(X) H (Xreg) are bijective if p+q

r =∼ r H(2)(X) H (Xreg) for r

r =∼ r H(2)(X) H0 (Xreg) for r>n.

Sketch of proof In [Sap], Saper proved that there exists a complete metric ω˜ on Xreg such that

r ˜ =∼ r H(2)(Xreg, ω) H (Xreg) for r

r ˜ =∼ r H(2)(Xreg, ω) H0 (Xreg) for r>n

r + ˜ p,q + ˜ hold. By analyzing the behavior of H(2)(Xreg,ω ω) and H(2) (Xreg,ω ω) as → 0, the required isomorphisms are obtained. Since Saper’s metric is Kählerian if so is ω, Theorem 2.32 naturally implies an extension of Hodge’s decomposition theorem to compact Kähler spaces with isolated singularities. On the other hand, Theorems 2.30 and 2.31 with the Kähler condition implies the following: Theorem 2.33 Let (X, ω) be a compact Kähler space of pure dimension n with dim Sing X = 0. Then

r =∼ p,q + H0 (Xreg) H0 (Xreg) for r>n 1 p+q=r

r ∼ p,q (resp.H(Xreg) = H (Xreg) for r

p,q =∼ q,p + + H0 (Xreg) H0 (Xreg) for p q>n 1 p,q ∼ q,p (resp.H (Xreg) = H (Xreg) for p + q

r ∞ r ∞ Since dim H0 (Xreg)< and dim H (Xreg)< for all r, the following is immediate from Theorem 2.33. 2.3 Finiteness Theorems 77

p,q ∞ + Theorem 2.34 In the situation of Theorem 2.33, dim H0 (Xreg)< for p p,q q>n+ 1 and dim H (Xreg)<∞ for p + q

p,q ∞ + + + dim H0 (Xreg)< for p q>n 1 k and

p,q dim H (Xreg)<∞ for p + q

Actually Theorem 2.35 is a special case of a more general ﬁniteness theorem due to Andreotti and Grauert in [A-G]. The L2 approaches towards it will be reviewed in the next section. Remark 2.7 Complete Kähler metrics naturally live on locally symmetric varieties, and the L2 cohomology is known to be isomorphic to the intersection cohomology there (cf. [Sap-St]). For basic theorems and the background on the L2 cohomology of such a distinguished class of metrics, see [Z, K-K, F].

2.3 Finiteness Theorems

Given a complex manifold M and a holomorphic vector bundle E over M,we 2 ¯ p,q have seen that certain L ∂-cohomology groups H(2) (M, E) vanish under some conditions on the metrics of M and E. Following a basic argument in [Hö-1, Theorem 3.4.1 and Lemma 3.4.2], we are going to see below that, by throwing away the positivity assumption on the curvature form of E, but only on a compact subset of M, one has ﬁnite dimensionality of the L2 ∂¯-cohomology instead of its vanishing. To derive the ﬁnite-dimensionality as well for the ordinary ∂¯-cohomology, a limiting procedure is applied which is reminiscent of Runge’s approximation theorem.

2.3.1 L2 Finiteness Theorems on Complete Manifolds

Let (M, ω) be a complete Hermitian manifold of dimension n and let (E, h) be a Hermitian holomorphic vector bundle over M. If there exists a compact set K0 ⊂ M such that dω = 0 holds on M \ K0, the basic inequality in Sect. 2.2.4 implies that 78 2 Analyzing the L2 ∂¯-Cohomology

¯ 2 ¯ ∗ 2 ∂u +∂ u ≥ (i(ΘhΛ − ΛΘh)u, u) (2.38)

∈ p,q \ holds for any u C0 (M K0,E). Therefore, for any neighborhood U ⊃ K0, one can ﬁnd a constant CU > 0 such that n ¯ 2 ¯ ∗ 2 CU 2 ω (1 + )(∂u +∂ u ) + |u| ≥ (i(ΘhΛ − ΛΘh)u, u) (2.39) U n!

∈ p,q holds for any >0 and u C0 (M, E). Recalling a basic part of real analysis, by the strong ellipticity of the differential operator ∂¯∂¯ $ + ∂¯ $∂¯ and Rellich’s lemma, one has: ¯ ¯ ∗ Lemma 2.9 For any compact subset K of M, any sequence uk ∈ Dom ∂∩Dom ∂ ∩ p,q L(2) (M, E) satisfying

¯ ¯ ∗ sup (uk+∂uk+∂ uk)<∞

∈ N admits a subsequence ukμ (μ ) such that ωn lim |u − u |2 = 0. →∞ kμ kν μ,ν K n!

Hence, in view of Proposition 2.2, we are naturally led to the following ﬁniteness theorem.

Theorem 2.36 Suppose that iΘh − c IdE ⊗ ω>0 holds for some c>0 outside a n,q ∞ compact subset of M. Then dim H(2) (M, E) < for all q>0.

Proof Let K0 be a compact subset of M such that the curvature form Θh satisfies iΘh − c IdE ⊗ ω ≥ 0onM \ K0 for some c>0. Then one can find a compact set ⊃ ∈ ¯ ∩ ¯ ∗ ∩ n,q K K0 and a constant C>0 such that for all u Dom ∂ Dom ∂ L(2) (M, E) n ∗ ω C ∂u¯ 2 +∂¯ u2 + |u|2 ≥u2 (2.40) K n! holds. Hence, by Lemma 2.9 one can see that the assumption of Proposition 2.2 is = n,q−1 = n,q = n,q+1 = satisfied for H1 L(2) (M, E), H2 L(2) (M, E), H3 L(2) (M, E), T ¯ n,q−1 = ¯ n,q ∂ on L(2) (M, E), and S ∂ on L(2) (M, E). Therefore, by Theorem 2.2, n,q ¯ ∩ ¯ ∗ ∩ n,q H(2) (M, E) is isomorphic to Ker ∂ Ker ∂ L(2) (M, E) and finite dimensional.

By virtue of the celebrated unique continuation theorem of Aronszajn [Ar], Theorem 2.36 implies the following. 2.3 Finiteness Theorems 79

Corollary 2.10 (cf. [Gra-Ri-1, Gra-Ri-2] and [T-1]) In the situation of Theo- rem 2.36 and K0 as above, assume moreover that M = K0, dω = 0, and that ≥ n,q = iΘh 0 holds everywhere. Then H(2) (M, E) 0 for q>0. Combining Corollary 2.10 with a theorem of Grauert on the coherence of the direct image sheaves of coherent analytic sheaves by proper holomorphic maps (cf. [Gra-4]), Takegoshi obtained in [T-2] the following: Theorem 2.37 Let M be a Kähler manifold and let π be a proper surjective holomorphic map from M to a complex space X. Then, for any Nakano semipositive q vector bundle (E, h) over M, the higher direct image sheaves R π∗O(KM ⊗ E) vanish for any q>n− dim X. Similarly, we obtain the ﬁniteness counterparts of Theorems 2.10, 2.12 and 2.18 and their strengthened versions as vanishing theorems. However, it is not known to the author whether or not Theorem 2.14 can also be strengthened to a reasonable ﬁnite-dimensionality theorem.

2.3.2 Approximation and Isomorphism Theorems

Once one knows the ﬁnite-dimensionality of the L2 cohomology groups, a natural question is to compare them with the ordinary ∂¯-cohomology groups, for instance as in the diagram below:

p,q p,q H(2) (M, E) H(2) (U, E)

H p,q(M, E) H p,q(U, E) (U M)

Here “the map” may not be well defined if the metric on U does not extend continuously to M. For the preparation of such a study, let us first go back to the setting of Sect. 2.1, and establish an abstract approximation theorem modelled on a beautiful argument of Hörmander [Hö-1, Proposition 3.4.5] generalizing a well-known proof of Runge’s approximation theorem. For that, the following slight extension of the notion of weak convergence is useful. Definition 2.4 Given a Hilbert space H and a dense subset V ⊂ H , a sequence uμ ∈ H is said to converge V -weakly to u ∈ H, denoted by wV -limμ→∞ uμ = u, if (u, v)H = limμ→∞ (uμ,v)H holds for any v ∈ V .

Let Hj (j = 1, 2) and T : H1 → H2 be as in Sect. 2.1.1. We consider a sequence of such triples (H1,H2,T),say(H1,μ,H2,μ,Tμ)(μ∈ N), together with bounded 80 2 Analyzing the L2 ∂¯-Cohomology

(C -linear) operators Pj,μ : Hj,μ → Hj such that the norms of Pj,μ are uniformly bounded and TP1,μ = P2,μTμ (in particular P1,μ(Dom Tμ) ⊂ Dom T ) holds for each μ. In this situation, we look for a condition for Ker Tμ to approximate Ker T in some appropriate sense. For that, we fix once for all a dense subset V ⊂ H1 and require the following: ∗ = ∈ (i) wV -limμ→∞ P1,μP1,μv v for any v H1. ∈ ∗ ∗ (ii) For any sequence uμ Dom Tμ such that w-lim P2,μuμ and wV -lim P1,μTμ uμ ∗ = ∗ both exist, wV -lim P1,μTμ uμ T (w-lim P2,μuμ) holds true. Moreover, in accordance with the situation of Sect. 2.1.1, we shall require also: ≤ ∗ (iii) There exists a constant C>0 such that u H2,μ C Tμ u H1,μ holds for all ∈ ∗ ∈ N ⊥ ∗ u Dom Tμ (μ ) satisfying P2,μu Ker T . Theorem 2.38 In the above situation, μ P1,μ(Ker Tμ) is dense in Ker T . ∈ ⊥ ∗ ⊥ Proof Take any v H1 such that v P1,μ(Ker Tμ) for all μ. Then P1,μv Ker Tμ so that, by the assumption (iii) and by virtue of Theorem 2.3 (ii), one can find a constant ˜ ∈ ∗ ∗ = ∗ ≤ ˜ C>0 and uμ Dom Tμ such that P1,μv Tμ uμ and uμ H2,μ C v H1 hold for all μ . Hence, by (i) and (ii), the weak limit, say u, of a subsequence of P2,μuμ satisfies T ∗u = v. Hence v⊥Ker T , which proves the assertion.

Now let (M, ω), (E, h) and K0 be as in the beginning, and let U ⊂ M be an open set containing K0. Given a complete Hermitian metric ωU on U and a ﬁber metric hU of E|U , we shall describe a condition for a sequence ωμ (μ ∈ N) of complete Hermitian metrics on M and a sequence hμ of ﬁber metrics of E such that the union ¯ ∩ p,q ∈ N of images of Ker ∂ L(2) (M,E,ωμ,hμ) for all μ by the restriction map

: p,q −→ p,q | ρU L(2),loc(M, E) L(2),loc(U, E U ) (2.41)

¯ ∩ p,q | is a dense subset of Ker ∂ L(2) (U, E U ,ωU ,hU ). = n,q ⊂ For simplicity, ﬁrst we assume that p n. Then, ρU (L(2) (M,E,ωμ,hμ)) n,q | ≤ ≥ L(2) (U, E U ,ωU ,hU ) as long as ωμ ωU and hμ hU hold on U (cf. the proof of Theorem 2.14). In this setting, a geometric variant of Theorem 2.38 can be stated as follows. Theorem 2.39 In the above situation, assume moreover the following:

(a) ωμ are all Kählerian on M \ K0. 1 (b) limμ→∞ ωμ|U = ωU and limμ→∞ hμ|U = hU locally in the C -topology. − ⊗ ≥ (c) There exists a constant c>0 such that iΘhμ c IdE ωμ 0 hold everywhere on M \ K0 for all μ. 2.3 Finiteness Theorems 81

Then, for all q ≥ 0, ¯ ∩ n,q ρU (Ker ∂ L(2) (M,E,ωμ,hμ)) μ∈N

¯ ∩ n,q | is dense in Ker ∂ L(2) (U, E U ,ωU ,hU ). = n,q = n,q+1 ∩ ¯ Proof Let H1 L(2) (U,E,ωU ,hU ), H2 L(2) (U,E,ωU ,hU ) Ker ∂,

= n,q H1,μ L(2) (M,E,ωμ,hμ), = n,q+1 ∩ ¯ H2,μ L(2) (M,E,ωμ,hμ) Ker ∂, ¯ ¯ T = ∂ : H1 → H2,Tμ = ∂ : H1,μ → H2,μ, and Pj,μ be the restriction maps. By the assumption that ωμ ≤ ωU and hμ ≥ hU , the uniform boundedness of Pj,μ = n,q is obvious. It is also clear that (i) and (ii) above hold for V C0 (U, E) follows from (b). (Note that ωU is also complete.) To see that (iii) is also true, (a), (b) and (c) are combined as follows. Suppose that the assertion were false. Then there would exist a nonzero n,q ∩ ¯ ¯ ∩ element of L(2) (U,E,ωU ,hU ) Ker ∂,sayv, orthogonal to ρU (Ker ∂ n,q ∈ L(2) (M,E,ωμ,hμ)) for all μ. Hence there would exist a sequence uμ H2,μ | ⊥ ¯ ∗ = ∗ = such that uμ U Ker ∂ , uμ H2,μ 1 and lim infμ→∞ Tμ uμ H1,μ 0, because ¯ ∗ otherwise v would belong to the image of ∂ . By (b), a subsequence of uμ|U would n,q ∩ ¯ ∩ ¯ ∗ weakly converge to some element of L(2) (U,E,ωU ,hU ) Ker ∂ Ker ∂ ,sayu. ¯ ∗ By (a) and (c), u = 0. But since uμ|U was in the orthogonal complement of Ker ∂ , so is u. An absurdity! Arguing similarly to the above, from Theorems 2.3 and 2.39 one has the following.

Proposition 2.14 In the situation of Theorem 2.39, there exists μ0 ∈ N such that n,q n,q | the natural homomorphisms from H(2) (M,E,ωμ,hμ) to H(2) (U, E U ,ωU ,hU ) induced by ρU are injective for all q>0 and μ ≥ μ0. Proof Let the notation be as in the proof of Theorem 2.39 for q ≥ 0. Suppose that there exist inﬁnitely many μ such that ρU induces noninjective homomorphisms n,q+1 n,q+1 | from H(2) (M,E,ωμ,hμ) to H(2) (U, E U ,ωU ,hU ). Then, in view of Theo- ¯ ¯ ∗ rem 2.3, there would exist a sequence uμ ∈ H2,μ such that uμ|U ⊥(Ker ∂ ∩ Ker ∂ ), = ∗ = uμ H2,μ 1 and lim infμ→∞ Tμ uμ H1,μ 0, which leads us to a contradiction, similarly to the above. Combining Proposition 2.14 with Theorems 2.39 and 2.38, we obtain:

Theorem 2.40 In the situation of Theorem 2.39, there exists μ0 ∈ N such that the n,q n,q | natural homomorphisms from H(2) (M,E,ωμ,hμ) to H(2) (U, E U ,ωU ,hU ) are isomorphisms for all q>0 and μ ≥ μ0. 82 2 Analyzing the L2 ∂¯-Cohomology

Remark 2.8 A natural question is whether or not the restriction μ ≥ μ0 is superﬂuous. It is not, as one can see from the following. Example 2.9 idz∧ dz (M, ω) = C, , (|z|2 + 1)(log (|z|2 + 2))2 E = {(ζ, ξ); zmζ − ξ = 0} (m ≥ 2), z∈C log (|z|2 + 2) |(ζ, ξ)|2 = (|ζ |2 +|ξ|2) , h |z|2 + 1 2 −2 (U, ωU ) = ({z;|z| < 1},i(1 −|z| ) dz ∧ dz)

⇒ 1,1 = 1,1 | = H(2) (M, E) 0butH(2) (U, E U) 0.

As this example shows, μ0 can be arbitrarily large depending on the choices of E, but we do not know any estimate for it in terms of the curvature of E. Let us brieﬂy illustrate how these approximation and isomorphism theorems are applied.

Proposition 2.15 Let (M, ω), (E, h), K0,U,ωU and hU be as above, such that U ={x ∈ M; φ(x) < d} for some C∞ plurisubharmonic function φ on M. Then, sequences ωμ and hμ satisfying ωμ ≤ ωU , hμ ≥ hU , (a), (b) and (c) exist if iΘh − c IdE ⊗ ω ≥ 0 on M \ K0 holds for some c>0. = + ¯ 1 = · − ∈ N Proof Put ωU ω i∂∂ log d−φ and hU h (d φ).Letλμ(t) (μ ) be ∞ a sequence of C convex increasing functions on R such that limμ→∞ λμ(t) = ∞ − log (−t) on (−∞, 0) locally in the C topology. Then it is easy to see that ωμ = ¯ −λ (φ−d) ω + i∂∂λμ(φ − d) and hμ = h · e μ satisfy the requirements. As is easily seen from the above, for pseudoconvex manifolds, the method of detecting the equivalence of L2 cohomology groups through the L2 estimates can be naturally extended to establish isomorphism theorems between the ordinary cohomology groups. For instance, let us prove the following. Theorem 2.41 (cf. [N-R]) Let (M, φ) be a pseudoconvex manifold of dimension n and let (E, h) be a holomorphic Hermitian vector bundle over M which is Nakano n,q positive on M \ Mc for some c. Then dim H (M, E) < ∞ for all q>0 n,q n,q and the restriction homomorphisms H (M, E) → H (Md ,E) (q > 0) are isomorphisms for all d ≥ c. Proof By Theorem 2.40 and Proposition 2.15, the natural restriction homomorphisms

d : n,q −→ n,q ρc H(2) (Md ,E) H(2) (Mc,E) (q>0) 2.3 Finiteness Theorems 83 are isomorphisms if d>c. Moreover, since λμ in the proof of Proposition 2.15 can be chosen to be of arbitrarily rapid growth, it follows that the natural maps

n,q −→ n,q H (Md ,E) H(2) (Mc,E) are also bijective. n,q → n,q ∈ n,q ∩ ¯ Injectivity of H (M, E) H (Mc,E):Letu L(2),loc(M, E) Ker ∂. ∈ n,q−1 = ¯ Suppose that there exists v L(2),loc(Mc,E)such that u ∂v holds on Mc. Then, d ∈ n,q−1 since ρc are known to be bijective, one can ﬁnd vd L(2),loc(Md ,E) such that ¯ ∂vd = u holds on Md . By Theorem 2.39, one can then deﬁne a sequence v˜μ (μ = 1, 2,...)inductively as follows:

v˜1 = v,

v˜2 = vc+1 − w1,

n,q−1 1 ∈ + ∩ ¯ − + − where w1 L(2),loc(Mc 1,E) Ker ∂ and w1 (vc 1 v1) Mc < 2 , and

vμ˜+1 = vc+μ − wμ,

n,q−1 1 ∈ + ∩ ¯ − + −˜ where wμ L(2),loc(Mc μ,E) Ker ∂ and wμ (vc μ vμ) Mc+μ−1 < 2μ . 2 · + ¯ 1 Here the L norm Md on Md is measured with respect to ω i∂∂ log d−φ and ¯ h · (d − φ). Then ∂(lim v˜μ) = u. n,q → n,q ∈ n,q ∩ ¯ Surjectivity of H (M, E) H (Mc,E):Letw L(2),loc(Mc,E) Ker ∂. ∈ n,q + ∩ ¯ By Theorem 2.39, one can ﬁnd wμ L(2),loc(Mc μ, E) Ker ∂ similarly to the n,q ∩ ¯ | = above, in such a way that lim wμ exists in L(2),loc(M, E) Ker ∂ and lim (wμ Mc ) w. q O =∼ 0,q K ⊗ K∗ ⊗ =∼ n,q K∗ ⊗ Since H (M, (E)) H (M, M M E) H (M, M E), one has: Corollary 2.11 For any strongly pseudoconvex manifold M and for any holomorphic vector bundle E over M, dim H q (M, O(E)) < ∞ for any q>0. In the situation of Theorem 2.41, it is clear that the above proof shows more precisely that there exists a Hermitian metric ω on M such that H n,q (M, E) =∼ n,q + ¯ 1 · − H(2) (Md ,E,ω i∂∂ log d−φ ,h (d φ)) for all q>0. Similarly, it can be n,q =∼ n,q shown also that H (M, E) H(2) (Md ,E,ω,h)for all q>0(cf.[Oh-7], where the smoothness assumption on ∂Md is superﬂuous). Therefore, one can infer from Corollary 2.11 the following vanishing theorem for ordinary cohomology groups. Theorem 2.42 (cf. [Gra-Ri-1] and [T-1]) Let (M, φ) and (E, h) be as in Theo- rem 2.41. Assume moreover that M = Mc, dω = 0 and Θh ≥ 0 on M. Then H n,q (M, E) = 0 for q>0. 84 2 Analyzing the L2 ∂¯-Cohomology

Let us also recall a well-known theorem of Grauert which was originally derived from Corollary 2.11. In view of the importance of the result in several complex variables, we shall give a proof as an application of Theorem 2.42. Theorem 2.43 (cf. [Gra-3]) Every strongly pseudoconvex manifold is holomorphically convex.

Proof Let M and Mc be as in Theorem 2.41, and let Γ ={xμ}μ=1,2,... be any sequence of points in M \ Mc which does not have any accumulation point. Let ˜ π : M → M be the blow-up of M along Γ and let IΓ be the ideal sheaf of the divisor π −1(Γ ). Then M˜ is pseudoconvex and it is easy to see that the line bundle ∗ (K ⊗ I )| −1 is positive. Hence M˜ Γ π (Γ )

∗ − ∗ − ∗ − ∗ H n,1(M,˜ K ⊗[π 1(Γ )] ) =∼ H n,1(π 1(M ), K ⊗[π 1(Γ )] ) M˜ c M˜ by Theorem 2.41. Since

∗ − ∗ − ∗ H n,1(M,˜ K ⊗[π 1(Γ )] ) =∼ H 0,1(M,˜ [π 1(Γ )] ) =∼ H 1(M,˜ I ), M˜ Γ

Γ and Mc ∩ Γ = ∅, it follows that the natural restriction map O(M) → C is surjective. This implies the assertion. We recall also that Corollary 2.11 was first proved in [Gra-3] by a sheaf theoretic method to derive Theorem 2.43 and later generalized to the following finiteness theorem which has already been mentioned in Chap. 1 (cf. Theorem 1.30). Theorem 2.44 (Andreotti–Grauert [A-G]) Let X be a q-convex space and let F be a coherent analytic sheaf over X. Then H p(X, F ) is finite dimensional for all p ≥ q. Although the above results obtained by the L2 method do not imply Theo- rem 2.44 in the full generality, it is by such L2 “representation” results that analytic methods work effectively in the study of cohomological invariants on complex manifolds and spaces. For instance, let us mention an application of Theorem 2.40 which was observed recently. Theorem 2.45 (cf. [Oh-34]) Let M be a compact complex manifold and let D be a smooth divisor of M such that [D] is semipositive. Then, for any holomorphic vector bundle E → M which is Nakano positive on a neighborhood of D, one can find 0 μ μ0 ∈ N such that the restriction homomorphism H (M, O(KM ⊗ E ⊗[D] )) → 0 μ H (D, O(KM ⊗ E ⊗[D] )) is surjective for any μ ≥ μ0.

To ﬁnd an effective bound for μ0 seems to be an interesting question. To the author’s knowledge, no purely algebraic proof of Theorem 2.45 is known for the projective algebraic case. It might be worthwhile to note that certain L2 cohomology is isomorphic to the ordinary cohomology on pseudoconvex manifolds. 2.3 Finiteness Theorems 85

Theorem 2.46 Let (M, φ) be a pseudoconvex manifold of dimension n and let (E, h) be a holomorphic vector bundle over M which is Nakano positive on M \Md for some d. Suppose that there exists c>0 such that i(Θh − c IdE ⊗ Θdet h) ≥ 0 on M \ Md . Then there exists a Hermitian metric ω on M such that

n,q =∼ n,q + ¯ 2 · −φ2 H (M, E) H(2) (M,E,ω i∂∂φ ,h e ) 1 =∼ H n,q (M ,E,ω+ i∂∂¯ log ,h· (d − φ)) =∼ H n,q (M ,E) (2) d d − φ d for all q>0.

Proof Let λμ (μ = 1, 2,...) be a sequence of convex increasing functions as in the proof of Proposition 2.15. We are allowed to choose λμ so that λμ(t) = + 1 + ≥−1 ˜ ∞ μ(t μ ) log μ holds for t μ . Then, for each μ let λμ be a C convex increasing function such that ≥ 2 ≤ ˜ = λμ(t) if λμ(t) t or t 0, λμ(t) 2 2 t if λμ(t) + 1

Then, one can ﬁnd a Hermitian metric ω on M such that

− 2 −˜ − n,q + ¯ 2 φ =∼ n,q + ¯ ˜ − λμ(φ d) H(2) (M,E,ω i∂∂φ ,he ) H(2) (M,E,ω i∂∂λμ(φ d),he ) for any μ. In fact, one may take iΘdet h as ω on M \ Md . Since

−˜ − n,q + ¯ ˜ − λμ(φ d) H(2) (M,E,ω i∂∂λμ(φ d),he ) 1 =∼ H n,q M,E,ω + i∂∂¯ log ,h(d − φ) (2) d − φ for sufﬁciently large μ, we are done. A stronger result holds when E is a line bundle. Namely: Theorem 2.47 Let (M, φ) be a pseudoconvex manifold of dimension n and let (B, a) be a holomorphic Hermitian line bundle over M which is positive on M \ Md for some d>0. Then there exists a Hermitian metric ω on M p,q =∼ p,q + ¯ 2 −φ2 =∼ p,q + such that H (M, B) H(2) (M,B,ω i∂∂φ ,ae ) H(2) (Md ,B,ω ¯ 1 − + i∂∂ log d−φ ,a(d φ)) if p q>n. Sketch of proof By assumption, there exists a Hermitian metric ω on M such that ω = iΘa holds on M \ Md . The rest is similar to the above. (For the detail, see [Oh-4].) If (M, φ) is strongly pseudoconvex, then Theorem 2.47 can be strengthened as follows. 86 2 Analyzing the L2 ∂¯-Cohomology

Theorem 2.48 Let (M, φ) be a strongly pseudoconvex manifold of dimension n and let (E, h) be a Hermitian holomorphic vector bundle over M. Then there p,q =∼ p,q + exists a Hermitian metric ω on M such that H (M, E) H(2) (Md ,E,ω ¯ 1 − + i∂∂ log d−φ ,h(d φ)) if p q>n.Hered is any number such that φ is strictly plurisubharmonic on M \ Md .

Taking the advantage of strict plurisubharmonicity of φ on ∂Md ,asimi- p,q =∼ p,q + lar argument can be applied to show that H (M, E) H(2) (Md ,E,ω ¯ 1 & + i∂∂ log ((d−φ)/R),h)(R 1) holds for p q>n(an exercise!). Combining the isomorphism between the L2 cohomology and ordinary cohomology with a classical theory of L2 harmonic forms (cf. [W-2]or[W]), we obtain the following.

Theorem 2.49 Let (M,φ,ω)be a pseudoconvex Kähler manifold of dimension n. If φ is strictly plurisubharmonic on M \ M , then H r (M, C) =∼ H p,q(M) c r=p+q p,q =∼ q,p + k : holds for r>nand H (M) H (M) for p q>n. Moreover, the map ω H p−k,q−k(M) → H p,q(M) deﬁned by u → k ω ∧ u induces an isomorphism p−k,q−k p,q + ≥ + = + − between H0 (M) and H (M) for p q n 1 and k p q n. Corollary 2.12 Let X be a complex space of dimension n which is nonsingular possibly except at x ∈ X, and let X˜ be a complex manifold which admits a Kähler : ˜ → | metric and a proper surjective holomorphic map π X X such that π X˜ \π−1(x) is a biholomorphic map. Then there exists a neighborhood U x such that the r-th Betti number of π −1(U) is even for r>n. Remark 2.9 In the assumption of Corollary 2.12, that X˜ admits a Kähler metric can be omitted, because there exist a Kähler manifold Xˆ and a proper bimeromorphic map πˆ ; Xˆ → X˜ obtained by a succession of blow-ups along nonsingular centers in virtue of Hironaka’s fundamental theory of desingularization (cf. [Hn]). Pursuing an extension of the Hodge theory of this type on strongly pseudoconvex domains, the following was observed in [Oh-6]. Proposition 2.16 ([Oh-6, Corollary 7 and Note added in proof]) In the situation of Corollary 2.12, there exists an arbitrarily small neighborhood V of π −1(x) such that the restriction homomorphisms H r (V, C) → H r (∂V, C) are surjective for all r ≥ n − 1. For the proof, the reader is referred to [Oh-5, Oh-6] and [Oh-13]. (See also [Dm-5, Sai, Oh-T-2].) ∂V is called the link of the pair (X, x) if ∂V = ρ−1(1) for some C∞ function ρ : U →[0, ∞) with V U and (dρ)−1(0) ∩ U = π −1(x). Corollary 2.13 (S1)2n−1 is not homeomorphic to any link if n>1. Remark 2.10 In [Ka], it was asked that those 3-manifolds be determined which can be realized as links of isolated hypersurface singularities in C3. According to [Ka], 2.3 Finiteness Theorems 87

Sullivan has shown that (S1)3 is not so. A celebrated theorem of Mumford [Mm] says that S3 ∂V if X is normal and singular at x. + ¯ 1 + ¯ 1 As well as the complete metric ω i∂∂ log ((d−φ)/R), the metric ω i∂∂ log d−φ is also naturally attached to (X, x) (cf. Chap. 4). With respect to this metric, by extending the Donnelly–Fefferman vanishing theorem (Theorem 2.18), one has the following in a way similar to that. Theorem 2.50 Let (M, φ) be a strongly pseudoconvex manifold of dimension n and let (E, h) be a Hermitian holomorphic vector bundle over M. Then there p,q =∼ p,q + exists a Hermitian metric ω on M such that H (M, E) H(2) (Md ,E,ω ¯ 1 + p,q =∼ p,q + ¯ 1 i∂∂ log d−φ ,h)if p q>nand H0 (M, E) H(2) (Md ,E,ω i∂∂ log d−φ ,h) if p + qc.In [Hö-1] the following was established by the L2 method. In fact, it is the prototype of the above arguments. Theorem 2.53 (Hörmander [Hö-1, Theorem 3.4.9]) Let (E, h) be a Hermitian holomorphic vector bundle over a complex manifold M of dimension n.IfM is q- convex with respect to an exhaustion function φ, then dim H q (M, O(E)) < ∞. 0,q ∼ Moreover, if φ is q-convex on M \ Md and ∂Md is smooth, H (M, E) = 0,q H(2) (Md ,E,ω,h)holds for any Hermitian metric ω on M. Furthermore, the image 0,q−1 → 0,q−1 of the restriction homomorphism H (M, E) H(2) (Md ,E,ω,h)is dense. If M is q-concave with respect to an exhaustion function ψ : M → (c, 0], then 88 2 Analyzing the L2 ∂¯-Cohomology dim H n−q−1(M, O(E)) < ∞ and there exists a Hermitian metric ω on M such 0,n−q−1 =∼ 0,n−q−1 d \ d that H (M, E) H(2) (M ,E,ω,h),ifψ is q-convex on M M and ∂Md is smooth. Here Md ={x; ψ(x) > d}. Corollary 2.14 Let M be a q-complete manifold. Then, for any holomorphic vector bundle E → M,

H 0,p(M, E) = 0 for p ≥ q.

Combining the techniques originating in [Hö-1] and [A-V-2], which have put the sheaf theoretic development of Oka’s solution of the Levi problem by Grauert [Gra-3] and Andreotti and Grauert [A-G] into the framework of the L2 theory, the following variant of Theorem 2.53 was proved in [Oh-7]. Theorem 2.54 Let (E, h), M, φ and ψ be as in Theorem 2.53.Ifφ is q-convex on M \ Md , then

0,q − α 0,q =∼ − 2 · d−φ & H (M, E) H(2) (Md ,E,ω/(d φ) ,h e ) for α 1. for any Hermitian metric ω on M.Ifψ is q-convex on M \ Md , then

∧ ¯ − − ∼ 0,n−q−1 ∂ψ ∂ψ − α H 0,n q 1(M, E) = H Md ,E,ω+ i ,h· e d−ψ for α & 1. (2) (d − ψ)2

Since the method of proof is more or less the same as in Theorems 2.41 and 2.46, we shall not repeat it here. The point is that one can choose a Hermitian metric on M in such a way that (i∂∂φΛu,¯ u) ≥u2 for the (n, q) forms u compactly supported ¯ 2 in M \ Md or (−iΛ∂∂ψu, u) ≥u for the (0,n− q − 1) forms u compactly supported in M \ Md . Remark 2.12 In [A-G], the above-mentioned L2 representation theorem for the ∂¯- cohomology is stated as a unique continuation theorem for the sheaf cohomology from sublevel sets (or superlevel sets) of q-convex functions to the whole space. Substantially, the point of the argument is also a Runge-type approximation. For application of the L2 approximation technique in (genuine) function theory, the reader is referred to [H-W] and [Sak], for instance.

2.4 Notes on Metrics and Pseudoconvexity

By the methods of L2 estimates, analytic invariants on complex manifolds have been analyzed above, particularly under the existence of positive line bundles, complete Kähler metrics and plurisubharmonic exhaustion functions. As examples of the 2.4 Notes on Metrics and Pseudoconvexity 89 situations to which they are applicable, a collection of questions and results in complex geometry related to these basic notions will be reviewed below, mostly without proofs.

2.4.1 Pseudoconvex Manifolds with Positive Line Bundles

We shall review a few results in which pseudoconvex manifolds arise naturally accompanied with positive line bundles. First, suppose that we are given a closed analytic subset S of a complex space X and a proper surjective holomorphic map π from S to a complex space T . Then a general question is whether or not there exist a complex space Y containing T as a closed analytic subset and a proper surjective holomorphic map from X to Y ,sayπ˜ such that π˜ |X \˜π −1(T ) is a biholomorphic map onto Y \T . If it is the case, we shall say that S is contractible to T in X. Note that the problem is local along T . Namely, if every point p ∈ T has a neighborhood U such that π −1(U) is contractible to U in some neighborhood of it in X, then S is contractible to T in X. When T is a finite set of points, S is the maximal compact analytic subset in its neighborhood in the sense that S is contractible, compact and nowhere discrete. There is a necessary and sufficient condition for the contractibility of compact analytic sets given by Grauert: Theorem 2.55 (cf. [Gra-5]) A compact analytic subset S of X is contractible to a point in X if and only if S admits a strongly pseudoconvex neighborhood system. Corollary 2.15 Let M be a complex manifold of dimension 2 and let C ⊂ M be a connected analytic subset of dimension one with irreducible components Cj (1 ≤ j ≤ m). Then C is contractible to a point in M if and only if the matrix [ ]| (deg ( Cj Ck ))1≤j,k≤m is negative definite. If X is a complex manifold and S is a compact submanifold, a sufficient (but not necessary) condition for S to have a strongly pseudoconvex neighborhood system is that the normal bundle of S is negative in the sense that its zero section has a strongly pseudoconvex neighborhood system (cf. [Gra-5, Satz 8]). This contracitibility criterion is essentially a corollary of Theorem 2.43. Hence a natural question arises whether or not the same is true for the case where S is not compact. When the codimension of S is one, the normal bundle of S is [S]|S, so that its negativity is equivalent to the positivity of [S]∗ on a neighborhood U of S which can be chosen to be pseudoconvex by the negativity of [S]|S, by localizing the situation if necessary. Proposition 2.17 Let S ⊂ X be as above. Then, for every point p ∈ T , there exists a pseudoconvex neighborhood U of π −1(p) such that [S]∗|U is positive. ∗ In this way, a pseudoconvex manifold U and a positive line bundle [S] |U arise. Let us mention three results in this situation. 90 2 Analyzing the L2 ∂¯-Cohomology

Theorem 2.56 (cf. [N-2] and [F-N]) Suppose that π : S → T is a complex m analytic ﬁber bundle with ﬁber CP , and that [S]|π−1(p) are of degree −1. Then S is contractible to T in X.(Y is actually a manifold.) This amounts to a characterization of the blowing-up of Y centered along T .Its proof is done by extending holomorphic functions on S to a neighborhood. In fact, it was for this purpose of contraction that a vanishing theorem like Theorem 2.19 was established. This procedure was generalized as follows.

Theorem 2.57 (cf. [Fj-1]) Suppose that S is holomorphically convex, [S]|S is negative, and that H 1(S, O(([S]∗)⊗μ)) = 0 for all μ>0. Then S is contractible to T in X. In contrast to Theorem 2.43, the condition on the vanishing of the first cohomology groups cannot be omitted. This was clarified by the following. Theorem 2.58 (cf. [Fj-1], Proposition 3) Let B be a positive line bundle over a compact complex manifold F such that H 1(F, O(B)) = 0. Then there exists a complex manifold X, a closed submanifold S of X, and a complex analytic fiber bundle π : S → T with fiber F such that S is not contactible to T in X and [ ]| ∼ ∗ S π−1(p) = B . Proof In the above situation, there exists an affine line bundle σ : Σ → B∗ such | =∼ × C = 0 O = that Σ F0 F (F0 the zero section) but H (U, (Σ)) 0 for any −1 −1 neighborhood U ⊃ F0. Then (S = σ (F0), X = σ (U), T = C) is such an example. Another example of a pseudoconvex manifold with positive line bundles is the quotient of Cn by the action of a discrete subgroup say Γ , satisfying a condition of Riemann type. First let us recall a classical theorem of Lefschetz. Theorem 2.59 (cf. [B-L, Kp]) Assume that X = Cn/Γ is compact and there exists a positive line bundle L → X. Then the following hold: 0 n (1) dim H (X, O(L)) = c1(L) /n!. (2) L⊗2 is generated by global sections. (3) L⊗3 is very ample. Since the complex semitori X = Cn/Γ are complex Lie groups, they are pseudoconvex (cf. [Mr]). Concerning the positive line bundles on X, they exist if X is compact and the following conditions are satisfied by Γ : there exists a Hermitian form H on Cn such that: (i) H is positive definite and ∼ (ii) the imaginary part A of H takes integral values on Γ × Γ(= H2(X, Z)). Then, the Appell–Humbert theorem says that, for each semicharacter χ,

χ(γ + γ ) = χ(γ )χ(γ )eπiA(γ,γ ),γ,γ ∈ Γ by deﬁnition, 2.4 Notes on Metrics and Pseudoconvexity 91 one can associate the so-called factor of automorphy

πH(z,γ)+ π H(γ,γ) j(γ,z) = χ(γ )e 2 (2.42) and a line bundle L = L(H, χ) of the form Cn × C/Γ , where the action of Γ is defined as γ : (z, t) ∈ Cn × C → (z + γ,j(γ,z)t) ∈ Cn × C. L is positive because so is H . (For the Appell–Humbert’s theorem, see also [B-L]or[Kp].) In [Ty-2], Theorem 2.59 was extended to the following. Theorem 2.60 Let L be a positive line bundle over X = Cn/Γ .IfX is noncompact, the following hold: (1) dim H 0(X, O(L)) =∞. (2) L⊗2 is generated by global sections. (3) L⊗3 is very ample. As in the compact case, a semitorus X admits a positive line bundle if its toroidal reduction Xˆ (X = Ca ×(C∗)b ×Xˆ and H 0(X,ˆ O) = C) satisfies a condition similar as above (cf. [A-Gh], [C-C, §2]). It is known that the ∂¯-cohomology group of the toroidal groups Xˆ reflect a certain Diophantine property of Γ (cf. [Kz-3] and [Vo]). For a general pseudoconvex manifold M of dimension n ≥ 2, positive line bundles are not necessarily ample (cf. [Oh-0]). Nevertheless, it was shown by Takayama ⊗ m 1 + [Ty-1] that KM L is ample if L is a positive line bundle and m> 2 n(n 1).Inthe proof of Takayama’s theorem, an extension theorem for L2 holomorphic functions plays an important role (cf. Sect. 3.1.3).

2.4.2 Geometry of the Boundaries of Complete Kähler Domains

In contrast to the vanishing theorems and finiteness theorems on pseudoconvex manifolds, the L2 vanishing theorems on complete Kähler manifolds were in part motivated by the following theorem of Grauert. Theorem 2.61 (cf. [Gra-2]) Let D be a domain in Cn with real analytic smooth boundary. Then the following are equivalent: (1) D admits a complete Kähler metric. (2) D is pseudoconvex. That (1) follows from (2) is contained in Proposition 2.12.Asfor(1)⇒ (2), Grauert showed it by approximating D locally by Reinhardt domains. Real analyticity of ∂D is needed for this argument. In [Oh-2] it was shown under the assumption (2) that, given any point p ∈ Cn \ D and a complex line intersecting with D and passing through p, there exists a holomorphic function f on ∩ D 92 2 Analyzing the L2 ∂¯-Cohomology which cannot be continued analytically to p, but extends holomorphically to D by establishing Theorem 2.14 in a special case. To apply this extension argument, C1- smoothness of ∂D suffices. Diederich and Pflug [D-P] proceeded further by showing that the purely topo- ◦ logical condition D = D (the interior of the closure of D) suffices. For that, they applied Skoda’s L2 division theorem which will be discussed in Chap. 3. In [Gra-2], it was also shown that the complement of a closed analytic subset of a Stein manifold admits a complete Kähler metric. Indeed, if A is a closed analytic subset of a Stein manifold D, then one can find finitely many holomorphic functions ∞ f1,...,fm on D such that A ={z ∈ D; fj (z) = 0 for all j}. Then, for any C function λ on D \ A for which there exists a neighborhood U of A such that λ(z) = 2 − log (− log ( |fj | )) holds on U \ A, there exists a strictly plurisubharmonic exhaustion function φ on D such that i∂∂(λ¯ + φ) is a complete Kähler metric on D \ A. In [Oh-3], the following was proved. Theorem 2.62 Let D be a pseudoconvex domain in Cn and let A ⊂ D be a closed C1-smooth real submanifold of (real) codimension 2. Then A is a complex submanifold if and only if D \ A admits a complete Kähler metric. Proof The “only if” part is already over. Conversely, suppose that D \ A admits a complete Kähler metric. To show that A is complex, let p ∈ A be any point and let be a complex line intersecting with A transversally at p. Take a Stein neighborhood W p such that (W \A)∩ is biholomorphic to the punctured disc {ζ ; 0 < |ζ | < 1} and W \ A is homotopically equivalent to (W \ A) ∩ .Letα : W\ A → W \ A be the double covering. Then, W\ A also admits a complete Kähler metric. By 1 the C -smoothness assumption on A√, one can apply Theorem 2.14 to extend the single-valued holomorphic function ζ on α−1((W \ A) ∩ ) to W\ A with an L2 growth condition. As a result, one has a holomorphic function on W\ A satisfying an irreducible quadratic equation over O(W) whose discriminant has zeros or poles along A. Hence A must be complex. Theorem 2.62 complements the following well-known result of Hartogs. Theorem 2.63 (cf. [H’09]) Let f be a continuous complex-valued function on a domain D ⊂ Cn. Then f is holomorphic if and only if the complement of its graph is a domain of holomorphy. Proof The “only if” part is trivial. If the graph of f has a Stein complement, so does the graph of ef , so that by Oka’s lemma (cf. (5) in Sect. 1.5) ± log |ef | is plurisubharmonic. Hence Re f is pluriharmonic. Similarly Im f, Re f 2 and Im f 2 are pluriharmonic. Therefore f must be holomorphic. We note that an alternate proof of Theorem 2.63 is to apply Theorem 2.14 to extend a holomorphic function on {(z0,ζ)∈ D × C; ζ = f(z0)} for z0 ∈ D with a pole at ζ = f(z0) to a holomorphic function on the complement of the graph of f as a meromorphic function on D × C. 2.4 Notes on Metrics and Pseudoconvexity 93

Anyway, from the viewpoint of Oka’s solution of the Levi problem, Hartogs’s theorem is about the +∞-singular set of plurisubharmonic functions. Conversely, Theorem 2.62 is closely related to the property of the preimages of −∞. Deﬁnition 2.5 A subset F of a complex manifold M is said to be pluripolar if there exists a plurisubharmonic function φ on M such that φ ≡−∞and F ⊂ {z; φ(z) =−∞}. Proposition 2.18 Closed nowhere-dense analytic sets in Stein manifolds are pluripolar. Proposition 2.19 Let D be a domain in Cn and let φ : D →[−∞, ∞) be a continuous plurisubharmonic function. Then there exists a plurisubharmonic function Φ on D such that Φ is C∞ on Φ−1(R) and φ−1(−∞) = Φ−1(−∞). Proof By a theorem of Richberg, every continuous plurisubharmonic function (with ﬁnite values) can be uniformly approximated by C∞ ones (cf. [R]; see also [Oh-21]). Corollary 2.16 For any pseudoconvex domain D ⊂ Cn and a continuous plurisubharmonic function φ on D with values in [−∞, ∞) but not in {−∞}, D \φ−1(−∞) admits a complete Kähler metric. Proof Take any C∞ function λ on D \ φ−1(−∞) satisfying λ(z) =−log (−φ(z)) on U \ φ−1(−∞). Then i∂∂(Φ¯ + Ψ) becomes a complete Kähler metric on D \ φ−1(−∞) for the above Φ and for some strictly plurisubharmonic exhaustion function Ψ on D. Therefore, although under a continuity assumption, Theorem 2.62 gives some information on pluripolar sets. In this direction, Shcherbina [Shc] has shown a remarkable result: Theorem 2.64 Let f be a continuous complex-valued function on a domain D ⊂ Cn. Then f is holomorphic if and only if its graph is pluripolar. The proof is based on a property of polynomially convex hulls. (See also [St].) Coming back to the boundary of complete Kähler domains, a natural question in view of Theorem 2.61 and subsequent remarks is whether or not Theorem 2.62 can be generalized for higher codimensional submanifolds. The answer is yes and no in the following sense. Theorem 2.65 (cf. [D-F-4, Theorem 1]) Let A be a closed real analytic subset of a pseudoconvex domain D ⊂ Cn. Suppose that codim A ≥ 3. Then A is complex analytic if and only if D \ A admits a complete Kähler metric. Theorem 2.66 (cf. [D-F-4, Theorem 2]) For any integer k ≥ 3, there exists a closed C∞ submanifold A of {z ∈ Cn;z < 1} such that there exists a complete Kähler metric on the complement of A but A is not complex. It was also shown in [D-F-3] that Shcherbina’s theorem cannot be generalized to vector-valued functions. Nevertheless, it was shown in [D-F-6] that the submanifold 94 2 Analyzing the L2 ∂¯-Cohomology

A in Theorem 2.66 still has some distinguished geometric structure. In this series of works, Diederich and Fornaess constructed a smooth real curve in C2 which is not pluripolar. As a development from Theorem 2.61, complete Kähler manifolds with curvature conditions have been studied in a wider scope. Some of the results of this type will be reviewed in the next subsection.

2.4.3 Curvature and Pseudoconvexity

If one wants to explore intrinsic properties of noncompact complete Kähler manifolds, it is quite unnatural to presuppose the existence of the boundaries. Namely, we do not see the boundaries of complete manifolds at first. In some cases the boundary appears as a result of compactification (cf. [Sat] and [N-Oh]). Accordingly, in this context concerning the relationship between pseudoconvexity and complete Kähler metric, questions naturally involve the curvature of the metric. It is expected that difference of metric structures implies that of complex structures. A prototype of such a question was solved by Huber [Hu] for Riemann surfaces: Theorem 2.67 Let (M, ω) be a noncompact complete Kähler manifold of dimension one whose Gaussian curvature is everywhere positive. Then M is biholomorphically equivalent to C. Since any simply connected open (=noncompact) Riemann surface is either C or D(={z ∈ C;|z| < 1}), it is natural to ask for a curvature characterization of the disc D. An answer was given by Milnor in the case where (M, ω) is rotationally symmetric, i.e. when there exists a point p ∈ M such that with respect to the geodesic length r from p and the associated geodesic polar coordinates r, θ,the Riemann metric associated to ω is of the form dr2 + g(r)dθ2. In this case, the Gaussian curvature K is given by K =−(d2g/dr2)/g. Theorem 2.68 (cf. [Ml]) Let (M, ω) be a simply connected complete Kähler manifold of dimension one. Assume that M is rotationally symmetric. Then the following hold: − (1) M =∼ C if K ≥ 1 holds for large r. r2 log r + (2) M =∼ D if K ≤− 1 for large r and g(r) is unbounded. r2 log r As for the extension of these results to higher-dimensional cases, Greene and Wu first established the following. Theorem 2.69 (cf. [G-W-2, Theorem 3]) Let (M, ω) be a complete noncompact Kähler manifold whose sectional curvature is positive outside a compact set. Then M is strongly pseudoconvex. 2.4 Notes on Metrics and Pseudoconvexity 95

Sketch of proof Taking the minimal majorant of the Buseman functions for all the rays emitted from some point, one has a convex exhaustion function on M.For Buseman functions and rays, see [Wu]. In [G-W-3], Greene and Wu raised several questions related to the extension of these results. One of them was eventually solved by themselves. The result is very striking: Theorem 2.70 (cf. [G-W-4], Theorem 4) Let (M, ω) be a simply connected noncompact complete Kähler manifold of dimension n ≥ 2. For a fixed p ∈ M, define k :[0, ∞) → R by k(s) = sup {|sectional curvature at q|; q ∈ M,dist(p, q) = s}. n Then M is isometrically equivalent to(C ,i dzj ∧ dz¯j ) if the sectional curvature ∞ ∞ of M is everywhere nonpositive and 0 sk(s)ds < . As for the nonnegatively curved case, they conjectured that a complete Kähler manifold with nonnegative sectional curvature and with positive Ricci curvature is holomorphically convex. Takayama settled it affirmatively in [Ty-3] based on the following. Theorem 2.71 (cf. [Ty-3, Main Theorem 1.1]) Pseudoconvex manifolds with negative canonical bundle are holomorphically convex. The proof of this beautiful result is actually beyond the scope of the theory presented in this section, and requires a more refined variant of Oka–Cartan theory including construction of specific singular fiber metrics, which will be discussed later in Chap. 3. As for the Ricci nonpositive case, the following was observed by Mok and Yau [M-Y] in the study of Einstein–Kähler metrics on bounded domains. Theorem 2.72 A bounded domain in Cn is pseudoconvex if it admits a complete Hermitian metric satisfying −c ≤ Ricci curvature ≤ 0 . The proof is based on Yau’s version of Schwarz’s lemma (cf. [Yau-1]), which is available without the Kählerianity assumption.

2.4.4 Miscellanea on Locally Pseudoconvex Domains

Here we shall collect some of the remarkable facts on locally pseudoconvex domains in or over complex manifolds. Since the proof of the fundamental fact that every locally pseudoconvex Riemann domain over Cn is an increasing union of strongly pseudoconvex domains depends essentially on the use of the Euclidean metric, it is natural that one needs more differential geometry to analyze locally pseudoconvex domains over complex manifolds. Let π : D → CPn be a locally pseudoconvex noncompact Riemann domain. For any z ∈ CPn,letB(z,r) be the geodesic ball of radius r centered at z with respect n to the Fubini–Study metric of CP ,sayωFS. For any x ∈ D we put 96 2 Analyzing the L2 ∂¯-Cohomology

δ(z) = sup{r; π maps a neighborhood of x bijectively to B(π(x),r)}.

A. Takeuchi extended Oka’s lemma (cf. Theorem 1.12)asfollows. Theorem 2.73 (cf. [Tk-1]) log δ(z)−1 is plurisubharmonic and i∂∂¯ log δ(z)−1 ≥ 1 3 ωFS holds on D. Corollary 2.17 (See also [FR]) Every noncompact locally pseudoconvex domain over CPn is a Stein manifold. It may be worthwhile to note that the solution of this Levi problem entails the following. Proposition 2.20 Let X be a connected compact analytic set of dimension ≥ 1 in CPn. Then every meromorphic function defined on a neighborhood of X can be extended to CPn as a rational function. Proof For any meromorphic function f defined on a domain in CPn, the maximal Riemann domain to which f is continued meromorphically, i.e. the envelope of meromorphy of f is locally pseudoconvex (see [Siu-2], for instance), so that over CPn (n ≥ 2) they are either CPn or Stein. Since it contains X it must be CPn. Theorem 2.73 was generalized to Riemann domains over Kähler manifolds (cf. [Tk-2, Suz, E]). Definition 2.6 The holomorphic bisectional curvature of a Hermitian manifold (M, ω) is a bihermitian form Θμ h ξ αηνξ β ηκ ((ξ α), (ην) ∈ Cn =∼ T 1,0 ,x∈ M) αβν μκ M,x μ

μ 1,0 associated to the curvature form (Θ ) of the associated fiber metric (h ¯ ) of T . αβν μν M Theorem 2.74 (cf. [E] and [Suz]) Let D be a locally pseudoconvex Riemann domain over a complete Kähler manifold M of positive holomorphic bisectional curvature, and let δ be defined for D → M similarly to Theorem 2.73. Then i∂∂¯ log δ−1 is strictly positive on D. Proof (Cf. [Oh-18]) Because of the local nature of the problem and by virtue of Oka’s lemma, we are allowed to assume that D is a domain with smooth boundary which is everywhere strongly pseudoconvex. Hence, it suffices to consider the situation that ∂D is a complex submanifold of codimension one, the metric on M is real analytic, and δ(z) is realized by a geodesic say γ :[0, 1]→M joining z = γ(0) and a point γ(1) ∈ ∂D, in such a way that the length of the geodesic from γ(0) to γ(s) is s. In this setting, on a neighborhood of γ([0, 1]) we take a local = = = [ ] coordinate t (t1,...,tn) (t ,tn) such that s Retn on γ( 0, 1 ), and look at + n−1 = the Taylor coefficients of the distance from t to tn j=1 cj tj 1, where cj are so chosen that γ([0, 1]) is orthogonal to ∂D at t = (0,...,0, 1). We may assume in advance that cj are all 0. Then by expressing the Kähler metric, say ω,as 2.4 Notes on Metrics and Pseudoconvexity 97

i n ω = g ¯ dt ∧ dt¯ (2.43) 2 j,k j k j,k=1 we have

n−1 n−1 ¯ 2 gn,n¯ (t) = 1 − λjktj tk − 4Re λjntj (Im tn) − 2λnn(Im tn) + (t). j,k=1 j=1 (2.44)

¯ ¯ Here λjk(= λjk(t , t , Re tn)) and (t) is of order at least 3 in (t , t ).From(2.38) one can directly read off that

n ∂2 1 1 (log )ξ ξ¯ ≥ κ|ξ|2 (2.45) ∂t ∂t¯ δ(t) j j 6 j,k j k

n holds for any ξ = (ξ1,...,ξn) ∈ C , where n ¯ ( j,k= λjk(t)ξj ξk) κ = inf 1 . (2.46) t∈γ([0,1]),ξ =0 ξ2

By the curvature condition on ω (for the complex 2-planes spanned by ∂/∂tn and ∂/∂tj ), κ>0, from which the desired conclusion is obtained. Corollary 2.18 Every noncompact pseudoconvex Riemann domain over a pseudoconvex Kähler manifold of positive bisectional curvature is Stein. Actually this does not generalize Takeuchi’s theorem so much, because it turned out that compact Kähler manifolds with positive holomorphic bisectional curvature is biholomorphically equivalent to CPn (cf. [M-1, S-Y]). Nevertheless, as one can see from the above proof, Theorem 2.73 can be immediately extended to the following. Proposition 2.21 Every locally pseudoconvex domain in a compact Kähler manifold of semipositive holomorphic bisectional curvature admits a continuous plurisubharmonic exhaustion function. Therefore an extension of Corollary 2.17 in this direction is naturally expected. Let us mention two typical results: Theorem 2.75 (cf. Hirschowitz [Hr]) Let X be a compact complex manifold whose tangent bundle is generated by global sections. Then every locally pseudoconvex domain in X admits a continuous plurisubharmonic exhaustion function. Theorem 2.76 (cf. Ueda [U-1]) Every noncompact locally pseudoconvex Riemann domain over a complex Grassmannian manifold is Stein. 98 2 Analyzing the L2 ∂¯-Cohomology

Although Hirschowitz proved more than Theorem 2.75, it is not known whether or not it can be generalized for an arbitary (infinitely sheeted) locally pseudoconvex Riemann domain. A related question of Shafarevitch [Sha] asks whether or not the universal covering spaces of projective algebaric manifolds (or more generally those of compact Kähler manifolds) are holomorphically convex. In the proof of Theorem 2.76, Ueda reduces the question to Oka’s theorem for the domains over Cn by exploiting a result of Matsushima and Morimoto [M-M] which was observed in the study of a question asked by J.-P. Serre. His question was as follows. By generalizing locally pseudoconvex Riemann domains over complex manifolds or complex spaces, one may consider a complex manifold M paired with a holomorphic map to some complex manifold N,sayf : M → N such that (M,f,N)is locally pseudoconvex in the sense that one can find an open covering −1 Uj of N such that f (Uj ) are all pseudoconvex. Then it is natural to ask whether or not M is also pseudoconvex (under some reasonable conditions). Within this general setting, the most closely studied case is when N is a Stein manifold and M → N is a holomorphic (= complex analytic) fiber bundle with Stein fibers. J.-P. Serre asked if M is also Stein. Concerning Serre’s problem, several counterexamples (cf. [Sk-3, Dm-1, C-L]) and useful partial answers are known. Theorem 2.77 (cf. [Dm-1]) There exists a holomorphic C2 bundle over the unit disc D which is not Stein. Skoda [Sk-5] raised a conjecture that C2 bundles over D with polynomial transition functions are Stein. Rosay [R’07] immediately refuted it. He found a counterexample in such a way that it can be extended as a bundle over Cˆ = CP1. One of the notable affirmative results is due to N. Mok: Theorem 2.78 (cf. [Mk]) Holomorphic fiber bundles over Stein manifolds with one-dimensional Stein fibers are Stein. The reader might notice that, as a variant of Serre’s problem we may ask whether or not holomorphic fiber bundles over compact complex manifolds are pseudoconvex. However, there is an immediate counterexample: (the total space of) the line bundle O(1) over CPn is 1-concave! Nevertheless, under some natural geometric circumstances, pseudoconvexity still holds true. Theorem 2.79 (cf. [D-Oh-2]) Every holomorphic D-bundle over a compact Käh- ler manifold is pseudoconvex. In this assertion, the fibers can be replaced by any symmetric bounded domain. By such a generalization, a link can be made with the Shafarevitch conjecture (cf. [E-K-P-R]). On the other hand, particularly interesting objects are D-bundles over compact Riemann surfaces since we have: Theorem 2.80 A holomorphic D-bundle over a compact Riemann surface is Stein if and only if it has no holomorphic section. In [Oh-32], Theorem 2.80 is applied to prove that certain covering spaces over a family of compact Riemann surfaces are holmorphically convex. 2.5 Notes and Remarks 99

Over higher–dimensional compact Kähler manifolds the D-bundles are never Stein as we shall show in Chap. 5 by using the L2 method. Hence the situation of Theorem 2.80 is really exceptional. The Kähler condition cannot be dropped in Theorem 2.79 because of the following example. n Example 2.10 (Cf. [D-F-5]) Let Ωn = H × (C \{0})/Γn (n ≥ 2), where Γn is generated by

(ζ, z1,...,zn) −→ (2ζ,2z1,...,2zn).

Then Ωn is a D-bundle over a Hopf manifold. Since

∼ 2 n Ωn = {exp(−2π / log 2)<|ζ | < 1}×(C \{0}),

n Ωn is not a domain of holomorphy in C , so that it does not admit any plurisubharmonic exhaustion function. Nevertheless, some Hopf manifolds contain open dense Stein subsets, which will also be described in Chap. 5.

2.5 Notes and Remarks

A germ of the L2 method can be seen already in Riemann’s thesis, where the solvability of a ∂¯ equation is asserted (without a rigorous proof) on a simply connected domain cut out from a given Riemann surface. Weyl’s method in [Wy-1] by orthogonal projection was applied to prove the existence of nonconstant meromorphic functions on one-dimensional complex manifolds. Hodge [Ho] and Kodaira [K-1, K-2, K-3] generalized it as a theory of harmonic forms on higher- dimensional manifolds. In particular, Kodaira [K-3, K-4] characterized projective algebraic manifolds as compact complex manifolds with positive line bundles, after Dolbeault’s isomorphism theorem [Dol’53] and Bochner’s technique of comparing two kinds of Laplace operators. As we have seen in this chapter, Kodaira’s method has further developed into the machinery of L2 estimates for the ∂¯ operator, thanks to [A-V-1, A-V-2] and [Hö-1]. We note that [Hö-1] was also preceded by Morrey’s work [Mry]. As a continuation of Morrey’s approach, deep analytic studies have been done on the L2 canonical solutions for the ∂¯ equation on strongly pseudoconvex domains, The principal general question is whether or not the set of C∞ (p, q)- forms on D,sayCp,q(D),foraC∞-smooth domain D in a Hermitian manifold M is stable under the orthogonal projection. Kohn initiated the research in this direction by establishing the following. 100 2 Analyzing the L2 ∂¯-Cohomology

Theorem 2.81 (cf. [K’63, K’64]) Let (M, g) be a Hermitian manifold, let D ⊂ M be a (bounded) strongly pseudoconvex domain with C∞-smooth boundary, and let v be a ∂¯-closed C∞ (0,q)-form on D. Then there exist u ∈ C0,q−1(D) and w ∈ C0,q (D) ∩ Ker ∂¯ ∩ Ker ∂¯ ∗ satisfying v = w + ∂u¯ . Kohn’s theory has established a link between function theory on D and geometry on ∂D, which had been predicted by Poincaré [P’1907] vaguely and was made stronger by the following.

Theorem 2.82 (cf. [F, B-L’80]) Let Dj (j = 1, 2) be strongly pseudoconvex ∞ domains with C -smooth boundary and let ϕ : D1 → D2 be a biholomorphic ∞ map. Then ϕ can be extended to a C diffeomorphism between D1 and D2 The proof of Theorem 2.82 in [B-L’80] can be deduced from a regularity property of an operator associated to ∂¯.1 Such a reasoning is centered around an operator

= : p,q −→ p,q N( NM,E) L(2) (M, E) L(2) (M, E)

| = ◦ ¯ ¯ ∗ + ¯ ∗ ¯ = satisfying N Ker ∂¯∩Ker ∂¯∗ 0 and N (∂∂ ∂ ∂) id on the orthogonal complement of Ker ∂¯ ∩ Ker ∂¯ ∗ in {u ∈ Dom ∂¯ ∩ Dom ∂¯ ∗; ∂u¯ ∈ Dom ∂¯ ∗ and ∂¯ ∗u ∈ Dom ∂¯}. N is unique if it exists and is called the Neumann operator. The main point of [K’63] and [K’64] is the existence and estimates of N for strongly pseudoconvex domains. Existence and estimates of N in the Sobolev spaces have been studied for some class of weakly pseudoconvex domains. For instance, it is known that ND,E is compact if D is a bounded convex domain in Cn with smooth boundary (cf. [F-S’98]). The counterpart of Theorem 2.10 for Nakano negative bundles is that 0,q = − ⊗ ≤ H(2) (M, E) 0 holds for q0ifM is compact and (E, h) is Nakano negative, although (E∗, t h−1) is not necessarily Nakano positive (cf. [Siu’84]). Vanishing theorems and their variants under some weaker negativity (or positivity) assumptions are useful to understand the rigidity of complex structures of irreducible locally symmetric varieties of dimension ≥ 2(cf. [C-V’60, Siu-5, Oh’87]). Concerning Theorem 2.14 which is a reﬁnement of Theorem 2.10 in the spirit of Hörmander [Hö-1], the works [Dm-2] and [Oh-2, Oh-8] were also preceded by Skoda’s work [Sk-2, Sk-4] on the solution of the division problem with L2 conditions, where the semipositivity of the curvature of E was ﬁrst effectively explored on pseudoconvex manifolds. For some detail of [Sk-4], see Sects. 3.2.1 and 3.2.2 in Chap. 3. On complete Kähler manifolds, Theorem 2.14 and its consequences on pseudoconvex manifolds play similar roles as Cartan’s theorem

1See also [Oh-21, Chapter 6]. 2.5 Notes and Remarks 101

B does on Stein manifolds. The proof of Theorem 2.56 in [N-2] and [F-N] is such an instance as well as the L2 proof of Theorem 2.61. The notion of singular fiber metric of a holomorphic vector bundle has been extended by de Cataldo [DC’98] to include a locally measureable map h with values in the space of semipositive, possibly unbounded Hermitian forms whose determinant det h satisfies 0 < det h<+∞ almost everywhere. Theorem 2.29 was recently generalized by Iwai [I’18] for the bundles E equipped with a singular fiber metric in this sense. In the situation of Theorem 2.29, it was proved by Takegoshi [T’95] that, for any Nakano semipositive vector bundle E → M, the direct image sheaves q O K ⊗ ≥ K = K ⊗ ∗K−1 R π∗ ( M/X E) are torsion free for q 0. Here M/X M π X .This is a natural generalization of Theorem 2.42 according to the idea of Kollár’s torsion freeness theorem in [Ko’86-1, Ko’86-2] for the algebraic case (see also [Ko’87]). There exist L2 cohomology vanishing theorems on complete Riemannian manifolds which are analogous to Akizuki–Nakano’s vanishing theorem (cf. [P-R-S’08, Oh’89, S’86, Ag’17]). Interpretation of the L2 ∂¯-cohomology into the ordinary ∂¯-cohomology has been studied in [P-S’91, R’14] and more recently in [B-P’17]. For the arithmetic ball quotients X = Γ \Bn, where Γ is a torsion-free discrete subgroup Γ ⊂ Aut Bn(= SU(n,1)),theL2 cohomology of X is closely related to the representation theory (cf. [Z, Sap, MS-Y-Z’12]). As in the case of vanishing theorems, the L2 method is effective in the finiteness theorems because the geometric properties of the vector bundles and underlying manifolds are sharply reflected on the Dolbeualt complexes. On the other hand, the proof of Theorem 2.44 in [A-G], which is a natural extension of Grauert’s proof for the case q = 1, has an advantage that it can be further generalized to relativize the result by virtue of the ideas of Malgrange and Grothendieck (cf. [F-K’72, K-V’71, Hl’73]). For instance one has the following relative version of Andreotti-Grauert’s theorem. Theorem 2.83 Let X be a complex space, let π : X → T be a holomorphic map, and let F → X be a coherent analytic sheaf. Suppose that there exists a continuous function ϕ : X → R and c ∈ R such that π|{x∈X;ϕ(x)≤c} is proper and ϕ{x∈X;ϕ(x)>c} is q-convex. Then Rpπ∗F are coherent for p ≥ q. The map π as in Theorem 2.83 is called a q-convex map. Since Stein factor- izations are q-convex for any q ≥ 1, Theorem 2.83 entails the Hausdorff property of H p(X, F ) for p ≥ q, similarly to the case of holomorphically convex spaces. Combining this fact with the L2 method, Takegoshi [T’99] obtained a generalization of Kollár’s torsion freeness theorem to q-convex maps as a continuation of [T’95]. We note that a “twisted variant” of (2.17)(cf.(3.5) in Chap. 3) is effectively used in [T’95, T’99]. Theorem 2.83 had been obtained in important special cases by Knorr[K’71] and Siu [Siu’70, Siu’72]. An application to the contraction problem in the theory of modification was given in [K-S’71]: 102 2 Analyzing the L2 ∂¯-Cohomology

Theorem 2.84 Let π : X → T be a 1-convex map. Then the union of the maximal compact analytic sets in the fibers of π,sayS, can be blown down to T in X. Namely, there exists a complex space Xˆ and a proper holomorphic map f : X → Xˆ such that f |X\S is biholomorphic and f |S = π|S. Knorr–Schneider’s theorem characterizes the contractibility geometrically and may well be regarded as a solution of the Levi problem. By the L2 method the following was obtained in [Oh’18-3]. Theorem 2.85 Let X be a weakly 1-complete manifold with a C∞ plurisubharmonic exhaustion function ϕ and a nonconstant holomorphic function f whose fibers are holomorphically convex. Assume that there exists an effective divisor δ on X such that f ||δ| is proper, ϕ|X\|δ| is strictly plurisubharmonic and the associated line bundle [δ]→X is negative. Then X is holomorphically convex. In addition to the Kodaira–Nakano’s method for the vanishing of the ∂¯- cohomology, the proof of finite-dimensionality of the L2 ∂¯-cohomology on complete manifolds rests on Proposition 2.2, which is a small but substantial improvement of a basic fact that a Banach space V is finite dimensional if and only if the unit ball of V is relatively compact. Unfortunately the argument is not so straightforward. In fact, to derive the finite dimensionality of the ordinary ∂¯-cohomology from that of the L2 ∂¯-cohomology, as in Theorem 2.53, one needs delicate injectivity results such as Proposition 2.14 and Theorem 2.40. This part is hard to relativize, so that the following very naturally expected statement remains an open question. Conjecture Let X be a pseudoconvex manifold, let π : X → T be a holomorphic map to a complex space T , and let E → X be a holomorphic vector bundle. Assume that E has a fiber metric whose curvature form is Nakano positive outside a subset q K ⊂ X such that π|K is proper. Then R π∗O(KX ⊗ E) are coherent analytic sheaves for all q>0. q K ⊗ = −1 Note that the upper semicontinuity of dim H (Xt , Xt E) (Xt π (t)) is true in the above situation, provided that T is nonsingular and π is everywhere of maximal rank. The proof is essentially contained in the theory of Kodaira and Spencer [Kd-S’58]. The vanishing of H 1(S, O([S∗])⊗μ) in Theorem 2.57 is a sufficient condi- O I μ tion for holomorphic functions on (S, X/ S ) to extend holomorphically to O I μ+1 (S, X/ S ). Therefore, the assertion can be phrased concisely as “contractibility can be formally detected”. This is a kind of formal principle which goes back to Cauchy’s solution of the initial value problem. A generalization of Theorem 2.57 is given in [B’81] from this viewpoint. The Hirzebruch–Riemann–Roch theorem (cf. [Hrz’56]) says that the equality (1) in Theorem 2.59 becomes 2.5 Notes and Remarks 103

n ⊗ χ(M, O(F μ)) := (−1)k dim H k(M, O(F μ)) k=0 n n−1 = a0μ + a1μ +···+an,μ= 1, 2,...,

n where aj ∈ Z and a0 = c1(F ) /n! if X is replaced by an n-dimensional compact complex manifold M and L by the powers L⊗μ of a positive holomorphic line bundle L → M.IfL is ample, Matsusaka’s big theorem (cf. [M’70, M’72, L-M’75]) says that, for every polynomial p(t) such that p(μ) = χ(M, O(L⊗μ)), there exists a constant c = c(p(t)) such that L⊗c is very ample. In [Ty-1] it was shown by the 2 ⊗m ⊗(n+2) L method that, for any pseudoconvex M and positive L → M, (KM ⊗L ) 1 + is very ample for m> 2 n(n 1).In[Gra’94], Grauert gave a comment on Theorem 2.61 as follows. The paper [5](=[Gra-2]) is my thesis. It contains (among others) the following results: 1. A Hermitian metric in a complex manifold is a Kähler metric if and only all local analytic sets are minimal surfaces. 2. If X is an unbranched domain over Cn with real-analytically smooth boundary, then X is pseudoconvex if and only if X has a complete Kähler metric. Of course, the condition on the boundary is very restrictive. But the theorem is not true in general if the boundary of X contains lower dimensional components. As a counterexample, a method was introduced in [5] to construct a complete Kähler metric on the complement of a proper subvariety in a compact projective algebraic manifold. There are two directions in which further work was done in this problem of characterizing Stein manifolds by complete Kähler metrics. Because of the counter-example, some additional conditions (on the boundary or the curvature of the metric on the interior) are needed to ensure that a complete Kähler manifold is Stein. In 1990, Remmert also gave a comment on [Gra-2] in an address celebrating Grauert’s 60th birthday: In 1952 Kähler manifolds entered the stage. GRAUERT got a small grant to study this new theory at Zürich (H. Hopf, B. Eckmann). He sat there in splendid isolation and came back to Münster with a manuscript: Charakterisierung der Holomorphiegebiete durch die vollständige Kählersche Metrik (Math. Ann. 131, 1956). Several of you will know this paper. Just to give all of you a ﬂavour of what he was dealing with let me state three results: a) Every Stein manifold carries a complete KÄHLER metric with global (real analytic) potential. b) If a domain X over Cn (unramiﬁed) has a real analytic boundary and a complete KÄHLER metric, X is Stein. n c) If a domain X over C is pseudoconvex then − log dX is plurisubharmonic in X (Satz 18)...... In August 1954 GRAUERT lectured about his results at Oberwolfach. Heiz HOPF was amazed that all these theorems had been obtained at Zürich. 104 2 Analyzing the L2 ∂¯-Cohomology

Of course (c) is Oka’s lemma (Theorem 1.12), so that (a) and (b) are what really counts. Anyway, it is the author’s priviledge to add another comment that an L2 proof of (b) found in [Oh-2] has led the author to Theorem 2.14 and eventually to [Oh-T-1]. JOHN ERIC BEDFORD remarked that the L2 proof of Theorem 2.61 in [Oh-2], which was sketched in Sect. 2.4.2 for the C1 case, works to prove the generalization in [D-P]. It is indeed the case as follows. Proposition 2.22 Let D be a domain in Cn with a complete Kähler metric. Assume that, for every point x0 ∈ ∂D and for every sequence xμ ∈ D(μ= 1, 2,...) 1 converging to x0, one can ﬁnd 0 << 2 such that there exist a subsequence = ∩ xμk (k 1, 2,...) which is contained in an open and closed subset Ω of D Bn := ∩{ ∈ (x0,), complex lines k xμk and holomorphic functions fk on Uk Ω z Cn; ∩ 1 } = + 2 dist(z, k D) < k such that f(xμk ) k/(log k 1) and 2(n−1) 2 lim k |fk(z)| dλ = 0 k→∞ Uk hold. Then D is a domain of holomorphy.

Proof Let χ : R →[0, 1] be a function satisfying χ|[0,1] = 1 and χ|[2,∞) = 0, and put ¯ dz1 ∧···∧dzn ∧ ∂(fkχ(k dist(z, k))), if z ∈ Uk vk = 0 if z ∈ D \ Uk.

¯ = 1 ≤ ≤ 2 Then vk is locally bounded and ∂vk 0. Note also that k dist(z, k) k on supp vk. n Let ω be any complete Kähler metric on D ∩ B (x0,)and put ¯ ωk = ω + i∂∂(− log (− log dist(z, k)))).

Clearly ωk is a complete Kähler metric on D \ k such that the length of ¯ ∂χ(k dist(z, k)) with respect to ωk is less than C log k for some constant C. Hence, 2 by the assumption on the L norm of fk, −2(n−1)| |2 n = lim dist(z, k) vk kωk 0. k→∞ D\ k

Here |·|k denotes the length with respect to ωk. 2 Therefore, by Theorem 2.14 one has a sequence of L (n, 0)-forms uk on D \ k ¯ satisfying ∂uk = vk and −2(n−1) lim dist(z, k) uk ∧ uk = 0. (2.47) k→∞ D\ k 2.5 Notes and Remarks 105

˜ Then we deﬁne fk by ˜ fk dz1 ∧···∧dzn = dz1 ∧···∧dzn · fkχ(k dist(z, k)) − uk. ˜ By (2.47) and Theorem 1.8, fk extends holomorphically to D, | ˜ |2 = ˜ = k lim fk dλ 0 and fk(xμk ) 2 . k→∞ D (log k + 1)

˜ = ˜ → Therefore one can find a subsequence fkm (m 1, 2,...)of fk and δm 0 such = ∞ ˜ that the series F m=1 δmfkm converges locally uniformly on D and satisfies | |=∞ lim supm→∞ F(xμkm ) . Hence D must be holomorphically convex by virtue of the local criterion of Oka (cf. Theorems 1.12, 1.17, and 2.43). The above method of extending holomorphic functions with a control of L2 norms can be refined to prove the following. Theorem 2.86 (cf. [Oh-T-1]) Let D be a bounded pseudoconvex domain in Cn | |≤ ={ ∈ ; = } such that supD zn 1 and let D z D zn 0 . Then, for any plurisubhar- ∈ O −ϕ| |2 ∞ monic function ϕ on D and for any f (D ) satisfying D e f dλn−1 < , there exists F ∈ O(D) such that F |D = f and −ϕ 2 −ϕ 2 e |F | dλn ≤ 1620π e |f | dλn−1. D D

A reﬁned version of Theorem 2.86 that replaces 1620 by 1 will be proved in Chap. 3. Local geometry of the boundary of complex domains was ﬁrst detected by Poincaré [P’1907]. He recognized a distinction between the sphere ∂B2 and the real hypersurface Im z2 = 0 ; the latter is foliated by complex curves but the former is not. An important thing is that this difference is invariant under biholomorphic maps so that it can be understood as an inequivalence of a geometric structure, which is now called the CR structure. A C2 manifold W of real dimension 2n + 1 is called a CR manifold (CR = Cauchy–Riemann or complex and real) if TW ⊗ C splits C 1,0 0,1 1,0 into the direct sum of three -subbundles TW ,TW and F such that TW is closed 0,1 = 1,0 = = 1,0 under the Lie bracket, TW TW , F F and rank F 1. TW is called a CR 2 2 structure of W. A C map α between two CR manifolds W1 and W2 is called a CR map if α∗T 1,0 ⊂ T 1,0. Similarly, a C1 map β from a CR manifold W to a complex W2 W1 ∗ 1,0 ⊂ 1,0 manifold M is called a CR map if β TM TW . CR functions are just CR maps to C. CR functions are characterized as the C1 functions that are annihilated by 0,1 2 TW .EveryC real hypersurface H of a complex manifold M is a CR manifold

2The condition rank F = 1 is sometimes not assumed. 106 2 Analyzing the L2 ∂¯-Cohomology

1,0 = ⊗ C ∩ 1,0 with respect to TH (TH ) TM . A basic CR invariant is the Levi form. It is deﬁned as the Hermitian form which can be expressed in terms of local frames 1,0 = ξ1,...,ξn of TW and θ with θ θ of F as √ ∗ ⊗ ∗ ≡ − [ ] 1,0 ⊕ 0,1 cjkξj ξ k, where cjkθ 1 ξj , ξ k mod TW TW .

If θ can be chosen around x ∈ W so that the Levi form is positive deﬁnite at x, W is said to be strongly pseudoconvex at x. W is called a strongly pseudoconvex CR manifold if so is W everywhere. For instance, the sphere ∂B2 is strongly pseudoconvex at the point (0,1) because one may take

2 ∂ ∂ρ ∂ 2 ξ1 = zk − ,ρ(z)=z − 1 ∂zk ∂z2 ∂z2 k=1 and ∂ ∂ θ = + ∂z2 ∂z2 to get c = 1 + 1 . 11 z2 z2 Akahori [A’87] proved that a C∞ strongly pseudoconvex CR manifold W of dimension 2n + 1 is locally embeddable into Cn+1 by a C∞ CR map if n ≥ 3. (See also [Ku’82-1, Ku’82-2, Ku’82-3] and [W’89].) It is known that there exists a non- embeddable W if n = 1(cf.[R’65]). For n = 2 the question is left open, whereas Boutet de Monvel [BM’74] proved that compact and strongly pseudoconvex C∞ CR manifolds of dimension ≥ 5 are CR embeddable into some CN . Relying on this result, it was shown in [Oh’84-1, Oh’84-2] that such a W can be realized as a real hypersurface of a complex manifold. This is the point in the proof of the following. Theorem 2.87 (cf. [N-Oh’84]) Let M be a connected pseudoconvex manifold of dimension n with a C∞ plurisubharmonic exhaustion function ϕ : M → [0,d) (d < ∞) such that there exists a Hermitian metric ω on M such that ω = i∂∂ϕ¯ holds outside a compact subset of M. Assume that the following conditions are satisfied. (a) The volume and the diameter of M are finite with respect to ω. (b) Tensor fields composed of successive derivatives of ϕ have bounded magnitudes, the magnitude being counted pointwise with respect to ω and boundedness referring to the change of the point in M. (c) |∂ϕ|ω is bounded and bounded from below by a positive constant outside a compact subset of M. (d) n ≥ 3. 2.5 Notes and Remarks 107

Then there exists a biholomorphic map γ from M to a domain D in a complex manifold, say N, with a C∞ function ϕ˜ : N →[0, ∞) such that D ={x ∈ −1 N;˜ϕ(x) < d} and ϕ˜|D = ϕ ◦ γ . It was remarked by Catlin that n ≥ 2 suffices because of the closedness of the ¯ range of ∂b. Note that the manifold (M, ϕ) in the above theorem admits a complete metric ω+i∂∂(¯ − log(d−ϕ)) which is Kählerian outside a compact set of M. Bland [Bl’85] considered a similar compactification problem for a complete Kähler manifold (M, ω). He assumed that M is simply connected and ω is of negative sectional curvature. In this very natural circumstance, the length function r(x) = dist(x, x0) −r from a fixed point x0 ∈ M induces a bounded exhaustion function ρ = 1 − e .By imposing on ρ a condition similarly to Theorem 2.87, Bland proved that a strongly pseudoconvex boundary can be attached to M. This research is obviously directed to a characterization of the ball as a complete Kähler manifold. The question is to n characterize (B ,ωB ) with ∂∂¯z2 ∂z2 ∧ ∂¯z2 ωB = i + . 1 −z2 (1 −z2)2

n Since U(n) is contained in Aut B as the isotropy group at 0 and ωB is invari- n ant under Aut B , the sectional curvature of ωB along complex lines of TBn (holomorphic sectional curvature) does not depend on the choice of lines. It is naturally expected that a simply connected complete Kähler manifold of constant n negative holomorphic sectional curvature is isometrically equivalent to (B ,cωB ) for some c>0, but it is not known except for the case of the Bergman metric (cf. = ¯ n [Lu’66]). It is readily seen that the Ricci form ( i∂∂ log ωB ) of ωB is proportional to ωB , i.e. ωB is an Einstein-Kähler metric. Recently Wu and Yau [W-Y’17] proved the following. Theorem 2.88 Let (M, ω) be a complete Kähler manifold of dimension n whose holomorphic sectional curvature lies between two negative constants, say c1 and c2. Then M admits a unique complete Kähler metric ωˆ whose Ricci form is equal to −ˆω. Inequality

− C 1ωˆ ≤ ω ≤ Cωˆ holds on M for some constant C>0 depending on n, c1 and c2. Moreover, the curvature tensor of ωˆ and all its covariant derivative are bounded. Some of the results in Sect. 2.4.4 have been continued by [N’12, G-M-O’13, G’17]. It is remarkable that there exist a complex manifold F of dimension 2 and a locally Stein map f : F → C2 such that F is not Stein. Fornaess [F’78] constructed such an example as follows. 108 2 Analyzing the L2 ∂¯-Cohomology

D ={ ∈ C;| | } ={1 ; ∈ N \{ }} Let z z < 1 ,letΛ n n 1 ,let ∞ 1 1 U = z ∈ C;|z − | < n n4 n=2 and let H : D →[−∞, 0) be a subharmonic function such that − 1 D \ ∞ D \ ∪{ } H(z)> 2 on U, H is C on (Λ 0 ) and 1 z − 1 H(z) = min log n − 1 n mn 2 holds on a neighborhood of Λ for some integers mn(& 1). Let Ω ={(z, w) ∈ D × (C \{0}); H(z) − log |w| < 0} and let sn be positive numbers with s < 1 such that n n4

− 1 1 z n 1 H(z) = log − 1on z;|z − |

Then

1 Ω ∩ (z, w);|z − |

={ ∈ C2;| | mn } = 1 + mn where Un (η, w) η < 2e and φn(η, w) ( n ηw ,w). It is classical that Ω is holomorphically convex (cf. [Ht-1] and [C-T]). Based on the properties of Ω, we deﬁne a proper holomorphic map σ : M → C2 \{(0, 0)}, obtained by a succession of blow up, such that | C2 \ ×{ }∪{ } (i) σ M\σ −1(Λ×{0}) is a holomorphic map onto (Λ 0 (0, 0) ) and ◦ (ii) σ −1(Ω) is a domain with smooth boundary. ◦ ◦ Here A denotes the closure of A and A the interior of A. We put F = σ −1(Ω) . Note that the closure of the preimage by σ of the set

∞ 1 ,w ; 0 < |w| < 1 n n=2 is contained in F and not relatively compact, but the preimage of ∞ { 1 ;| |= } n=2 ( n ,w) w 1 is relatively compact in F . Therefore F is not 2 holomorphically convex because of the maximum principle. Hence (F, σ|F , C ) is locally Stein but F is not Stein. σ : F → C2 can be deformed to a holomorphic map σ : F → C2 with discrete ﬁbers in such a way that the closure of the preimage by σ of the set 2.5 Notes and Remarks 109

∞ 1 ,w ; 0 < |w| < 1 n n=2 ∞ { 1 ;| |= is contained in F and not relatively compact, but the preimage of n=2 ( n ,w) w 1} is relatively compact in F . In particular, locally Stein ramified Riemann domains over Cn are not necessarily Stein if n ≥ 2. For an explicit construction of F ,see [F’78]. It is not known whether or not Fornaess’s domain F is holomorphically separable, i.e. whether F is embeddable into some domain in C3 as a hypersurface. This question is closely related to the following. Conjecture Let f : M → N be a locally pseudoconvex map such that f is injective and N is Stein. Then M is Stein. This question was raised by Ph. A. Griffiths in 1977 at RIMS in Kyoto. Theorem 2.77 shows that locally pseudoconvex maps of maximal rank can be globally bizarre. However, Theorems 2.78, 2.79, 2.80 show that something is there if the fibers are one-dimensional. In connection to this, after some exploration in [Oh’18-1], an L2 approach was found in [Oh’18-2] to prove a local property of certain locally pseudoconvex maps. It is an application of the following generalization of Theorem 2.86 in [Oh’88]. Theorem 2.89 Let X be a Stein manifold of dimension n,letY ⊂ X be a closed complex submanifold of codimension m,let(E, h) be a Nakano semipositive vector bundle over X,lets1,...,sm be holomorphic functions on X vanishing on Y , and let ϕ be a plurisubharmonic function on X. Then, for any holomorphic E-valued (n − m)-form g on Y satisfying − e ϕh(g) ∧ g < ∞ Y and for any >0, there exists a holomorphic E-valued n-form G on X which satisfies G = g ∧ ds1 ∧···∧dsm on Y and −ϕ 2 −m− Cm −ϕ e (1 +|s| ) h(G) ∧ G ≤ e h(g) ∧ g, X Y | |2 = m | |2 where s j=1 sj and Cm is a positive constant depending only on m. In [N’69] Nishino proved the following. Theorem 2.90 Let f : M → N be a locally pseudoconvex map whose fibers are isomorphic to C. Then f is locally trivial, i.e. every point y ∈ N has a neighborhood | C × → U such that f f −1(U) is equivalent to the projection U U. For the proof Nishino took a very natural approach. By fixing a holomorphic local section of f : M → N,says over U, and a local coordinate around 110 2 Analyzing the L2 ∂¯-Cohomology s(U) one has a canonically defined map Φ : f −1(U) → C mapping s(U) to 0 and mapping the fibers biholomorphically to C by canonically specifying the derivatives of f along s(U). By Koebe’s distorsion theorem it is easy to see that Φ is continuous. Unfortunately, the proof of the analyticity of Φ is delicate and rather tricky. Yamaguchi [Y’76] exploited a variational property of Robin constants to give an alternate proof of Theorem 2.90.Chirka[Ch’12] also gave a proof by using holomorphic motions. Here a holomorphic motion means a holomorphic submersion f : M → N such that M is continuously foliated by holomorphic sections over a neighborhood of each point of N. An L2 proof is as follows.

Proof of Theorem 2.90 Let y0 ∈ N and let U be a Stein neighborhood of y0 such −1 that there exists a holomorphic section s : U → M. We put My = f (y) and y˜ = s(y) for y ∈ U.LetΔ be a neighborhood of y˜0 such that one can ﬁnd a holomorphic submersion π : Δ → D such that π −1(0) ={˜y; y ∈ f(Δ)} and | ∈ D f π−1(ζ) are proper onto f(Δ) for all ζ . To obtain the conclusion we may assume in advance that M = f −1(f (Δ)). Then we put − log |f(z)| for z ∈ Δ ϕ(z) = 0forz ∈ M \ Δ.

∈ =∼ C Clearly ϕ PSH(M). Since My0 by assumption, there exists a holomorphic \{˜} 2 1-form ω0 on My0 y0 such that ω0ζ /dζ extends to a holomorphic function without zeros on a neighborhood of y˜0. It is clear that such ω0 is unique up to a multiplicative constant if one imposes that

~~i ω0 ∧ ω0 < ∞. \ My0 Δ~~

Let us ﬁx ω0. Then, since M is Stein, by Theorem 2.89 there exists a holomorphic −1 2-form ω˜ on M \ π (0) satisfying ω˜ | \{ ˜ } = dz ∧···∧dzm ∧ ω and My0 y0 1 0 − e 3ϕω˜ ∧ ω<˜ ∞ (2.48) X

Here (z1,...,zm) denotes a local coordinate around y0. = \{˜} ˜ | = ∧···∧ Let us deﬁne holomorphic 1-forms ω y on My y by ω Mt \{y ˜} dz1 dzm ∧ ωy. Then (2.48)impliesthatωy has a pole of order 2 at ζ = 0 and that ωy is nowhere zero on My for all y. Therefore ωy admits a primitive for every y. Let s : U(= π(Δ) = N) → M be any holomorphic section satisfying s (U) ∩ s(U) =∅, where we shrink U if necessary. Then we put References 111

~~x σ(x) = ωf(x), s (f (x))~~

where the integration is along any path in Mf(x) \{f(x)} connecting s (f (x)) to x. ˆ Then the map (σ, f ) gives an equivalence between M and C×U \s∞(U) for the ˆ ˆ section s∞ : U → C×U defined by {s∞(y)}=C\{σ(x); x ∈ My}, which is holo- ˆ ∼ morphic because M is Stein (cf. Theorem 2.63). Since C × U \ s∞(U) = C × U, we obtain the conclusion. Theorem 2.90 can be generalized to an assertion on certain locally Stein maps by exploiting a straightforward consequence of Cartan’s theorem B that locally Stein maps with smooth fibers are locally trivial on a neighborhood of any compact subset in a fiber (cf. [D-G’60] and [A-V’62]). More precisely we have the following. Corollary 2.19 Every locally Stein and locally topologically trivial family of a finite Riemann surface is locally analytically trivial outside a subset lying properly over the parameter space. Here a finite Riemann surface means the complement of a finite set of points in a compact Riemann surface. The L2 proof of Nishino’s rigidity theorem can be generalized. For instance, a rigidity criterion for a Stein family of Cn was obtained in [Oh’18-4] by extending the above-mentioned method.

~~References~~

[Ag’17] Akagawa, S.: Vanishing theorems of L2-cohomology groups on Hessian manifolds. arXiv:1702.06678v1 [math.DG] [A’87] Akahori, T.: A new approach to the local embedding theorem of CR-structures for ¯ n ≥ 4 (the local solvability for the operator ∂b in the abstract sense). Mem. Am. Math. Soc. 67(366), xvi+257 (1987) [A-V’62] Andreotti, A., Vesentini, E.: On the pseudo-rigidity of Stein manifolds. Ann. Scuola Norm. Sup. Pisa (3) 16, 213–223 (1962) [B-P’17] Bei, F., Piazza, P.: On the L2-∂¯-cohomology of certain complete Kähler metrics. arXiv:1509.06174v3 [math.DG] [B-L’80] Bell, S., Ligocka, E.: A simpliﬁcation and extension of Fefferman’s theorem on biholomorphic mappings. Invent. Math. 57(3), 283–289 (1980) [B’81] Bingener, J.: On the existence of analytic contractions. Invent. Math. 64(1), 25–67 (1981) [Bl’85] Bland, J.S.: On the existence of bounded holomorphic functions on complete Kähler manifolds. Invent. Math. 81(3), 555–566 (1985) [BM’74] Boutet de Monvel, L.: Opérateurs hypoelliptiques à caractéristiques doubles. Sémi- naire Goulaouic-Schwartz 1973–1974: Équations aux dérivées partielles et analyse fonctionnelle, Exp. Nos. 1 et 2, 16 pp. Centre de Mathématiques, École Polytech- nique, Paris (1974) 112 2 Analyzing the L2 ∂¯-Cohomology

[C-V’60] Calabi, E., Vesentini, E.: On compact, locally symmetric Kähler manifolds. Ann. Math. (2) 71, 472–507 (1960) [Ch’12] Chirka, E.: Holomorphic motions and the uniformization of holomorphic families of Riemann surfaces (Russian). Uspeki Mat. Nauk 67(6) (408), 125–202 (2012); translation in Russian Math. Surv. 67(6), 1091–1165 (2012) [DC’98] de Cataldo, M.: Singular Hermitian metrics on vector bundles. J. Reine Angew. Math. 502, 93–22 (1998) [D-G’60] Docquier, F., Grauert, H.: Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten. Math. Ann. 140, 94–123 (1960) [Dol’53] Dolbeault, P.: Sur la cohomologie des variétés analytiques complexes. C. R. Acad. Sci. Paris 236, 175–177 (1953) [F’78] Fornaess, J.E.: A counterexample for the Levi problem for branched Riemann domains over Cn. Math. Ann. 234(3), 275–277 (1978) [F-K’72] Forster, O., Knorr, K.: Ein Beweis des Grauertschen Bildgarbensatzes nach Ideen von B. Malgrange. Manuscripta Math. 5, 19–44 (1971) [F-S’98] Fu, S.Q., Straube, E.: Compactness of the ∂¯-Neumann problem on convex domains. J. Funct. Anal. 159(2), 629–641 (1998) [G’17] Gilligan, B.: Levi’s problem for complex homogeneous manifolds. arXiv:1607.04310v4 [math.CV] [G-M-O’13] Gilligan, B., Miebach, C., Oeljeklaus, K.: Pseudoconvex domains spread over complex homogeneous manifolds. Manuscripta Math. 142, 35–59 (2013) [Gra’94] Grauert, H.: Selected papers. I. With commentary by Y. T. Siu et al. Reprint of the 1994 edition [MR1314425]. Springer Collected Works in Mathematics, pp. xiv+439. Springer, Heidelberg (2014) [H’09] Hartogs, F.: Über die aus den singulären Stellen einer analytischen Funktion mehrerer Veränderlichen bestehenden Gebilde. Acta Math. 32(1), 57–79 (1909) [Hrz’56] Hirzebruch, F.: Neue topologische Methoden in der algebraischen Geometrie. Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 9, pp. viii+165. Springer, Berlin/Göttingen/Heidelberg (1956) [Hl’73] Houzel, C.: Espaces analytiques relatifs et théorème de ﬁnitude. Math. Ann. 205, 13–54 (1973) [I’18] Iwai, M.: Nadel-Nakano vanishing theorems of vector bundles with singular Hermi- tian metrics. arXiv:1802.01794v1 [math.CV] [K-V’71] Kiehl, R., Verdier, J.-L.: Ein einfacher Beweis des Kohärenzsatzes von Grauert. Math. Ann. 195, 24–50 (1971) [K’71] Knorr, K.: Die Endlichkeitssätze für q-konvexe und p-konkave Abbildungen und einige Anwendungen. 1971 Espaces Analytiques (Séminaire, Bucharest, 1969), pp. 9–15. Editura Acad. R.S.R., Bucharest [K-S’71] Knorr, K., Schneider, M.: Relativexzeptionelle analytische Mengen. Math. Ann. 193, 238–254 (1971) [Kd-S’58] Kodaira, K., Spencer, D.C.: On deformations of complex analytic structures. I, II. Ann. Math. (2) 67, 328–466 (1958) [K’63] Kohn, J.J.: Harmonic integrals on strongly pseudo-convex manifolds. I. Ann. Math. (2) 78, 112–148 (1963) [K’64] Kohn, J.J.: Harmonic integrals on strongly pseudo-convex manifolds. II. Ann. Math. (2) 79, 450–472 (1964) [Ko’86-1] Kollár, J.: Higher direct images of dualizing sheaves. I. Ann. Math. (2) 123(1), 11–42 (1986) [Ko’86-2] Kollár, J.: Higher direct images of dualizing sheaves. II. Ann. Math. (2) 124(1), 171– 202 (1986) [Ko’87] Kollár, J.: Vanishing theorems for cohomology groups. In: Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985). Proceedings of Symposia in Pure Math- ematics, vol. 46, Part 1, pp. 233–243. American Mathematical Society, Providence (1987) References 113

[Ku’82-1] Kuranishi, M.: Strongly pseudoconvex CR structures over small balls. I. An a priori estimate. Ann. Math. (2) 115(3), 451–500 (1982) [Ku’82-2] Kuranishi, M.: Strongly pseudoconvex CR structures over small balls. II. A regularity theorem. Ann. Math. (2) 116(1), 1–64 (1982) [Ku’82-3] Kuranishi, M.: Strongly pseudoconvex CR structures over small balls. III. An embedding theorem. Ann. Math. (2) 116(2), 249–330 (1982) [L-M’75] Lieberman, D., Mumford, D.: Matsusaka’s big theorem. In: Algebraic Geometry. Proceedings of Symposia in Pure Mathematics, vol. 29, Humboldt State University, Arcata, 1974, pp. 513–530. American Mathematical Society, Providence (1975) [Lu’66] Lu, Q.-K.: On Kähler manifolds with constant scalar curvature. Acta Math. Sinica 16, 269–281 (1966) Chinese) (= Chinese Math. 9, 283–298, 1966) [M’70] Matsusaka, T.: On canonically polarized varieties. II. Am. J. Math. 92, 283–292 (1970) [M’72] Matsusaka, T.: Polarized varieties with a given Hilbert polynomial. Am. J. Math. 94, 1027–1077 (1972) [MS-Y-Z’12] Müller-Stach, S., Ye, X.-M., Zuo, K.: Mixed Hodge complexes and L2-cohomology for local systems on ball quotients. Doc. Math. 17, 517–543 (2012) [N-Oh’84] Nakano, S., Ohsawa, T.: Strongly pseudoconvex manifolds and strongly pseudoconvex domains. Publ. Res. Inst. Math. Sci. 20(4), 705–715 (1984) [N’12] Nemirovski, S.: Levi problem and semistable quotients. arXiv:1006.4245v2 [math.CV] [N’69] Nishino, T.: Nouvelles recherches sur les fonctions entières de plusieurs variables complexes. II, Fonctions entières qui se réduisent à celles d’une variable. J. Math. Kyoto Univ. 9, 221–274 (1969) [Oh’84-1] Ohsawa, T.: Global realization of strongly pseudoconvex CR manifolds. Publ. Res. Inst. Math. Sci. 20(3), 599–605 (1984) [Oh’84-2] Ohsawa, T.: Holomorphic embedding of compact s.p.c. manifolds into complex manifolds as real hypersurfaces. In: Differential Geometry of Submanifolds, Kyoto, 1984. Lecture Notes in Mathematics, vol. 1090, pp. 64–76. Springer, Berlin (1984) [Oh’87] Ohsawa, T.: On the rigidity of noncompact quotients of bounded symmetric domains. Publ. Res. Inst. Math. Sci. 23(5), 881–894 (1987) [Oh’88] Ohsawa, T.: On the extension of L2 holomorphic functions. II. Publ. Res. Inst. Math. Sci. 24(2), 265–275 (1988) [Oh’89] Ohsawa, T.: A generalization of the Weitzenböck formula and an analytic approach to Morse theory. J. Ramanujan Math. Soc. 4(2), 121–144 (1989) [Oh’18-1] Ohsawa, T.: On the local pseudoconvexity of certain families of C. Ann. Inst. Fourier (To appear) [Oh’18-2] Ohsawa, T.: L2 proof of Nishino’s rigidity theorem. Kyoto J. Math. (To appear) [Oh’18-3] Ohsawa, T.: Application of an L2 extension theorem to the Levi problem on complex manifolds. Annales Polonici Mathematici (To appear) [Oh’18-4] Ohsawa, T.: A generalization of Nishino’s rigidity theorem to Stein families of Cn, preprint [P-S’91] Pardon, W., Stern, M.: L2 ∂¯-cohomology of complex projective varieties. J. Am. Math. Soc. 4(3), 603–621 (1991) [P’1907] Poincaré, H.: Les fonctions analytique de deux variables et la re présentation conforme. Rend. Circ. Math. Palermo 23, 59–107 (or Oeuvres. IV, 244–289) (1907) [P-R-S’08] Pigola, S., Rigoli, M., Setti, A.: Vanishing and Finiteness Results in Geometric Analysis. A Generalization of the Bochner Technique. Progress in Mathematics, vol. 266, pp. xiv+282. Birkhäuser Verlag, Basel (2008) [R’15] Rauﬁ, H.: Singular Hermitian metrics on holomorphic vector bundle. Ark. Mat. 53(2), 359–382 (2015) [R’07] Rosay, J.-P.: Extension of holomorphic bundles to the disc (and Serre’s problem on Stein bundles). Ann. Inst. Fourier (Grenoble) 57(2), 517–523 (2007) 114 2 Analyzing the L2 ∂¯-Cohomology

[R’65] Rossi, H.: Attaching analytic spaces to an analytic space along a pseudoconcave boundary. In: 1965 Proceedings of the Conference on Complex Analysis, Minneapo- lis, pp. 242–256. Springer, Berlin (1964) [R’14] Ruppenthal, J.: L2-theory for the ∂¯-operator on compact complex spaces. Duke Math. J. 163(15), 2887–2934 (2014) [S’86] Shima, H.: Vanishing theorems for compact Hessian manifolds. Ann. Inst. Fourier 36(3), 183–205 (1986) [Siu’70] Siu, Y.-T.: The 1-convex generalization of Grauert’s direct image theorem. Math. Ann. 190, 203–214 (1970/1971) [Siu’72] Siu, Y.-T.: A pseudoconvex-pseudoconcave generalization of Grauert’s direct image theorem. Ann. Scuola Norm. Sup. Pisa (3) 26, 649–664 (1972) [Siu’84] Siu, Y.-T.: A vanishing theorem for semipositive line bundles over non-Kähler manifolds. J. Differ. Geom. 19(2), 431–452 (1984) [T’95] Takegoshi, K.: Higher direct images of canonical sheaves tensorized with semipositive vector bundles by proper Kähler morphisms. Math. Ann. 303(3), 389–416 (1995) [T’99] Takegoshi, K.: Torsion freeness theorems for higher direct images of canonical sheaves by a certain convex Kähler morphism. Osaka J. Math. 36(1), 17–26 (1999) [W’89] Webster, S.: On the proof of Kuranishi’s embedding theorem. Ann. Inst. H. Poincaré Anal. Non Linéaire 6(3), 183–207 (1989) [W-Y’17] Wu, D.-M., Yau, S.-T.: Invariant metrics on negatively pinched complete Kähler manifolds. arXiv:1711.09475v1 [math.DG] [Y’76] Yamaguchi, H.: Parabolicité d’une fonction entière. J. Math. Kyoto Univ. 16, 71–92 (1976) Chapter 3 L2 Oka–Cartan Theory

Abstract Oka–Cartan theory is mainly concerned with the ideals of holomorphic functions on pseudoconvex domains over Cn. To describe how one can ﬁnd global generators of the ideals, the application of extension theorems and division theorems is indispensable. From the viewpoint of the ∂¯-equations, these questions amount to solving those of very special type. Making use of the speciﬁc forms of these ∂¯-equations, they are solved with precise L2 norm estimates, yielding optimal quantitative variants of Oka–Cartan theorems.

~~3.1 L2 Extension Theorems~~

From a general point of view, existence theorems and uniqueness theorems for the extension of holomorphic functions are equivalent to the vanishing of cohomology groups with certain boundary conditions, which has already been discussed in Chap. 2. When one wants to study more specific questions of extending functions with growth conditions, such a connection is lost in the sense that the vanishing of cohomology with growth conditions does not imply the existence of extension with growth conditions, except for very special situations. It turns out that there exists a refined L2 estimate for the ∂¯ operator which implies an extension theorem with a right L2 condition. A general L2 extension theorem of this kind is formulated on “quasi-Stein” manifolds. They have significant applications in complex geometry.

~~3.1.1 Extension by the Twisted Nakano Identity~~

~~Let (M, ω) be a Kähler manifold of dimension n and let (E, h) be a holomorphic Hermitian vector bundle over M. First let us recall Nakano’s identity:~~

~~[¯ ¯ $] −[ $ ] =[ ] ∂,∂h gr ∂ ,∂h gr iΘh,Λ gr (3.1)~~

© Springer Japan KK, part of Springer Nature 2018 115 T. Ohsawa, L2 Approaches in Several Complex Variables, Springer Monographs in Mathematics, https://doi.org/10.1007/978-4-431-56852-0_3 116 3 L2 Oka–Cartan Theory

~~(cf. Theorem 2.7) and a subsequent formula~~

~~¯ ∗ $ ¯ [∂,θ ]gr +[∂ , θ]gr =[i∂θ,Λ]gr (3.2) for any θ ∈ C0,1(M) (cf. Theorem 2.8). To derive a variant of (3.1), we write it as~~

~~¯ ◦ ¯ $ + ¯ $ ◦ ¯ − $ ◦ − ◦ $ = − ∂ ∂h ∂h ∂ ∂ ∂h ∂h ∂ i(ΘhΛ ΛΘh). (3.3)~~

~~Let η be any positive C∞ function on M. Then, as a modiﬁcation of (3.3), one has~~

¯ ◦ ◦ ¯ $ + ¯ $ ◦ ◦ ¯ − $ ◦ ◦ − ◦ ◦ $ ∂ η ∂h ∂h η ∂ ∂ η ∂h ∂h η ∂ = ¯ ◦ ¯ $ − ¯ ∗ ¯ + ∗ ◦ − ◦ $ + − ∂η ∂h (∂η) ∂ (∂η) ∂h ∂η ∂ iη(ΘhΛ ΛΘh).

~~Hence, applying (3.2)forθ = ∂η¯ , we obtain~~

¯ ◦ ◦ ¯ $ + ¯ $ ◦ ◦ ¯ − $ ◦ ◦ − ◦ ◦ $ ∂ η ∂h ∂h η ∂ ∂ η ∂h ∂h η ∂ = ¯ ◦ ¯ $ + ¯ ¯ ∗ + ∗ ◦ + $ ◦ − ¯ − ¯ − − ∂η ∂h ∂(∂η) (∂η) ∂h ∂ ∂η i(∂∂ηΛ Λ∂∂η η(ΘhΛ ΛΘh)).

~~∈ n,q Therefore, for any u C0 (M, E), √ √ ¯ 2 + ¯ $ 2 ≥ ¯ $ ¯ ∗ + − ¯ + η∂u η∂hu 2Re(∂hu, (∂η) u) (i( ∂∂η ηΘh)Λu, u). (3.4)~~

Hence, by the Cauchy–Schwarz inequality one has √ ¯ 2 + + −1 ¯ $ 2 ≥ − ¯ − ∧ ¯ + η∂u η c ∂hu (i( ∂∂η c∂η ∂η ηΘh)Λu, u) (3.5) for any positive continuous function c on M. We ﬁrst infer from (3.5) the following. The proof is similar to Theorem 2.14 and may well be left to the reader. Proposition 3.1 Let (M, ω) be a Kähler manifold of dimension n,let(E, h) be a holomorphic Hermitian vector bundle over M, and let η be a bounded positive C∞ function on M. Suppose that there exist a complete Kähler metric on M and positive −1 continuous functions c1 and c2 on M such that c1 is bounded and ¯ ¯ ¯ (i(ηΘh − ∂∂η − c1∂η ∧ ∂η)Λu, u) ≥ (ic2∂η ∧ ∂ηΛu, u)

~~∈ n,q ∈ ¯ ∩ n,q holds for any u C0 (M, E). Then, for any v Ker ∂ L(2),loc(M, E) of the form ¯ ∂η ∧ v0 such that 3.1 L2 Extension Theorems 117~~

~~¯ −1 ((ic2∂η ∧ ∂ηΛ) v,v) < ∞, one can ﬁnd a solution to ∂w¯ = v satisfying~~

~~ + −1 −1/2 2 ≤ ∧ ¯ −1 (η c1 ) w ((ic2∂η ∂ηΛ) v,v).~~

−1 n,q We note that the boundedness of η and c1 is required so that C0 (M, E) is √ ¯ ∩ ¯ ◦ + −1 ∗ dense in Dom( η∂) Dom(∂ η c1 ) with respect to the graph norm. Reﬁning Proposition 3.1 we shall prove an L2 extension theorem for holomorphic functions at ﬁrst in a somewhat abstract form. Let M be a connected Stein manifold of dimension n,let(E, h) be a Hermitian vector bundle over M,let(B, b) be a Hermitian line bundle and let σ be a holomorphic section of B which is not identically zero. Let S = σ −1(0). | ≥ Theorem 3.1 In the above situation, assume that iΘh Sreg 0 and that there exist positive C∞ functions η, c on M \ S such that ¯ ¯ i(ηΘh − (∂∂η + c∂η ∧ ∂η) ⊗ IdE) ≥ 0

+ | |2 and η log σ b is bounded on some neighborhood of S. Then, for any positive continuous function Q on M \ S satisfying | |2 + −1 ≤ σ b(η c ) Q, the following is true: For any holomorphic section f of KM ⊗ B ⊗ E (= KS ⊗ E ˜ on S) over S, there exists a holomorphic section f of KM ⊗ B ⊗ E satisfying ˜ f |S = f ∧ dσ and − 1 ˜ 2 2 Q 2 f ≤ 2πf .

Here f is identiﬁed with an E-valued (n − 1) form on Sreg. Let us show ﬁrst how it works to prove the following, whose validity itself should be obvious to everybody who knows the area of discs and a few elementary properties of holomorphic functions. Theorem 3.2 There exists a holomorphic function f on D ={z ∈ C;|z| < 1} such that f(0) = 0 and |f(z)|2 dx dy ≤ π|f(0)|2. D

~~Proof To apply Theorem 3.1, we put M = D, (E, h) = (D × C, 1 −|z|2), 118 3 L2 Oka–Cartan Theory~~

~~− log |z|2 η(z) = 1 −|z|2 and~~

~~|z|2(1 −|z|2) c(z) = . 1 −|z|2 +|z|2 log |z|2~~

~~Then~~

~~− ¯ −| |2 + | |2 | |2 + −| |2 | |2 ∂∂η = 1 z 2 z log z (1 z ) log z dz ∧ dz (1 −|z|2)3 −| |2 + +| |2 | |2 = 1 z (1 z ) log z (1 −|z|2)3 and~~

~~1−|z|2 ∂η ∧ ∂η¯ |z log |z|2 − |2 (|z|2(1 − log |z|2) − 1)2 = z = . dz ∧ dz (1 −|z|2)4 |z|2(1 −|z|2)4~~

~~Hence~~

~~−∂∂η¯ − c∂η ∧ ∂η¯ log |z|2 = . dz ∧ dz (1 −|z|2)3~~

~~Therefore ¯ ¯ i(ηΘh − ∂∂η − c∂η ∧ ∂η) = 0. On the other hand, by letting (B, b) be the trivial Hermitian line bundle, σ = z and Q = 1, one has~~

~~2 2 2 2 − − log |z| 1 −|z| +|z| log |z| |σ |2(η + c 1) =|z|2 + = 1 = Q. b 1 −|z|2 |z|2(1 −|z|2)~~

Thus we obtain the assertion from Theorem 3.1. Proof of Theorem 3.1 In view of the Steinness of M and because of the required properties of f˜, we may assume in advance that S is nonsingular and σ has only simple zeros. Moreover, it sufﬁces to ﬁnd for each bounded open set D of M,a − 1 2 2 holomorphic extension of f ∧ dσ to D,sayfD such that Q 2 fD ≤ 2πf . In this situation, let fˆ be any holomorphic extension of f ∧ dσ to M,let(η, c) be as in the assumption, and let D be any bounded pseudoconvex open set. We shall solve a set of ∂¯-equations ¯ ˆ¯ ∂u = f ∂χ(− log |σ|b) on D \ S, (3.6) 3.1 L2 Extension Theorems 119

1 where 0 << e and ⎧ ⎨ 0 if t<− log = − − − ≤ ≤− χ(t) ⎩ log t log( log ) if log t e log 1 if t>−e log . with side conditions of u such that u are extendable to D continuously in such a way that u|S∩D = 0 and − 1 2 2 lim inf Q 2 u ≤ 2πf . (3.7) →0

ˆ Then a subsequence of fχ(− log |σb|) − u will converge to a desired extension. To find u, we fix a Kähler metric on D \ S and apply a general formula (3.5)for ∈ n,q \ u C0 (D S,E) by modifying the given η and c to η and c as follows. We first put ⎧ | | ⎨⎪ η(z) (< σ(z) b) | | 2 log σ(z) b e ηˆ(z)= min η(z), − log |σ(z)| + log |σ(z)|b log ( ≤|σ(z)|b ≤ ) ⎩⎪ b log e −e log (|σ(z)|b< )

2 Then, by using iΘh|S ≥ 0, the boundedness of η + log |σ | near S and the ∞ b Steinness of M to obtain positive C functions η, c and d which approximate ηˆ, c and 0 on M \ S in such a way that they satisfy − − ¯ − ∧ ¯ ≥ i ηΘh⊗b|σ| 2 ∂∂η c∂η ∂η 0, b ¯ ¯ i ηΘ ⊗ | |−2 − ∂∂η − c∂η ∧ ∂η h b σ b ¯ e ≥ i((1 − d)∂ log |σ|b ∧ ∂ log |σ|b/(− log |σ |b)) ( < |σ(z)| <), and − + −1 | |2 ≤ (1 d)(η c ) σ b Q. (3.8)

~~Thus we obtain u such that u|D∩S = 0 and − − − 1 − 1 + 1 2 2 ≤ 2 (1 d) (η c ) u ⊗ | |−2 2π f . (3.9) h b σ b~~

~~Combining (3.8) and (3.9) we obtain the required estimate~~

~~− 1 2 2 ≤ 2 Q u h⊗b 2π f .~~

~~Theorem 3.1 implies the following similarly to Theorem 3.2. 120 3 L2 Oka–Cartan Theory~~

~~Theorem 3.3 (L2 extension theorem) Let D be a pseudoconvex domain in Cn | | such that supz∈D zn < 1,letϕ be a plurisubharmonic function on D, and let~~

D ={z ∈ D; zn = 0}. Then, for any holomorphic function f on D satisfying − |f |2e ϕ dλ < ∞, D one can ﬁnd a holomorphic extension f˜ of f to D satisfying − − |f˜|2e ϕ dλ ≤ π |f |2e ϕ dλ. D D

Proof Since D is pseudoconvex in Cn, there exist an increasing sequence of relatively compact subdomains Dμ, μ = 1, 2,...whose union is D, and a decreasing ∞ sequence of C plurisubharmonic functions ϕν on D converging (pointwise) to ϕ. Therefore, it sufﬁces to show that, for each μ and ν, one can ﬁnd a holomorphic ˜ | extension fμ,ν of f Dμ to Dμ such that ˜ 2 −ϕν 2 −ϕ |fμ,ν| e dλ ≤ π |f | e dλ.

~~Dμ D~~

This can be shown similarly to the proof of Theorem 3.2 as follows. −ϕ−φ In Theorem 3.1, put M = D, (B,b) = (1D, 1), (E, h) = (1D,e )(ϕ,φ∈ ∞ C ,ϕ ∈ PSH,φ|D = 0) and σ = zn. It sufﬁces to show that there exist η and c 2 −1 −φ satisfying |zn| (η + c ) ≤ e such that ¯ ¯ i(ηΘh − ∂∂η − c∂η ∧ ∂η) ≥ 0

~~2 and η + log |zn| is bounded. To see this, we put~~

~~−t t − 1 − e s(t) = ,u(t)=−log(1 − e t ), v(t) = 1 − e−t et − 1 − t and = 1 = 1 = 1 η(z) s log 2 ,φ(z) u log 2 ,c(z) v log 2 . |zn| |zn| |zn|~~

~~Then √ ∧ − ¯ − ∧ ¯ ≥ − − − 2 | dzn dzn i(ηΘh ∂∂η c∂η ∂η) 1(su s vs ) t=log 1 2 . |zn|2 |zn| 3.1 L2 Extension Theorems 121~~

~~Since t − s(t) = eu(t) e u(x)dx 0~~

~~s 1 = (log s) = u + , s s~~

~~Hence s − su = 1,~~

2 t − − s =−s = e 1 t = 1 su − s u 1 − e−t v so that √ ¯ ¯ −1(ηΘh − ∂∂η − c∂η ∧ ∂η) ≥ 0 is obtained. Remark 3.1 Theorem 3.3 was ﬁrst proved by Błocki [Bł-2] to settle a question posed by Suita [Su-1]. Related materials will be discussed in more detail in the next chapter. (See also 3.4 in this chapter.)

~~3.1.2 L2 Extension Theorems on Complex Manifolds~~

We shall formulate L2 extension theorems in more general settings in such a way that they include some of the interpolation theorems in classical complex analysis in one variable. The proofs are essentially the same as in Theorem 3.3. Deﬁnition 3.1 A complex manifold M with a closed subset A is said to be a quasi- Stein manifold if A satisﬁes the following two conditions: (1) M \ A is a Stein manifold. (2) Each point x ∈ A has a fundamental neighborhood system Ux in M such that, for every U ∈ Ux, the natural restriction map

~~O −→ { ∈ 0,0 ; | } (U) f L(2),loc(U) f U\A is holomorphic~~

is surjective. Example 3.1 A complex manifold M with a nowhere-dense closed analytic subset A is quasi-Stein if M \ A is a Stein manifold. In particular, a nonsingular projective algebraic variety with a hyperplane section is quasi-Stein. The following is obvious. 122 3 L2 Oka–Cartan Theory

Proposition 3.2 Let (M, A) be a connected quasi-Stein manifold of dimension ≥ 1. Then A is nowhere dense in M. Theorem 3.4 Let (M, A) be a quasi-Stein manifold of dimension n,letw be a −1 −1 −1 holomorphic function on M,letH = w (0), and let H0 = w (0) \ dw (0). Suppose that H0\A is dense in H. Let φ and ψ be plurisubharmonic functions on M such that sup (ψ + 2log|w|) ≤ 0. Then, for any holomorphic (n − 1)-form f on H satisfying | e−φf ∧ f¯| < ∞, there exists a holomorphic n-form F on M 0 H0 such that F = dw ∧ f holds at any point of H0 and − + − e φ ψ F ∧ F¯ ≤ 2π e φf ∧ f¯. (3.10) M H0

Proof By the condition (2) on A, it suffices to show the assertion for the manifold M \ A. Since M \ A is Stein, it is an increasing union of strongly pseudoconvex domains. Therefore, we may assume that φ and ψ are smooth and it suffices to find, for each relatively compact Stein domain Ω in M \ A, a holomorphic n-form FΩ on Ω such that FΩ = dw ∧ f holds at any point of H0 ∩ Ω and −φ+ψ −φ ¯ e FΩ ∧ FΩ ≤ 2π e f ∧ f . (3.11) Ω H0

To prove this, one has only to repeat the argument of the proof of Theorem 3.2. Namely, by letting ψ+log |w|2 play the role of log |z|2 in defining η and by applying 2 the formula (3.5)forh = e−φ−log |w| , we are done. Remark 3.2 Theorem 3.3 was first established for the case ψ ≡ 0in[Oh-T-1], where the estimate for F is not sharp. The motivation of introducing ψ in [Oh-17] was to improve the known estimate for the Bergman kernel from below. (For the detail, see the next chapter.) In [Oh-17], φ−ψ was assumed to be plurisubharmonic. The present form with an optimal estimate was first established in [G-Z-1].

Now let us proceed to a more general case. Given a positive measure dμM on M, 2 2 we shall denote by A (M,E,h,dμM ) the space of L holomorphic sections of E over M with respect to h and dμM .LetS be a closed complex submanifold of M 2 and let dμS be a positive measure on S. A (S,E,h,dμS ) will stand for the space 2 of L holomorphic sections of E over S with respect to h and dμS. 2 Deﬁnition 3.2 (S, dμS) is said to be a set of interpolation for A (M,E,h,dμM ) 2 2 if there exists a bounded linear operator I from A (S,E,h,dμS) to A (M,E,h, dμM ) such that I(f)|S = f holds for any f .

~~Let dVM be any continuous volume form on M. Then we consider a class of continuous functions Ψ from M to [−∞, 0) such that: (1) Ψ −1(−∞) ⊃ S and 3.1 L2 Extension Theorems 123~~

~~(2) if S is k-dimensional around a point x, there exists a local coordinate z = (z1,z2,...,zn) on a neighborhood U of x such that zk+1 = ··· = zn = 0 on S ∩ U and~~

~~n 2 sup |Ψ(z)− (n − k)log |zj | | < ∞. \ U S k+1~~

The set of such functions Ψ will be denoted by (S). Clearly, the condition (2) does not depend on the choices of local coordinates. For each Ψ ∈ (S) we deﬁne a positive measure dVM [Ψ ] on S as the minimum element of the partially ordered set of positive measures dμ satisfying − 2(n k) −Ψ fdμ≥ lim sup fe χR(Ψ, t) dVM →∞ Sk t σ2n−2k−1 M for any nonnegative continuous function f with supp f M, where Sk denotes the m+1 k-dimensional component of S, σm denotes the volume of the unit sphere in R , and χR(Ψ, t) the characteristic function of the set

~~R(Ψ,t) ={x ∈ M;−t − 1 <Ψ(x)<−t}.~~

Note that the coefficient 2(n − k)/σ2n−2k−1 is chosen in such a way that 2 dλz[log |zn| ]=dλz for z = (z ,zn). Theorem 3.5 (L2 extension theorem on manifolds) Let M be a complex manifold with a continuous volume form dVM ,let(E, h) be a Nakano semipositive vector bundle over M,letS be a closed complex submanifold of M,letΨ ∈ (S) ∩ ∞ C (M \ S) and let KM be the canonical bundle of M. Then (S, dVM [Ψ ]) is a 2 −1 set of interpolation for A (M, E ⊗ KM ,h⊗ (dVM ) ,dVM ) if the following are satisfied: (1) There exists a closed subset A such that S∩A is nowhere dense in S and (M, A) is quasi-Stein. (2) There exists a positive number δ such that he−(1+δ)Ψ is a Nakano semipositive singular fiber metric of E. If, moreover, Ψ is plurisubharmonic on M, the interpolation operator from 2 ⊗K ⊗ −1 [ ] 2 ⊗K ⊗ −1 A (S, E M ,h (dVM ) ,dVM Ψ ) to A√(M, E M ,h (dVM ) ,dVM ) can be chosen so that its norm does not exceed π. Sketch of proof. Similarly to Theorem 3.4 it suffices to prove that, for every relatively compact Stein domain Ω in M \A, there exists a bounded linear map IΩ : 2 −1 2 −1 A (S, E ⊗ KM ,h⊗ (dVM ) ,dVM [Ψ ]) → A (Ω, E ⊗ KM ,h⊗ (dVM ) ,dVM ) whose norm is bounded by a constant independent of Ω such that IΩ (f )|S∩Ω = 2 −1 f |S∩Ω holds for all f ∈ A (S, E ⊗ KM ,h ⊗ (dVM ) ,dVM [Ψ ]). When Ψ is plurisubharmonic, the main difference from the situation of Theorem 3.4 is that 124 3 L2 Oka–Cartan Theory

S ∩ Ω is not necessarily defined as the zero set of a single holomorphic function. However, the assumption on Φ was made in such a way that, as was indicated above 2 2 by the equation dλz[log |zn| ]=dλz , Ψ plays the same role as log |w| in the proof of Theorem 3.4. For the proof of the first part, see [Oh-19]. √ 4 Remark 3.3 Theorem 3.5 with a weaker bound 2 π of the interpolation√ operator for the last statement was obtained in [Oh-19]. A proof for the bound π was found in [G-Z-1]. Compared to Theorem 3.4, the advantage of Theorem 3.5 is that the condition on the set S is stated in terms of a measure on S and a function with value −∞ along S, since it is often hard to find a generator of the ideal of holomorphic functions vanishing along S. The prototype of Theorem 3.5 is a theorem on interpolation and sampling in one variable due to K. Seip, which we shall recall below. For the proofs the reader is referred to [Sp-1, Sp-2] and [S-W]. Definition 3.3 A subset Γ ⊂ C is said to be uniformly discrete if

~~inf {|z − w|; z, w ∈ Γ,z = w} > 0.~~

~~The upper uniform density of a uniformly discrete set Γ is deﬁned to be {z ∈ Γ ;|z − w|~~

Theorem 3.6 Let Γ be a uniformly discrete subset of C and let δΓ be the Dirac 2 mass supported on Γ . Then, (Γ, δΓ ) is a set of interpolation for Aα if and only if α>πD+(Γ ). For the unit disc D we put

~~2 = 2 D D × C −| |2 α Aα,D A ( , ,(1 z ) ,dλz).~~

~~Deﬁnition 3.4 A subset Γ ⊂ D is said to be uniformly discrete if z − w inf ; z, w ∈ Γ,z = w > 0. 1 − zw¯~~

~~=|z−w | Letting ρ(z,w) 1−zw¯ we put ∈ 1 log ρ(z,ξ) + ξ Γ,2 <ρ(z,ξ)~~

2 Theorem 3.7 Let Γ be a uniformly discrete subset of D. Then (Γ, (1 −|z| )δΓ ) is 2 + a set of interpolation for Aα,D if and only if α>2DD (Γ ). Theorems 3.6 and 3.7 are related to Theorem 3.5 through an interpretation of the density concepts in the following manner. For any discrete set Γ ⊂ C we set

~~+ C (Γ ) = inf {α; there exists a Ψ ∈ (Γ ) such that Ψ + α|z|2 is subharmonic on C}.~~

~~For a discrete set Γ ⊂ D we put~~

~~+ 2 CD (Γ ) = inf {α; there exists a Ψ ∈ (Γ ) such that Ψ − α log (1 −|z| ) is subharmonic on C}.~~

~~Theorem 3.8 For any uniformly discrete subset Γ ⊂ C (resp. Γ ⊂ D) one has + + + πD+(Γ ) = C (Γ ) (resp. 2DD (Γ ) = CD (Γ )). For the proof, see [Oh-19].~~

~~3.1.3 Application to Embeddings~~

Although it might look unlikely from the above exposition, L2 extension theorems have applications to algebraic geometry. They are used effectively in the induction on the dimension. Uniformity of the estimate in the weight factor ϕ is important for that purpose. A typical example of such application is to a question of T. Fujita: Fujita’s Conjecture (cf. [F-2]) Let M be a nonsingular projective variety over C of dimension n, and let L → M be an ample line bundle. Then: ⊗(n+1) (1) KM ⊗ L is generated by global sections and ⊗(n+2) (2) KM ⊗ L is very ample. It is obvious that Fujita’s conjecture is true if M = CPn. A theorem of Lefschetz implies its validity for the case of Abelian varieties. There have been a number of results on part (1) of Fujita’s conjecture, including its full veriﬁcation for n ≤ 5(cf. [Rd, Ko, E-L, F-3, Km-4, Y-Z’15]) and its partial veriﬁcation for general n which requires m to be greater than a constant of order n2 (cf. [A-S, H-1, H-2, Hei]). As for the connections of Fujita’s conjecture to other questions of algebraic geometry, see [M-2] and [Km-1, Km-2, Km-4]aswellas[F-2]. In [A-S], Theorem 3.3 (with a weaker estimate) is used to show a semicontinuity property for multiplier ideal sheaves which naturally yields the following effective partial solution to Fujita’s conjecture. 126 3 L2 Oka–Cartan Theory

Theorem 3.9 (cf. Angehrn and Siu [A-S]) Let M and L be as above and let κ be a positive number. Suppose that, for any irreducible subvariety W of dimension 1 ≤ 1 ≤ d · | d · d ≥ 1 + − + d n in M, the degree L W of L W satisﬁes (L W) 2 n(n 2r 1) κ. Then the global sections of OM (KM ⊗ L) separate any set of r distinct points P1,...,Pr of M. K ⊗ m ≥ 1 2 + + Corollary 3.1 M L is generated by global sections if m 2 (n n 2).

Combining Corollary 3.1 with the following lemma, it is known that KM ⊗ L ⊗ K ⊗ m n+1 ≥ 1 2 + + ( M L ) is very ample for m 2 (n n 2). Lemma 3.1 Let L be a positive line bundle over a compact complex manifold M of n+1 dimension n such that OM (L) is generated by global sections. Then, KM ⊗L ⊗A is very ample for any positive line bundle A. The method of Angehrn and Siu was applied by Takayama [Ty-1] to establish a projective embedding theorem for pseudoconvex manifolds with positive line bundles, which amounts to a partial solution to the following generalization of Fujita’s conjecture: Let M be a pseudoconvex manifold of dimension n and let L be a positive line bundle over M. Generalized Fujita’s Conjecture ⊗(n+1) (1) KM ⊗ L is generated by global sections and ⊗(n+2) (2) KM ⊗ L is very ample. Theorem 3.10 (cf. [Ty-1]) Let M be an n-dimensional pseudoconvex manifold m with a positive line bundle L. Then KM ⊗ L is generated by global sections if 1 + K ⊗ m 1 + + m> 2 n(n 1), and ( M L ) is very ample if m> 2 n(n 1) and >n 1. The proofs of Theorems 3.9 and 3.10 follow from Nadel’s vanishing theorem once one has appropriate singular fiber metrics. Such singular fiber metrics are constructed from the sections of sufficiently high tensor powers of the line bundle L. To obtain a “sufficiently singular” fiber metric by this method, one needs an induction, and at this step the L2 extension theorem plays a crucial role. Let us have a glance at this argument by tracing a lemma on the semicontinuity of multiplier ideal sheaves in Siu’s exposition [Siu-6]. In the following, “s is a a multivalued section of the fractional bundle L b ” means that sb is a section of the bundle La. Lemma 3.2 Let M be a compact complex manifold of complex dimension n and let L be a positive line bundle over M. Let P0 be a point of M and U be a local holomorphic curve in M passing through P0 with P0 as the only (possible) singularity. Let U be the open unit disc in C and σ : U → U be the normalization of U so that σ(0) = P0. Let β be a positive rational number. Let s1,...,sk be mul- ∗ β × tivalued holomorphic sections of pr1 (L ) over M U (pr1 denotes the projection 3.1 L2 Extension Theorems 127 to the first factor.). Suppose that for almost all u ∈ U \{0} (in the sense that the statement is true up to a subset of measure zero) the point (σ (u), u) belongs to the k | |2 −1| zero-set of the multiplier ideal sheaf of the singular metric ( ν=1 sν ) M×{u} β = ∗ β | k | |2 −1 · of L pr1 (L ) M×{u} (i.e. the function ( ν=1 sν ) ( ,u) is not locally integrable at σ(u)). Then (P0, 0) belongs to the zero-set of the multiplier ideal k | |2 −1| β = ∗ β | sheaf of the singular metric ( ν=1 sν ) M×{0} of L pr1 (L ) M×{0} (i.e. k | |2 −1 · the function ( ν=1 sν ) ( , 0) is not locally integrable at P0).

Proof Assume the contrary. Then for some open neighborhood V of P0 in M the k | |2 −1 · function ( ν=1 sν ) ( , 0) is integrable on V . We can assume without loss of ∗ | generality that pr1 L V ×U is holomorphically trivial and V is biholomorphic to a bounded pseudoconvex domain in Cn. We apply Theorem 3.3 to the domain D = × = Cn ×{ } = V U and the hyperplaneH 0 for zn 0. For the plurisubharmonic = k | |2 function we use ϕ log ( ν=1 sν ) and for the function to be extended we use f ≡ 1. Let F be the holomorphic function on V × U such that

~~k −1 2 2 |F | |sν| < ∞ × V U ν=1~~

~~and F(·, 0) = f on V . There exist an open neighborhood V of P0 in V and an open neighborhood W of 0 in U such that |F | is bounded from below on~~

V × W by some positive number. There is a set E of measure zero in W such k 2 −1 that ( |s | ) is ﬁnite for u ∈ W \ E, contradicting the assumption V ×{u} ν=1 ν k | |2 −1 · ∈ \{ } that ( ν=1 sν ) ( ,u)is not locally integrable at σ(u)for almost all u U 0 .

~~3.1.4 Application to Analytic Invariants~~

0,0 Km ∈ N Given a compact complex manifold M,dimH (M, M )(m ) is called the (m-th) plurigenus of M and denoted by Pm = Pm(M). S. Iitaka asked whether or not Pm is invariant under deformation (cf. [Nk]). Namely, let π : M → D be −1 a smooth family of compact complex manifolds and let Mt = π (t). Then is it true that Pm(Mt ) = Pm(M0) for all t? Nakamura [Nk’72] found that there exists a compact 3-fold M0 with trivial canonical bundle which admits a small deformation 3,0 M such that H (M ) = 0, so that at least one plurigenus of M0, namely P1,is not invariant under deformation. Since Nakamura’s example M0 is non-Kähler, the invariance of Pm for compact Kähler manifolds was left open. Although the full conjecture is still open, it was veriﬁed for algebraic manifolds in [Siu-7, Siu-9] (see also [T’02]). Siu’s proof combines the L2 extension theorem and Skoda’s division theorem (see the next section for Skoda’s division theorem). An alternate proof using only the L2 extension was given by Paun˘ [P]. Moreover, Paun’s˘ method was generalized by Claudon [Cl’07] to prove an extension theorem which is a relative version of the following. 128 3 L2 Oka–Cartan Theory

Theorem 3.11 (cf. [Ty’06]) Let X be a projective algebraic manifold, let S ⊂ X be a nonsingular irreducible hypersurface and let L → X be a holomorphic line bundle equipped with a singular ﬁber metric h such that ∞ (i) Θh ≥ ω (with >0 for some C Hermitian metric ω on X) and I = O (ii) the restriction hS of the metric h to S is well deﬁned and hS S. Then, for any integer m ≥ 1 the natural restriction map

~~0 ⊗m 0 ⊗m H (X, O(KX ⊗[S]⊗L) ) −→ H (S, O(KS ⊗ L) )~~

is surjective. The circumstance of Claudon’s theorem is as follows. Let π : M → D be a proper holomorphic submersion and let L → M be a positive line bundle with a singular ﬁber metric h such that

~~(i) Θh ≥ 0((L ,h)is pseudo-effective) := | (ii) h0 h M0 is well-deﬁned as a singular ﬁber metric and I = O (iii) h0 M0 . Theorem 3.12 In the above situation, the restriction map~~

~~0 M O K ⊗ L ⊗m −→ 0 O K ⊗ L ⊗m H ( , ( M ) ) H (M0, ( M0 ) )~~

is surjective. Obviously the invariance of plurigenera is an immediate corollary of Theorem 3.12. The main tool of the proof is the following variant of Theorem 3.3. Theorem 3.13 (cf. [Siu’02]) In the situation of Theorem 3.12, there exists a ∈ 0 O K ⊗ L (universal) constant C0 such that for every section σ0 H (M0, ( M0 )) 0 satisfying |σ ∧ σ|h < ∞, there exists σ˜ ∈ H (M , O(KM ⊗ L )) with M0 σ˜ | = σ ∧ dt and |˜σ ∧ σ˜ | ≤ C |σ ∧ σ| . M0 M h 0 M0 h

~~3.2 L2 Division Theorems~~

n Given holomorphic functions g1,...,gm on a pseudoconvex domain D over C , Oka [O-2] proved that, for any holomorphic function h on D which is locally in the ≤ ≤ ≤ ≤ ideal generated by gj (1 j m), one can ﬁnd holomorphic functions fj (1 j = m m) on D such that h j=1 fj gj holds on D. In Cartan’s terminology, this is due to the coherence of the kernel of the sheaf homomorphism 3.2 L2 Division Theorems 129

m m g : O −→ O,g(u1,...,um) = uj gj j=1 and the vanishing of the first cohomology of D with coefficients in the coherent analytic sheaves (cf. [G-R]). The L2 method of Hörmander [Hö-1, Hö-2]was applied by Skoda [Sk-2] to obtain an effective quantitative refinement of Oka’s theorem. After reviewing Skoda’s theory in its generalized form (cf. [Sk-4]), we shall show that an L2 extension theorem on complex manifolds can be applied to prove an L2 division theorem. This approach has an advantage that it yields a division theorem with an optimal L2 estimate in some cases.

~~3.2.1 A Gauss–Codazzi-Type Formula~~

Let M be a complex manifold and let (Ej ,hj )(j = 1, 2) be two Hermitian holomorphic vector bundles. By a morphism between (E1,h1) and (E2,h2),we : → | shall mean a holomorphic bundle morphism γ E1 E2 such that γ (Ker γ)⊥ ﬁberwise preserves the length of vectors. Here (Ker γ)⊥ denotes the orthogonal complement of Ker γ .Let

0 −→ S −→ E −→ Q −→ 0 (3.12) be a short exact sequence of holomorphic vector bundles over M and let h be a fiber Q metric of E. Then one has fiber metrics of S and Q,sayhS and h respectively, for which the arrows in (3.12) become morphisms of Hermitian holomorphic vector Q bundles. There is a relation between the curvature forms of h, hS and h which is similar to the classical Gauss–Codazzi formula. It was found by Griffiths [Gri-1, Gri-3]. The presentation below follows [Gri-1, Gri-3] and [Sk-4]. Let DE be the Chern connection of (E, h) (cf. Chap. 1). Then DE is decomposed according to the orthogonal decomposition E = S ⊕ Q: ! " ∗ DS −B DE = , (3.13) BDQ where DS and DQ denote respectively the Chern connections of(S, hS) and Q = α ∈ 1,0 ∗ = ∗ α ∈ (Q, h ), B Bα dz C (M, Hom(S, Q)), and B Bα dz 0,1 ∗ C (M, Hom(Q, S)), where Bα denotes the adjoint of Bα. With respect to a local frame (s1,...,sm) of E extending a local frame (s1,...,s ) of S,

~~ = μ = μσ ∂hνσ Bα (Bαν) h . ∂zα σ=1 130 3 L2 Oka–Cartan Theory~~

~~In other words, if s and t are respectively smooth sections of S and Q over an open set of M,~~

~~∗ DE(s + t) = DSs − B t + Bs + DQt.~~

B is called the second fundamental form of S ⊕ Q. Note that the second fundamental forms of S ⊕ Q and (S ⊗ L) ⊕ (Q ⊗ L) are equal for any holomorphic Hermitian line bundle L. Example 3.2 Let M = C,letE = C × C2,letQ = C × C,letg : E → Q be given by (z, (ζ, ξ)) → (z, (zζ, ξ)), let h = (δμν¯ ), and let S = Ker g. Then

~~S ⊕ Q = C · (1, −z) ⊕ C · (z,¯ 1), |1|2 = 1 +|z|2, hS | |2 = | |2 + −1 1 hQ ( z 1) , B(1, −z) = (0, −dz) and~~

~~∗ B (1) = dz.¯~~

~~Recall that the curvature form Θh is deﬁned as a Hom(E, E)-valued (1,1)-form on M satisfying~~

2 = DEs Θhs (3.14) for any local C∞ section s of E. Therefore, from (3.13) one immediately obtains # $ 2 − ∗ ∧ − ∗ = DS B B D(B ) Θh 2 − ∧ ∗ (3.15) D(B) DQ B B where D(B) denotes the derivative of B with respect to the connection of Hom(S, Q) associated to DS and DQ, namely

DQ(rs) = (Dr)s + r(DSs) for local sections r of Hom(S, Q) and s of S. In particular, one has ∂B¯ ∗ = 0. From (3.15) we obtain the vector bundle version of the Gauss–Codazzi formula: = | + ∗ ∧ ΘhS Θh S B B, = | + ∧ ∗ (3.16) ΘhQ Θh Q B B . 3.2 L2 Division Theorems 131

= Here the restrictions of Θh are with respect to the orthogonal decomposition E ⊕ = = κ α S Q. In terms of the local coordinates such that h (hμν) and B ( α Bανdz ), % μ μ ¯ ρ (Θ ) = (Θ ) − Bκ hμσ B h ¯, hS ναβ¯ h ναβ¯ κ,σ,ρ αν βσ κρ κ κ κ νμ¯ σ (3.17) (Θ Q ) = (Θ ) + B h B h ¯ . h ραβ¯ h ραβ¯ μ,ν,σ αν βμ ρσ

Note that the cohomology class in H 1(M, Hom(Q, S)) represented by B∗ vanishes if and only if (3.12) splits as a short exact sequence of holomorphic vector bundles. To see this, let g be the given morphism from E to Q and let j be the section of Hom(Q, E) which satisﬁes g◦j = IdQ and embeds Q into E isometrically. From (3.13), for any C∞ section t of Q one has ∂(jt)¯ =−B∗t + ∂t,¯ (3.18) ∂(jt)¯ = (∂j)t¯ + j∂t.¯

~~Hence~~

~~∗ ∂j¯ =−B .~~

~~If there exists A ∈ C∞(M, Hom(Q, S)) satisfying~~

∗ ∂A¯ =−B , then j − A is a holomorphic bundle morphism from Q to E such that g ◦ (j − A) = idQ. Now, given a holomorphic section f of Q, one has ∂(jf)¯ =−B∗f , so that ∂(B¯ ∗f)= 0. If there exists a solution u ∈ C∞(M, S) to the equation

~~∗ ∂u¯ =−B f, (3.19) f − u will then be holomorphic and satisfy~~

~~g(f − u) = f. (3.20)~~

Therefore, the problem of lifting holomorphic sections of Q to those of E is reduced to solving the ∂¯ equations of the form (3.19) with values in S. In the next subsection, we shall review how Skoda solved a division problem under this formulation.

~~3.2.2 Skoda’s Division Theorem~~

~~As well as in the case of extension theorems, it is most appropriate to state a general L2 division theorem for the bundle-valued (n, 0)-forms on n-dimensional complete 132 3 L2 Oka–Cartan Theory~~

Kähler manifolds. Let (M, ω) be a complete Kähler manifold of dimension n and let 0 → S → E → Q → 0, g : E → Q, h and B∗ be as above. From now on, the ranks of S,E and Q will be denoted by s,p and q, respectively. For the division theorem, Nakano’s identity is combined with the following lemma which Skoda called “LEMME FONDAMENTAL”. Lemma 3.3 Let r = min {n, s}=min {n, p − q}. For any form v ∈ Cn,1(M, S) and β ∈ C1,0(M, Hom(S, Q)) one has:

∗ 2 r!i Tr ββ ⊗ IdSΛv, v"≥|βv| (3.21) at every point of M, where Tr ββ∗ denotes the trace of β ∧ β∗ ∈ C1,1(M, Hom (Q, Q)) and β the adjoint of exterior multiplication by β∗. Proof By using the local orthonormal frames, (3.21) follows from the Cauchy– Schwarz inequality

~~ s s 2 2 ak ≤ s |ak| k=1 k=1 if r = s.Ifr = n,(3.21) is reduced to the inequality~~

~~s 2 2 n βkλv λ ≥ βkλvkλ . k, =1 λ k,λ~~

For the detail, see [Sk-4, pp. 591–594]. In the sense of Nakano positivity similar to the case of curvature form of vector ∗ ∗ bundles, the lemma says that irTr ββ ⊗ IdS + iββ ≥ 0. Hence, in view of (3.16) and

= Tr ΘhQ Θdet hQ one has the following L2 estimate. Proposition 3.3 Assume that (E, h) is Nakano semipositive and let (L, b) be a Hermitian holomorphic line bundle over M whose curvature form Θb satisﬁes

~~≥ + iΘb i(r )Θdet hQ (3.22) for some >0. Then~~

~~∗ ∂u¯ 2 +∂¯ u2 ≥ Bu2 (3.23) r~~

~~∈ n,1 ⊗ holds for any u C0 (M, S L). 3.2 L2 Division Theorems 133~~

Example 3.3 In the case of Example 3.2,(3.22) holds for (L, b) = (C × C,(|z|2 + 1)−1−). Hence, solving the equation (3.19) with an L2 estimate based on (3.23), Lemma 2.1 and Theorem 2.3, one has: Theorem 3.14 Let the situation be as in Proposition 3.2. Then, for any Q ⊗ L- valued holomorphic n-form f on M which is square integrable with respect to ω and hQ ⊗ b, there exists an E ⊗ L-valued holomorphic n-form e such that f = g · e and r e2 ≤ 1 + f 2. (3.24)

~~Corollary 3.2 Suppose that (E ⊗ det E,h ⊗ det h) is Nakano semipositive and a Hermitian holomorphic line bundle (L, b) over M satisﬁes~~

~~− − + ≥ iΘb iΘdet h i(r )Θdet hQ 0 for some >0. Then the map~~

n,0 ⊗ −→ n,0 ⊗ H(2) (M, E L) H(2) (M, Q L) induced from g is surjective. Proof It sufﬁces to apply Theorem 3.11 for the morphism E ⊗ det E → Q ⊗ det E and the line bundle (det E)∗ ⊗ L. Remark 3.4 It was recently shown by Liu, Sun and Yang [L-S-Y] that ample vector bundles have ﬁber metrics such that Θh⊗det h is Nakano positive. satisfying the condition of h in Corollary 3.2. See also [Dm-S]. Applying Corollary 3.2 when M is a bounded pseudoconvex domain in Cn and h = (δμν¯ ), one has the followng. Corollary 3.3 Let D be a bounded pseudoconvex domain in Cn,letφ be a plurisubharmonic function on D and let g = (g1,...,gp) be a vector of holomorphic functions on D.Iff is a holomorphic function on D such that − − − − |f |2|g| 2k 2 e φ dλ < ∞ D holds for k = min {n, p − 1} and some >0, there exists a vector of holomorphic functions a = (a1,...,ap) satisfying

~~p f = aj gj j=1 134 3 L2 Oka–Cartan Theory and − − − |a|2|g| 2k e φ dλ < ∞. D~~

An advantage of Corollary 3.3 is that it has the following division theorem as an immediate consequence. This special case is useful for the construction of integral kernels (cf. [He]). Theorem 3.15 Let D be a bounded domain in Cn which admits a complete Kähler n metric and let z = (z1,...,zn) be the coordinate of C . Then, for any positive number , there exists a constant C such that, for any holomorphic function f on D satisfying − − |f(z)|2|z| 2n dλ < ∞, D one can ﬁnd a system of holomorphic functions a = (a1,...,an) satisfying

~~n f(z)= zj aj (z) j=1 and 2 −2n+2− 2 −2n− |a(z)| |z| dλ ≤ C |f(z)| |z| dλ. D D~~

~~Corollary 3.4 (cf. [D-P]) Let D be a domain in Cn which admits a complete ◦ Kähler metric. If D = D , then D is a domain of holomorphy.~~

~~Proof Replacing D by the bounded domains D ∩{|z|~~

~~n − 0 = (zj zj )aj (z) 1. j=1~~

0 ¯ ◦ Hence not all of aj can be analytically continued to z . Since D = D , this means that D is a domain of holomorphy. Since the method of Skoda is very natural, the estimate in Theorem 3.12 is expected to be optimal. It is indeed the case in some situations as the following example shows. 3.2 L2 Division Theorems 135

Example 3.4 The L2 division problem zu + v = dz on C: It has a solution (u, v) = (0,dz). The squared L2 norm of this solution (0,dz) with respect to the above- ⊗ 2π Q 2π mentioned ﬁber metric of E L is , while that of dz with respect to h b is 1+ . Hence (3.24) is an equality in this case. However, it is remarkable that Theorem 3.12 is not optimal in the sense that the following is true. Theorem 3.16 Let D be a bounded pseudoconvex domain in Cn. Then there exists a constant C depending only on the diameter of D such that, for any plurisubharmonic function φ on D and for any holomorphic function f on D satisfying − − | | |f(z)|2e φ 2n log z dλ < ∞, D there exists a vector-valued holomorphic function a = (a1,...,an) on D satisfying

~~n f(z)= zj aj (z) j=1 and − − − | | − − | | |a(z)|2e φ(z) 2(n 1) log z dλ ≤ C |f(z)|2e φ(z) 2n log z dλ. D D~~

~~The purpose of the following two subsections is to give a proof of Theorem 3.13 after [Oh-20] as an application of Theorem 3.5.~~

~~3.2.3 From Division to Extension~~

~~For the proof of Theorem 3.13, we need the following special case of Theorem 3.5.~~

Theorem 3.17 (Corollary of Theorem 3.5) Let M,E,S and dVM be as in Theorem 3.5. If moreover S is everywhere of codimension one and there exists a ∗ ﬁber metric b of [S] such that Θh + IdE ⊗ Θb and Θh + (1 + δ)IdE ⊗ Θb are both Nakano semipositive for some δ>0, then there exists, for any canonical section s of [S] and for any relatively compact locally pseudoconvex open subset Ω of M,a 2 −1 2 bounded linear operator I from A (S ∩ Ω,E ⊗ KM ,h⊗ (dVM ) ,dVM [log |s| ]) 2 −1 to A (Ω, E ⊗ KM ,h⊗ (dVM ) ,dVM ) such that I(f)|S = f . Here the norm of I | | does not exceed a constant depending only on δ and supΩ s . Let us describe below how the division problem in Theorem 3.13 is reduced to an extension problem which can be solved by Theorem 3.14.LetN be a complex manifold and let F be a holomorphic vector bundle of rank r over N.LetP(F)be the projectivization of F , i.e. we put

~~P(F)= (F \ zero section)/(C \{0}). 136 3 L2 Oka–Cartan Theory~~

Then P(F)is a holomorphic fiber bundle over N whose typical fiber is isomorphic to CPr−1.LetL(F ) be the tautological line bundle over P(F)i.e. L(F ) = ∈P(F) where the points of P(F)are identified with complex linear subspaces of dimension one in the fibers of F . Let O(F ) denote the sheaf of germs of holomorphic sections of F . Then we have a natural isomorphism

~~∗ ∗ ∗ H 0(N, O(F )) =∼ H 0(P (F ), O(L(F ) )) which arises from the commutative diagram~~

~~L(F ∗)∗ π∗F F~~

P (F ∗) N where π denotes the bundle projection (the bottom arrow). Let γ : F → G be a surjective morphism from F to another holomorphic vector bundle G. Then one has the induced injective holomorphic map

~~∗ ∗ P(G )−→ P(F ) and a commutative diagram:~~

∗ ∗ ∗ ∗ One may identify L(F ) |P(G∗) with L(G ) by this isomorphism. Hence, for any holomorphic line bundle L over N, one has a commutative diagram which transfers division problems to extension problems: 3.2 L2 Division Theorems 137

∗ Here ργ denotes the natural restriction map. Note that P ((F ⊗ L) ) is naturally identiﬁed with P(F∗). By this diagram, L2 division problems are also transferred to L2 extension problems. If γ is a morphism between Hermitian holomorphic vector bundles, one has the following L2 counterpart of the above:

~~Given a volume form dV on N and a ﬁber metric h of F , the volume form on P(F∗) associated to dV and h is deﬁned as~~

r −1 = ¯ | |2 ∧ dVh (i∂∂ log ζ h) dV where ζ denotes the ﬁber coordinate of F . In order to apply Theorem 3.14 for M = P(F∗) and S = P(G∗), the condition on the codimension is missing in general. To ﬁll this gap, let us replace P(F∗) by its monoidal transform σ : P(F∗) → P(F∗) along P(G∗) and consider the restriction map

~~∗ ∗ ∗ − ∗ ∗ ∗ ∗ A2(P(F∗), σ L(F ) ) −→ A2(σ 1(P (G )), σ L(G ) ) or equivalently the map~~

~~+ − ∗ ∗ ∗ ∗ H n r 1,0(P(F∗), σ L(F ) ⊗ K ) (2) P(F∗) −→~~

~~0,0 −1 ∗ ∗ ∗ ∗ ∗ H (σ (P (G )), σ L(G ) ⊗ K ⊗ K∗ ) (2) P(F∗) P(F )~~

~~∗ Here the volume form on P(F ) is induced from dVh and a ﬁber metric b of the bundle [σ −1(P (G∗))] via the isomorphism~~

∼ ∗ − ∗ ⊗ − K = σ K ∗ ⊗[σ 1(P (G ))] (k 1), P(F∗) P(F ) where k is the codimension of P(G∗) in P(F∗). Accordingly, as the ﬁber metric of ∗ ∗ ∗ ∗ ∗ ∗ k−1 σ L(F ) ⊗ K we take σ (π h · dVh) · b . P(F∗) 138 3 L2 Oka–Cartan Theory

~~3.2.4 Proof of a Precise L2 Division Theorem~~

Let the situation be as in the hypothesis of Theorem 3.13. We may assume that φ is smooth since D is Stein. Since the assertion is obviously true if n = 1(evenfor any φ), we assume that n ≥ 2. For simplicity we shall assume that φ = 0, since the proof is similar for the general case. To apply Theorem 3.14 we put

~~N = D \{0}, n dV = (i∂∂(¯ |z|2 + log |z|2)),~~

~~F = N × Cn,~~

~~h = (δμν¯ ), G = N × C and~~

~~n γ(z,ζ)= z, zj ζj . j=1~~

~~To ﬁnd a right ﬁber metric b of [σ −1(P (G∗))] one needs a little more geometry. First we consider the extensions~~

πˆ : Fˆ = Cn × Cn −→ Cn, Gˆ = (Cn \{0}) × C of the above bundles F and G and note that the closure of the image of P(Gˆ ∗) in P(Fˆ ∗),sayP(Gˆ ∗), is nothing but the monoidal transform of Cn with center 0. Let σˆ : P(Fˆ ∗) → P(Fˆ ∗) be the monoidal transform along P(Gˆ ∗). Observe that, for any complex line in the projectivization of πˆ −1(0), the normal bundle N satisﬁes P(Gˆ ∗)/P (Fˆ ∗)

~~∼ n−2 O(N )| = O ⊕ O(1). P(Gˆ ∗)/P (Fˆ ∗)~~

~~Therefore [ˆσ −1(P(Gˆ ∗))]∗ admits a ﬁber metric b such that, with respect to the induced ﬁber metric hˆ of σˆ ∗L(Fˆ ∗)∗,~~

~~+ + iΘb (1 )iΘhˆ > 0 (3.25) holds for any >0. 3.2 L2 Division Theorems 139~~

~~On the other hand,~~

~~∼ ∗ −1 ˆ ∗ ⊗(n−2) K = σˆ K ˆ ∗ ⊗[ˆσ (P (G ))] , P(Fˆ ∗) P(F ) so that~~

ˆ ∗ ˆ ∗ ∗ ⊗ K∗ =∼ ˆ ∗ ˆ ∗ ∗ ⊗ˆ∗K ⊗[ˆ−1 ˆ ∗ ]⊗(n−2) σ L(F ) σ L(F ) σ P(Fˆ ∗) σ (P (G )) . P(Fˆ ∗) ≥ Since iΘσdVˆ h niΘhˆ follows immediately from the deﬁnition of dVh, one has

~~iΘˆ ∗ n−2 + (1 + δ)iΘ ≥ (1 + n)iΘ ˆ + (n − 1 + δ)iΘ h⊗ˆσ dVh⊗b b h b~~

on P(Fˆ ∗) \ˆσ −1(πˆ −1(0)). By (3.25) the right-hand side of the above inequality is positive if 1 − n<δ<2. Hence Theorem 3.14 is applicable if n ≥ 2. Remark 3.5 From the above proof, the difference of the weights in the L2 estimate for the solution is geometrically understood as the singularity of the induced ﬁber metric of N × C over 0. It might be worthwhile to compare Theorem 3.13 with its predecessor obtained by Skoda in [Sk-2]: Theorem 3.18 Let D be a pseudoconvex domain in Cn,letφ be a plurisubharmonic function on D and let g = (g1,...,gp) be a vector of holomorphic functions on D and let f be a holomorphic function on D such that − − − |f |2|g| 2k 2(1 + Δ log |g|)e φ dλ < ∞ D holds for k = min {n, p − 1}, where Δ denotes the Laplacian. Then there exists a vector of holomorphic functions a = (a1,...,ap) satisfying

~~p f = aj gj j=1 and − − − |a|2|g| 2k(1 +|z|2) 2e φ dλ < ∞. D~~

The author does not know whether or not one can get rid of the factor 1 + Δ log |g| from the above condition, although it is certainly the case when g = z as Theorem 3.13 shows. Skoda’s L2 division theory, as well as the L2 extension theorems inspired by it, was meant to reﬁne the Oka–Cartan theory of ideals of analytic functions. As a 140 3 L2 Oka–Cartan Theory result, it has applications to subtle questions in algebra. For instance, Theorem 3.11 can be applied to estimate the degrees of the polynomial solutions f = (f1,...,fp) to p fj gj = 1 j=1

n for the polynomials gj without common zeros in C (cf. [B-G-V-Y]). In the next section, we shall give a survey on applications of the L2 method to the ideals in C{z} which started from the breakthrough in [B-Sk].

~~3.3 L2 Approaches to Analytic Ideals~~

Beginning with a celebrated application of Skoda’s division theorem to a reﬁnement of Hilbert’s Nullstellensatz, we shall review subsequent results on the ideals in C{z} 2 obtained by the L method, particularly those on the multiplier ideal sheaves in OCn . They are initiated by Nadel [Nd] and enriched by Demailly and Kollár [Dm-K] and Demailly, Ein and Lazarsfeld [Dm-E-L]. Recent activity of Berntdsson [Brd-2] and Guan and Zhou [G-Z-2, G-Z-3, G-Z-5] settled a question posed in [Dm-K] (see also Hiep [Hp]).

~~3.3.1 Briançon–Skoda Theorem~~

~~In [B-Sk], Briançon and Skoda extended Euler’s identity~~

n ∂f rf = zj ∂zj j=1 which holds for any homogeneous polynomial f of degree r, by establishing a remarkable result on the integral closure of ideals in C{z}. Recall that the integral closure I of an ideal I of a commutative ring R is deﬁned as

~~q q+1 j I = x ∈ R ; there exists a monic polynomial b(X) = X + bj X j=0~~

~~q+1−j such that bj ∈ (I ) (j = 0,...,q) and b(x) = 0 . (3.26) 3.3 L2 Approaches to Analytic Ideals 141~~

~~Theorem 3.19 (Briançon–Skoda theorem) For any ideal I ⊂ C{z} which is generated by k elements, I k+ −1 ⊂ I holds for any ∈ N. Moreover, I n+ −1 ⊂ I if k ≥ n.~~

Corollary 3.5 Let f be any element of C{z} without constant term and let If be ∂f ∂f +n−1 the ideal generated by z ,...,zn . Then f ∈ (If ) for any nonnegative 1 ∂z1 ∂zn integer . Corollary 3.5, which we shall prove below, had been conjectured by J. Mather (cf. [Wl]). For the systematic treatment including the proof of Theorem 3.16,the reader is referred to [B-Sk]or[Dm-8]. For non-L2 proofs, see [L-T, S] and [Sz].

Lemma 3.4 Let f, g1,...,gk be germs of holomorphic functions vanishing at 0 ∈ Cn. Suppose that for every holomorphic map γ : D → Cn with γ(0) = 0 one can ﬁnd a positive number Cγ such that |f ◦γ |≤Cγ |g ◦γ | holds on a neighborhood of 0 ∈ D. Then there exists a constant C such that |f |≤C|g| holds on a neighborhood of 0 ∈ Cn. Proof Let (A, 0) be the germ of an analytic set in (Cn+k, 0) deﬁned by

~~gj (z) = f(z)zn+j , 1 ≤ j ≤ k.~~

If one could not find C, there would exist a sequence pμ converging to the origin such that f(pμ) = 0 and lim |g(pμ)|/|f(pμ)|=0. Then, taking a germ of a holomorphic map from (C, 0) to (Cn+r , 0) whose image is contained in (A, 0) but not in f −1(0), one has a holomorphic curve as the projection to the first n factors, which contradicts the assumption. ∂f ∂f Proof of Corollary 3.5 Let g = (z ,...,zn ). By Lemma 3.4 and the chain 1 ∂z1 ∂zn rule for differentiation, it is easy to see that |f |≤C|g| holds on a neighborhood of 0. Hence the conclusion follows from Corollary 3.2, since − |g| dλ < ∞ U for sufficiently small for a sufficiently small neighborhood U 0iff is any nonzero element of C{z} without constant term. Remark 3.6 A connection between Corollary 3.5 and a topological theory of isolated hypersurface singularities was suggested by E. Brieskorn and established ∂f ∂f by J. Scherk [Sch]. The ideal Jf generated by ,..., is called the Jacobian ∂z1 ∂zn ideal of f . Jf plays an important role in the theory of period mappings (cf. [Gri-1]). As for Theorem 3.16, it was extended by Demailly [Dm-8] to a result on I ⊂ I −n multiplier ideal sheaves. A weak form of it says that a a for any singular Hermitian line bundle (B, a) on a complex manifold of dimension n, and a strengthened version established in [Dm-E-L] says that 142 3 L2 Oka–Cartan Theory

I ⊂ I · I a1a2 a1 a2 for any (B1,a1) and (B2,a2) (subadditivity theorem). Since results of this kind have applications to algebraic geometry, there have been subsequent developments in the theory of multiplier ideal sheaves. In the next subsection, we shall review some of them which are related to the L2 theory.

~~3.3.2 Nadel’s Coherence Theorem~~

Before describing the results on the multiplier ideal sheaves, let us present the most basic result by Nadel [Nd]. It is the coherence of multiplier ideal sheaves. Concerning the vanishing theorems for the ∂¯-cohomology, global theorems look similar to local theorems from the L2 viewpoint, since the geometric conditions needed are positivity of the bundle metric and (complete) Kählerianity of the base. On the other hand, in the finite-dimensionality theorems, geometry is involved in a subtler way. (Recall Theorems 2.32 and 2.36, for instance.) Nadel’s coherence theorem is a local theorem attached to his vanishing theorem (cf. Theorem 2.24). Theorem 3.20 (Nadel’s coherence theorem [Nd]) For any singular fiber metric a of a holomorphic line bundle B over a complex manifold M, Ia is a coherent ideal sheaf of OM . Proof Since the assertion is local, we may assume that M is a bounded Stein domain in Cn and a = e−ϕ for some plurisubharmonic function ϕ.LetI denote the ideal sheaf generated by the global sections of Ia. Since the ideal sheaves generated by finitely many global sections of Ia are coherent (Oka’s coherence theorem), and since C{z} is a Noetherian ring, I is coherent. Therefore it remains to show that ∼ Ia,x = Ix holds for any x ∈ M. Since OM,x(= C{z}) is Noetherian, by the intersection theorem of Krull it suffices to show that

~~I + I ∩ k = I x a,x mx a,x (3.27) for every k ∈ N (cf. [Ng, Chapter 1, Theorem 3.11]). But (3.27) is obtained immediately by applying Theorem 2.24. ~~

~~3.3.3 Miscellanea on Multiplier Ideal Sheaves~~

Since the results on the multiplier ideal sheaves are all local in this subsection, we shall consider only trivial line bundles over complex manifolds and denote the sheaves Ie−ϕ by I (ϕ) for simplicity. 3.3 L2 Approaches to Analytic Ideals 143

A striking variant of Briançon–Skoda’s theorem is a subadditivity theorem due to Demailly, Ein and Lazarsfeld [Dm-E-L]. It is obtained by combining the following two basic formulae. Restriction Formula Let M be a complex manifold, let ϕ be a plurisubharmonic function on M, and let S be a closed complex submanifold of M.Then

~~I (ϕ|S) ⊂ I (ϕ)|S.~~

~~Proof A direct consequence of Theorem 3.3. ~~

~~Addition Formula Let M1,M2 be complex manifolds, πj : M1 ×M2 → Mj ,j= 1, 2 the projections, and let ϕj be a plurisubharmonic function on Mj . Then~~

~~I ◦ + ◦ = ∗I · ∗I (ϕ1 π1 ϕ2 π2) π1 (ϕ1) π2 (ϕ2).~~

Proof It sufﬁces to show the assertion when Mj are bounded Stein domains in complex number spaces. Consider the ideal sheaf, say J generated by global ∗I · ∗I sections of π1 (ϕ1) π2 (ϕ2). By Fubini’s theorem, it is easy to see that the orthogonal complement of the subspace of the square integrable sections of J ∗I · ∗I consisting of the square integrable sections of π1 (ϕ1) π2 (ϕ2) is 0. Hence, similarly to the proof of Theorem 3.14, one has the asserted equality in the sheaf level. Theorem 3.21 (Subadditivity theorem) Let M be a complex manifold and let ϕ,ψ be plurisubharmonic functions on M. Then

~~I (ϕ + ψ) ⊂ I (ϕ) · I (ψ)~~

Proof Applying the addition formula to M1 = M2 = M and the restriction formula to S = the diagonal of M × M, one has I (ϕ + ψ) = I ((ϕ ◦ π1 + ψ ◦ π2)|S) ⊂ I ◦ + ◦ | = ∗I · ∗I | = I · I ( (ϕ π1 ψ π2)) S (π1 (ϕ) π2 (ψ)) S (ϕ) (ψ). Since I t makes sense for any ideal I ⊂ C{z} and any nonnegative real number t, it is natural to ask whether or not the subadditivity theorem can be generalized to

~~I (tϕ) ⊂ I (ϕ)t .~~

Prior to [Dm-E-L], in the study of an invariant closely related to the existence of a Kähler–Einstein metric on a complex manifold M, Demailly and Kollár [Dm-K] raised a question on the sheaf I+(ϕ) := I ((1 + )ϕ). >0 144 3 L2 Oka–Cartan Theory

~~Openness Conjecture Assume that I (ϕ) = OM . Then~~

~~I+(ϕ) = I (ϕ).~~

It is easy to see that the above extension of subadditivity theorem will follow in the right sense if the openness conjecture is true. Besides this, the question is undoubtedly of a basic nature. In [Dm-K], quantities of central interest are the log canonical threshold and the complex singularity exponent of plurisubharmonic functions. Given a plurisubharmonic function ϕ,thelog canonical threshold cϕ of ϕ at a point z0 is deﬁned as

~~−2cϕ 1 cϕ(z0) = sup {c>0; e is L on a neighborhood of z0}∈(0, +∞].~~

~~For any compact set K ⊂ M,thecomplex singularity exponent cK (ϕ) is deﬁned as~~

~~−2cϕ 1 cK (ϕ) = sup {c; e is L on a neighborhood of K}.~~

For the two-dimensional case, the openness conjecture was proved by Favre and Jonsson in [F-J]. For arbitrary dimension it has been reduced to a purely algebraic statement by Jonsson and Mustata(cf.[˘ J-M]). In [Brd’13], Berndtsson solved the openness conjecture afﬁrmatively by using symmetrization of plurisubharmonic functions. The strong openness conjecture which implies the openness conjecture was posed by Demailly in [Dm-8, Dm-9]. It is stated as follows. Strong Openness Conjecture. For any plurisubharmonic function ϕ on M, one has

~~I+(ϕ) = I (ϕ).~~

~~The strong openness conjecture was solved by Guan and Zhou [G-Z-2]by applying Theorem 3.3. A related theorem for the weighted log canonical threshold~~

2 −2cϕ 1 cϕ,f (z0) = sup {c>0;|f | e is L on a neighborhood of z0} for a holomorphic function f was obtained in [G-Z-3]. Its effective version was proved by Hiep [Hp] by combining Theorem 3.3 with a generalization of the Weierstrass division theorem due to Hironaka (cf. [H-U]). We shall follow Hiep’s proof below. In [Hp], the main theorem is stated as follows. Theorem 3.22 Let f be a holomorphic function on an open set Ω in Cn and let ϕ ∈ PSH(Ω). −2cϕ (i) (“Semicontinuity theorem”) Assume that e dλ < ∞ on some open Ω subset Ω ⊂ Ω and let z0 ∈ Ω . Then, for any ψ ∈ PSH(Ω ), there exists 3.3 L2 Approaches to Analytic Ideals 145

= − ≤ δ δ(c,ϕ,Ω ,z0)>0 such that ψ ϕ L1(Ω ) δ implies cψ (z0)>c. Moreover, as ψ converges to ϕ in L1(Ω ), the function e−2cψ converges to − e 2cϕ in L1 on every relatively compact open subset Ω of Ω . | |2 −2cϕ ∞ (ii) (“Strong effective openness”) Assume that Ω f e dλ < on some open subset Ω ⊂ Ω. When ψ ∈ PSH(Ω ) converges to ϕ in L1(Ω ) with ψ ≤ ϕ, the function |f |2e−2cψ converges to |f |2e−2cϕ in L1 norm on every relatively compact open subset of Ω. Corollary 3.6 (“Strong openness”) For any plurisubharmonic function ϕ on a n neighborhood of a point z0 ∈ C ,theset

2 −2cϕ 1 {c>0;|f | e is L on a neighborhood of z0 } is an open interval (0,cϕ,f (z0)). Corollary 3.7 (“Convergence from below”) If ψ ≤ ϕ converges to ϕ in a n neighborhood of z0 ∈ C , then cψ,f (z0) ≤ cϕ,f (z0) converges to cϕ,f (z0). The proof is done by induction on n which is run using Hironaka’s division theorem and the L2 extension theorem (Theorem 3.3) as machinery. To state Hironaka’s division theorem, we ﬁrst make C{z} an ordered set. The homogeneous α = α1 ··· αn α1 ··· αn lexicographical order of monomials z z1 z means that z1 z < β1 ··· βn | |= + ··· + | |= + ··· + z1 z if and only if α α1 αn < β β1 βn or |α|=|β| and αj <βj for the ﬁrst index j with αj = βj . Then, for each = α1 + α2 +··· C{ } = ≥ α1 α2 ··· f aα1 z aα2 z in z with aαj 0, j 1 and z

~~= IC (f ) aα1 , 1 IM (f ) = zα and~~

~~= α1 IT (f ) aα1 z , and the support of f by~~

~~1 2 SUPP(f ) ={zα ,zα ,...}.~~

For any ideal I ⊂ C{z},IM(I ) will denote the ideal generated by {IM(f ); f ∈ I }. Hironaka’s Division Theorem (cf. [G, By, B-M-1, B-M-2, Eb]. See also [H-U].) Let f, g1,...,gk ∈ C{z}. Then there exist h1,...,hk,s ∈ C{z} such that f = h1g1 +···+hkgk + s, and 146 3 L2 Oka–Cartan Theory

~~SUPP (s) ∩!IM (g1),...,IM(gk)"=∅, where !IM(g1),...,IM(gk)" denotes the ideal generated by the family (IM(g1),...,IM(gk)).~~

Standard basis Let I be an ideal of C{z} and let g1,...,gk ∈ I be such that IM(I ) =!IM(g1), . . . , IM(gk)". Then, by Hironaka’s division theorem it is easy to see that gj s are generators of I . One may choose such gj s in such a way that IM(g1)

n ={ ∈ Cn;| | } ΔR z zj 0 such that for some c>0 − |f(z)|2e 2cϕ(z) dλ < ∞. n ΔR ≤ ≥ n Let ψμ 0,μ 1, be a sequence of plurisubharmonic functions on ΔR with → 1 n = ≤ ψμ ϕ in Lloc(ΔR), and assume that either f 1 identically or ψμ ϕ for all ≥ ∈ 1 ] ∈ \{ } μ 1. Then for every rcand a sequence of holomorphic functions Fμ on Δr , μ ≥ μ0, such that IM(Fμ) ≤ IM(f), α Fμ(z) = f(z)+ (zn − wn) aμ,αz

−|α| n with |wn||aμ,α|≤r for all α ∈ N , and 2 2 −2cψ˜ μ(z) |Fμ(z)| e dλ ≤ < ∞ n |w |2 ΔR n for all μ ≥ μ0. Moreover, one can choose wn in a set of positive measure in the punctured disc Δ \{0}. 3.3 L2 Approaches to Analytic Ideals 147

~~Proof By Fubini’s theorem, the function~~

2 −2cϕ(z ,zn) |f(z,zn)| e dλz n−1 ΔR in the variable zn is integrable on ΔR. Therefore, for any η>0 and 0 > 0, one can ﬁnd wn ∈ Δη \{0} of positive measure such that 2 2 −2cϕ(z ,wn) 0 |f(z,wn)| e dλz < . n−1 |w |2 ΔR n

~~Since Theorem 3.19 is assumed to hold for n − 1, for any ρcsuch that 2 2 −2cψ˜ μ(z ,wn) 0 |f(z,wn)| e dλz < n−1 | |2 Δρ wn~~

2 for all μ ≥ μ0. Hence, by extending f(z,wn) with the L estimate, one has a n−1 × = holomorphic function Fμ on Δρ ΔR such that Fμ(z ,wn) f(z,wn) for all ∈ n−1 z Δρ , and − ˜ |F (z)|2e 2cψμ(z) dλ − μ z Δn 1×Δ ρ R

~~2 2 −2cψ˜ μ(z ,wn) ≤ CnR |f(z,wn)| e dλz n−1 Δρ C R22 ≤ n 0 2 , |wn|~~

2 where Cn is a constant which depends on n. Since |Fμ(z)| is plurisubharmonic, one has 1 |F (z)|2 ≤ |F |2 dλ μ n −| | 2 ··· −| | 2 μ z π (ρ z1 ) (ρ zn ) −| | ×···× −| | Δρ z1 (z1) Δρ zn (zn) C R22 ≤ n 0 n 2 2 2 , π (ρ −|z1|) ···(ρ −|zn|) |wn| where Δρ(z) denotes the disc of radius ρ centered at z. = 1 + Hence, for any r

~~1 n 2 2 Cn R0 ∞ ≤ Fμ L (Δn) n . (3.28) r n π 2 (R − r) |wn| 148 3 L2 Oka–Cartan Theory = α n−1 × Let gμ(z) α∈Nn aμ,αz be functions on Δr ΔR satisfying~~

~~Fμ(z) = f(z)+ (zn − wn)gμ(z).~~

~~Then, by (3.28) one has~~

|α| |wn||aμ,α|r ≤ C 0 for some constant C depending only on n, r, R and f . This yields the required estimates for 0 := C with C sufﬁciently small. As for the inequality IM(Fμ) ≤ IM(f), they are achieved since one may take

|α| |wn||aμ,α|r ≤ and arbitrarily small. Before going to the proof of Theorem 3.19, Let us note that the L1 convergence of ψ to ϕ implies that ψ → ϕ almost everywhere, and that the assumptions guarantee that ϕ and ψ are uniformly bounded on every relatively compact subset of Ω . In particular, after shrinking Ω and substracting constants if necessary, we may assume that ϕ ≤ 0onΩ. Since the L1 topology is metrizable, we may eventually restrict ourselves to a nonpositive sequence (ψμ)μ≥1 almost everywhere converging to ϕ in L1(Ω ). It sufﬁces to show (i) and (ii) for some neighborhood of a given ∈ = n n ⊂ point z0 Ω . For simplicity we assume z0 0 and ΔR such that ΔR Ω .In · → · 1 n−1 this situation, ψ( ,zn) ϕ( ,zn) in the topology of L (ΔR ) for almost every zn ∈ ΔR. Proof of statement (i) By Lemma 3.5 with f = 1, for every r0, there ∈ \{ } ˜ n exist wn Δ 0 ,μ0, c>cand a sequence of holomorphic functions Fμ on Δr , α −|α| μ ≥ μ0, such that Fμ(z) = 1 + (zn − wn) aμ,αz , |wn||aμ,α|r ≤ and 2 2 −2cψ˜ μ(z) |Fμ(z)| e dλz ≤ n |w |2 Δr n for all μ ≥ μ0. ≤ 1 | |=| − |≥ 1 ≥˜ Choosing 2 , one has Fμ(0) 1 wnaμ,0 2 so that cψμ (0) c>cand the ﬁrst part of (i) is proved. The second assertion of (i) follows from the estimate 3.3 L2 Approaches to Analytic Ideals 149 −2cψμ −2cϕ |e − e | dλz

~~Ω −2cψμ −2cϕ ≤ |e − e | dλz Ω ∩{|ψ |≤A} μ −2cϕ −2(c˜−c)A −2cψ˜ μ + e dλz + e e dλz,~~

~~Ω ∩{|ψμ|<−A} Ω ∩{|ψμ|<−A}~~

~~since ψμ → ϕ almost everywhere on Ω . ~~

Proof of statement (ii) Take f1,...,fk ∈ C{z} so that (f1,...,fk) is a standard I ··· n basis of (cϕ)0 with IM (f1)< < IM (fk), and take a polydisc ΔR in such a way that 2 −2cϕ(z) |f (z)| e dλz < ∞, = 1,...,k. n ΔR

Similarly to the above, one has Fμ, ∈ I (cψ˜ μ)0 ⊂ I (cϕ)0 from f . Namely, by ∈ \{ } = Lemma 3.5, for every rcand a sequence of holomorphic functions Fμ, on Δr ,μ μ0, such that α −|α| Fμ, (z) = f + (zn − wn, ) aμ, ,αz , |wn, ||aμ, ,α|r ≤ and 2 2 −2cψ˜ μ(z) |Fμ, (z)| e dλ ≤ (3.29) n |w |2 ΔR n, for all = 1,...,k and μ ≥ μ0. Since ψμ ≤ ϕ and c>c˜ ,wehaveFμ, ∈ I (cψ˜ μ)0 ⊂ I (cϕ)0. The next step of the proof is to modify (Fμ, )1≤ ≤k into a standard basis of I (cϕ)0. By virtue of (3.29) and Cauchy’s estimate, by taking & & ··· & ∈ \{ } 1 2 k and suitable wn, Δ 0 , one can inductively ﬁnd ≤ ≤ Fμ, and polynomials Pμ, ,m for 1 m< k possessing uniformly bounded coefﬁcients and degrees, such that the linear combinations = − Fμ, Fμ, Pμ, ,mFμ,m 1≤m≤ −1

|IC (F )| satisfy IM (F ) = IM (f ) and μ, ∈ ( 1 , 2) for all and μ & 1. In this way μ, |IC (f )| 2 I one ﬁnds a sequence (Fμ,1,...,Fμ,k) of standard bases of (cϕ)0. This procedure is elementary but long, so that the reader is referred to [Hp] for the detail. Then, by the previleged neighborhood theorem of Siu (cf. [Siu-1]), one can ﬁnd ρ,K > 0 n with ρ

~~= + +···+ n f hμ,1Fμ,1 hμ,2Fμ,2 hμ,kFμ,k on Δρ~~

~~ ∞ n ≤ ∞ n and hμ, L (Δρ ) K f L (Δr ) for all .By(3.29) this implies a uniform bound 2 −2cψ˜ μ(z) |f (z)| e dλz ≤ M<∞ n Δρ~~

1 2 −2cψ 2 −2cϕ for some c>c˜ and all μ ≥ μ0.TheL convergence of |f | e μ to |f | e is similar to the last part of the proof of statement (i). Guan–Zhou’s solution of the strong openness conjecture is by induction on the dimension n. It is done ﬁrst for the case n = 1 and using Theorem 2.86.This part will be of special interest in the present context so it will be reviewed below following [G-Z-2]. Lemma 3.6 For any f ∈ O(D) \{0}, there exist constants C>0 and 0 C|a| D holds. m Proof Let m and 0 0.

~~It sufﬁces to show that one can ﬁnd C1 > 0 and 0~~

~~a ∈ rΔ,ha ∈ A(Δ), ha(a) = 1,ha(0) imply m 2 2 −2 |z | |ha| dλ > C1|a| r Δ~~

~~For that, we put~~

~~∞ j ha(z) = cj z j=1 to obtain ∞ 2j+2m+2 m 2 2 2 r |z | |ha| dλ = 2π |cj | . 2j + 2m + 2 r Δ j=1 3.3 L2 Approaches to Analytic Ideals 151 ∞ j = Since j=1 cj a 1 by assumption, we have~~

~~∞ ∞ r 2j+2m+2 2j + 2m + 2 −1 |c |2 ≥ |a|2j j 2j + 2m + 2 r 2j+2m+2 j=1 j=1 by the Cauchy–Schwarz inequality. Since~~

~~∞ 2j + 2m + 2 a 2 (2m + 2)r −2m−2 2r −2m−2 | |2j = + 2j+2m+2 a a 2 a 2 2 r r 1 −| | (1 −| | ) j=1 r r~~

~~holds, C1 as above exists if 0 < 2r~~

We want to show that for some 0 1 it holds that | |2 −pϕ ∞ rD F e dλ < . Clearly, it sufﬁces to show that by assuming ϕ<0 and −1 F (0) = 0. By (3.30), there exists a sequence aj converging in D to 0 such that, ϕ(aj ) =−∞and

~~2 2 −ϕ(aj ) lim |aj | |F(aj )| e = 0. j→∞~~

~~Therefore a sequence pj > 1 can be chosen so that~~

2 2 −pj ϕ(aj ) lim |aj | |F(aj )| e = 0 j→∞ is satisﬁed. By Theorem 2.86, there exists a constant C>0 such that, for each j one can ﬁnd Fj ∈ A(D) such that Fj (aj ) = F(aj ) and 2 −pj ϕ 2 −pj ϕ(aj ) |Fj | e dλ ≤ C|F(aj )| e . (3.31) Δ

Taking pj closer to 1 if necessary, we may assume that 2 −pj ϕ 2 −ϕ(aj ) |Fj | e dλ ≤ 2C|F(aj )| e D is satisﬁed. Since ϕ<0, By (3.30) and (3.31), 2 2 lim |aj | |Fj | dλ = 0. (3.32) j→∞ D 152 3 L2 Oka–Cartan Theory

We are going to show that if − |F |2e pj ϕ dλ =∞ (3.33) rD were true for all 0 0 j→∞ rD must hold, but this contradicts (3.31). The solution of the strong openness conjecture entails a basic result on Lelong numbers which measures the singularity of plurisubharmonic functions. Let Ω be any open set in Cn. Deﬁnition 3.5 Given ϕ ∈ PSH (Ω) and x ∈ Ω,theLelong number of ϕ at x is deﬁned as ϕ(z) supBn ϕ ν(ϕ,x) := lim inf ;z − x

~~Example 3.5~~

~~m 2 n ν log |zk| , 0 = 2 (z ∈ C and m ≤ n) k=1 and ∞ − 1 1 − ϕ(z) = 2 k log z − (z ∈ C) ⇒ ν ϕ, = 2 k. k k k=1~~

Skoda’s L2 division theorem implies the following. Theorem 3.23 (cf. [Sk-1]) If ν(ϕ,x) < 2 then e−ϕ is integrable in a neighborhood of x. Recently it was reﬁned to: Theorem 3.24 (cf. [F-J](forn = 2) and [G-Z-4] (in general)) If ν(ϕ,x) = 2 and ({z; ν(ϕ,z) ≥ 1},x) is not a germ of regular hypersurface, then e−ϕ is integrable on a neighborhood of x. Lelong numbers are related to the multiplier ideal sheaves as follows. 3.4 Notes and Remarks 153

Theorem 3.25 (cf. [B-F-J] and [G-Z-2, G-Z-3, G-Z-5]) For any ϕ,ψ ∈ PSH (Ω) and x ∈ Ω, the following are equivalent: : → Cn | (1) For any proper holomorphic map π X such that π X\π−1 (0) is a local homeomorphism and for any p ∈ π −1(x), ν(ϕ ◦ π, p) = ν(ψ ◦ π, p) holds true. (2) I (tϕ) = I (tψ) for all t>0.

~~3.4 Notes and Remarks~~

Theorem 3.3 was ﬁrst established by Błocki [Bł-2] in the following form. Theorem 3.26 Let D be a bounded domain in C containing 0 and let Ω ⊂ Cn−1 × D be a pseudoconvex domain. Then for any holomorphic function f on Ω := Ω ∩{zn = 0} and for any ϕ ∈ PSH(Ω), there exists a holomorphic extension F of f to Ω satisfying | |2 −ϕ ≤ π | |2 −ϕ F e 2 f e Ω (cD(0)) Ω .

~~Here cD is the logarithmic capacity on D deﬁned by cD(z) = exp lim (gD(ζ, z) − log |ζ − z|) , ζ→z where~~

gD(ζ, z) := sup{u(ζ ) | u ≤ 0, ∈ PSH, sup (u(ξ) − log |ξ − z|)<∞} ξ∈D\{z} (the Green function of D). ζ−z If D = D, gD(ζ, z) = log so that cD(0) = 1. Hence Theorem 3.3 is a 1−ζz corollary of Theorem 3.26. Błocki’s proof is based on Chen’s approach [Ch-2]of using Berndtsson-Charpentier’s technique [B-C’00] and refines a method in [G-Z- Z-1,2]. The proof presented in 3.1 is more in the spirit of [G-Z-1], but the analysis is essentially the same as in [Bł-2]. The main motivation to prove Theorem 2.86 was to explore what is left after the results of Hörmander [Hö-1] and Pflug [Pf]fortheBergman kernel (cf. Theorems 4.3 and 4.6 in Chap. 4). In [Oh’84] the method of [Oh-2] was applied to obtain a growth estimate for the Bergman kernel of weakly pseudoconvex domains. It turned out that, for any bounded pseudoconvex domain D ⊂ Cn with C2-smooth boundary, the Bergman kernel kD and the distance δD to ∂D, one can find C > 0 for any >0 such that 154 3 L2 Oka–Cartan Theory

~~−1 −2+ −n−1 C δD(z)~~

~~= {| |2; 2 = ∈ 0,0 } kD(z) sup f(z) f 1,f H(2) (D)~~

(cf. (4.7) in Chap. 4), it is straightforward from Theorem 2.86 that (3.34) with = 0 holds if ∂D is Lipschitz continuous in the sense that ∂D is locally the graph of Lipschitz continuous function of 2n − 1 real variables. After establishing Theorem 2.86 in [Oh-T-1] and succeeding in removing from (3.34), the author has been trying to generalize the result to extend its applications. Some of the materials related to such activity will be reviewed below. As was mentioned in Sect. 2.2.5, Skoda had already applied the L2 method to solve a division problem with growth conditions (cf. Theorems 3.11 and 3.15). Skoda’s theory had applications to commutative algebra (cf. Theorem 3.16) and stimulated the introduction of an important notion of tight closure (see [L-T] and [H-S’06]). This was a big encouragement because there is an intimate relation between the division problem and the extension problem since the theory of Weierstrass and Mittag–Lefﬂer. More recently, this connection can be seen in Carleson’s solution [C’62] of Kakutani’s corona problem, a deep result of harmonic analysis in one complex variable. Carleson established that, given any ideal I of ∞ D the ring H of bounded holomorphic functions on generated by f1,...,fn, = ∞ n | | 1 I H holds if and only if infz∈D j=1 fj (z) > 0. In order to solve the corona problem, Carleson had to settle an extension problem in the following way.

~~Theorem 3.27 (cf. [C’58]) Let zk ∈ D (k = 1, 2,...)be a sequence satisfying zk − zj inf > 0. k 1 − z z j =k j k~~

∞ N Then the image of the map ρ : H → C deﬁned by ρ(f )(k) = f(zk) coincides ∞ ={ ∈ CN; | | ∞} with c supk c(k) < . Theorem 3.26 was extended to a characterization of images of the restriction p p maps from H ={f ∈ O(D); D |f | dλ < ∞} for 0

~~1 T.Wolff gave an L2 proof to this theorem (cf. [K’80]and[G’80]). 3.4 Notes and Remarks 155~~

˜ n ˜ ˜ function f on X does there exist f ∈ O(C ) satisfying f |X = f such that |f(z)|≤ n C1exp(C2p(z)) holds for all z ∈ C for some constants C1 and C2? Secondly, under which condition on X is it true that for every holomorphic function f on X satisfying |f(z)|≤Aexp(Bp(z)), z ∈ X, there exists f˜ ∈ O(Cn) satisfying the above mentioned requirements? That the latter holds if X is algebraic follows from results of Ehrenpreis [E’70, chap.4] and Palamodov [P’70, chap.4]. As a result closer in the spirit of Carleson’s interpolation theorem for H p, Beat- rous [B’85] proved an interpolation theorem on strongly pseudoconvex domains by applying the method of integral kernels. Let D ⊂ Cn be a bounded strongly pseudoconvex domain, let S be a closed m-dimensional complex submanifold of a neighborhood of D which intersects ∂D transversally and let D = S ∩ D.For − p = O ∩ p s = = s> 1,p>0, let As (X) (X) L (X, δX dλ) for X D or X D , p p where δX(z) = dist(z, ∂X) and L (·, ·) denotes the Lebesgue space of L functions. p Further, let A−1(X) be the space of all holomorphic functions on X with boundary p p values in L (∂X). Put A(X) = As (X) (s ≥−1,p >0). Theorem 3.28 (cf. [B’85]) In the above situation the following hold. p (a) The restriction map R : O(D) → O(D ) maps As (D) continuously to P An−m+s(D ). (b) There exists a linear map

~~L : A(D ) −→ A(D)~~

p p = which maps An−m+s(D ) continuously into As (D) such that R(L(f)) f holds for all f . The proof of Theorem 3.28 consists of finding a current T satisfying fT = 1in terms of the integral kernel of Henkin [H’70] and Ramirez [R’70]. The L2 method can be applied to extend Theorem 3.27.LetD be as above, let ϕ ∈ PSH(D) ∩ C0(D),letH be a smooth hypersurface in a neighborhood of D p which intersects ∂D transversally, and let Aα,ϕ be the space of f ∈ O(D) satisfying − − e ϕδα|f |pdλ < ∞ resp. lim (α + 1) e ϕδα|f |pdλ < ∞ D α −1 D for α>−1 (resp.α =−1) and p>0. : 2 ∩ Theorem 3.29 (cf. [Oh’03]) There exists a continuous linear map L A0,ϕ(D → 2 | = H) A1,ϕ(D) such that L(f ) D∩H f for all f . Between [Oh-T-1] and [Oh’03], the author experienced a short period of excitement around 1993, when he found a partial answer to Suita’s conjecture in [Oh-16, Addendum] (see Chap. 4 for the detail of this problem). This discovery was greatly influenced by [Sp-1, Sp-2] and [S-W], which suggested the author what to expect beyond (3.34). The following passage in the introduction to [Oh’94] reflects such a situation. 156 3 L2 Oka–Cartan Theory

Usually the conditions of interpolatability have been described in terms of highly analytic terms, like the convergence of a sequence associated to the Weierstrass canonical product or that of the Blaschke product. Generalizations to the several variables seem to have been done in the same vein. Nevertheless, the recent works of Seip and Wallstén show a remarkable similarity with our previous results. Namely, they explored a more geometric concept of density to solve certain interpolation problems, and the density is nothing but the curvature in many situations, not to mention Einstein’s theory of gravity. The manuscript of [Oh’94] was written while the author was staying at Harvard university hosted by Yum Tong Siu. Einstein’s name is here probably because the author had a chance to talk in the seminar of Shing Tung Yau. A primitive form of Theorem 3.5 was formulated in [Oh’94]. Since it may have some instructive nature, let us present it here. Let M be a Stein manifold equipped with a positive Radon measure dμM . 2 By A (M, dμM ) we denote the set of holomorphic functions f on M satisfying 2 := | |2 ∞ ⊂ f M f dμM < .LetS M be a closed complex submanifold and let dμS be a positive Radon measure on S.Wesay(S, dμS) is a set of interpolation for 2 A (M, dμM ) if there exists a bounded linear operator

2 2 I : A (S, dμS) −→ A (M, dμM ) satisfying I(f)|S = f for all f .Letdk(z, S) denote the distance from z ∈ M to the union of k-dimensional components of S, measured by any ﬁxed Hermitian metric on M. A continuous function g : M →[−∞, ∞) will be said to be a polarization function of S if g is of class C2 outside g−1(−∞) and the function

n−1 g(z) − 2 (n − k)log dk(z, S) k=0 is bounded on the compact subsets of M. Clearly the concept of the polarization function does not depend on the metric. The set of polarization functions of S will be denoted by Π(S). We put

~~Πb(S) ={g ∈ Π(S); sup g<∞}. M~~

~~For any g ∈ Π(S) and for any volume form dVM on M,letdVM [g] be the minimal element of the set of positive Radon measures dμ on S satisfying ≥ −g fdμ lim sup fe χ{−t−1~~

Theorem 3.30 Let M be a Stein manifold equipped with a volume form dVM of class C2 and let S be a closed complex submanifold of M equipped with a positive 2 Radon measure dμ. Then (S, dμ) is a set of interpolation for A (M, dVM ) if there exists a g ∈ Πb(S) such that dVM [g]≤dμ and ¯ {; i(∂∂(1 + )g + κM ) ≥ 0 on M \ S}⊃[0,0) for some 0 > 0.HereκM denotes the curvature form of dVM . Another moment of excitement came when Siu gave a talk on [A-S] in March 1995 at the first Hayama conference in complex analysis of several variables (cf. [Siu’96]). He gave an elegant application of Theorem 2.86 to the Fujita conjecture and gave an alternate proof with a better constant. In the same workshop the author gave a survey talk presenting five types of applications of Theorem 2.86 and its variants. However, BEDFORD remarked that there are too many theorems in the talk (cf. [Oh’96-2]). The author must admit that an application to Teichmüller spaces might have been superfluous. In [Oh’96-1] it was shown by the L2 method that Teichmüller spaces of Riemann surfaces can be biholomorphically nonequivalent even if they are infinite dimensional. This answered a question raised by S. Mukai (personally to the author), but Earle and Gardiner [E-G’96], renowned experts of Teichmüller theory, subsequently gave much better treatment of this problem. It must be noted that one receives various error messages when one wants to extend Theorem 2.86 to other situations. The following example is in [Oh’05]. 2 2 Example 3.6 Let D = B (= B (0, 1)),letS ={(z, w) ∈ D; zw = 0} and let ϕk be a decreasing sequence of C∞ plurisubharmonic functions on D converging poitwise to log |z − w|2. Then there exists no universal constant C such that, for any k and − for any holomorphic function f on S = S \{(0, 0)} such that e ϕk |f |2 < ∞, 0 S0 there exists a holomorphic extension fk of f to D satisfying −ϕk 2 −ϕk 2 e |fk| ≤ C e |f | . D S0

Here the integrals are with respect to the Lebesgue measures. Indeed, if one takes the function z(z − w)/(z + w) as f , a subsequence of such extensions fk would converge to a holomorphic function f˜ on D satisfying f˜ (z, z) = 0, f(z,˜ 0) = z and f(˜ 0,w)= 0, which is clearly impossible. This example shows that Theorem 2.86 has no simple-minded generalization to complex spaces with singularities, unlike the case of Oka–Cartan theory. Absence of the universal constant in the L2 extension has been more clearly shown by the following result of Guan and Li [G-L’18]. It is remarkable that Skoda’s L2 division theorem (Theorem 3.14 and Corollary 3.3) and the solution of Demailly’s strong openness conjecture (cf. Theorem 3.22) are combined in the proof. 158 3 L2 Oka–Cartan Theory

Theorem 3.31 Let Ω ⊂ Cn (n ≥ 2) be a domain, A ⊂ Ω an analytic set through the origin 0. Then, for sufﬁciently small balls Bn(0,r) ⊂ Ω,theL2 extension theorem holds for (Bn(0,r),A)if and only if 0 is a regular point of A. Proof It is enough to prove the necessity. Without loss of generality, we can assume 1 ≤ dim0 A ≤ d ≤ n − 1, and the germ(A, 0) is irreducible, because the reducible case is easy as in the above example. Suppose that 0 is a singular point of A. Then, by applying the Weierstrass preparation theorem it is easy to see that there is a local coordinate system (z ; z ) = (z1,...,zd ; zd+1,...,zn) around 0 such that for some constant C>0wehavez ≤Cz for any z ∈ A near 0. Let I ⊂ OA,0 be the ideal generated by germs of holomorphic functions zˆ1,...,zˆd ∈ OA,0, where OA = OΩ /IA|A and zˆk are the residue classes of zk in OA,0. Since 0 is a singularity + ≤ ≤ ˆ ∈ I ≤ of A, there must exist d 1 k0 n such that zk0 / A. Since z C z ∈ | |≤ 2 ≤ + 2 2 ∩ for any z A near 0, zk0 C z and z (1 C ) z on U A for some neighborhood U of 0. On the other hand it is easy to verify that U can be chosen in such a way that −2(d−1) z dVA < ∞ U∩Areg = d | ! = i n ∧ where dVA ω Areg /d and ω 2 k=1 dzk dzk, so that | |2 −2d ≤ 2 + 2 d−1 −2(d−1) ∞ zk0 z dVA C (1 C ) z dVA < . U∩Areg U∩Areg

~~Thus, if the L2 extension theorem were true, there would exist a holomorphic ∈ O Bn | =ˆ function F ( (0,r))such that F A zk0 and 2 −2d |F | z dλn < ∞. Bn(0,r)~~

By Theorem 3.22, for sufﬁciently small >0 and smaller Bn(0,r)we have 2 −2(d+) |F | z dλn < ∞. Bn(0,r) ∈ Then we infer from Corollary3.3 that there exist holomorphic functions fk O Bn = d ˆ ∈ I ( (0,r)) such that F k=1 fkzk. By restricting to A we have zk0 , ˆ ∈ I which contradicts zk0 / . So the universal constant in the estimate for the L2 extension exists only for the extension from nonsingular sets. It must also be noted that the L2 extension can fail even for a ﬁxed weight if functions must be extended from a hypersurface with singularities (cf. [D-M’00]). In [G-Z-1] Theorem 3.3 was further extended as follows. 3.4 Notes and Remarks 159

Theorem 3.32 For any pseudoconvex domain D ⊂ Cn, for any ϕ ∈ PSH(D),for any >0 and for any holomorphic function f on D = D ∩{zn = 0}, there exists ˜ ˜ a holomorphic function f on D such that f |D = f and | ˜|2 f −ϕ ≤ π | |2 −ϕ 2 1+ e f e . D (1 +|zn| ) D

There is a generalized variant of Theorem 3.32 in [G-Z’17] containing the following. Theorem 3.33 (see also [Oh’17]) Let α>0. Then, in the situation of Theo- rem 3.32, every holomorphic function f on D extends to a holomorphic function f˜ on D satisfying − | |2− π − e α zn ϕ|f˜|2 ≤ e ϕ|f |2. D α D

Proof of Theorem 3.32 In Theorem 3.1 we put X = D, σ = zn, (B, b) = (D × 2 −1 2 −1− −ϕ −1 2 C,(1 +|zn| ) ), (E, h) = (D × C,(1 +|zn| ) e ) and Q = (1 +|zn| ). Then, with respect to t − − − s(t) = (1 + e t ) (1 + e x) dx, −∞ − u(t) = log (1 + e t ),

~~2 φ(z) =− log (1 +|zn| ) and~~

~~2 η(z) = s(log |zn| ), similarly as in the proof of Theorem 3.3 one has~~

~~¯ − ≥ − | 2 ∧ ηΘh ∂∂η (su s ) t=log |zn| dzn dzn~~

~~=− | 2 ∧ u s t=log |zn| dzn dzn.~~

The semipositivity of the last term follows from u < 0 (obvious) and s ≥ 0. The latter inequality holds because t t − − − − s = eu e u(x) dx = e2u (e u) e u(x) dx − e 2u , −∞ −∞ − − − − − − − K := ((e u) )2 − (e u) e u = e t (1 + e t ) 2 2 > 0 160 3 L2 Oka–Cartan Theory t −u(x) and because the function r = ∞ e dx satisﬁes r 2 − r r r(r − r r ) rK = = > 0. r r 2 r 2

~~− Hence we are allowed to set c(z) = v(log |z |2), where v = su s . Finally n s 2~~

~~2 − s 1 η + c 1 = s + = ( +|z |2), − 1 n su s t=log |zn|2 since~~

2 s s su − s 1 1 − s + = s − = =− = et (1 + e t ), su − s u u u from which the conclusion follows immediately. For the proof of Theorem 3.33, we put t t − u(t) = ee ,s(t)= e u(t) eu(x)dx −∞ and apply Theorem 3.1 similarly to the above. As for other generalizations and refinements of Theorem 2.86, the reader is referred to [M’93, P’05, M-V’07, M-V’17, B-L’16, H’17-1, H’17-2, C-D-M’17] and [Dem’17], for instance. Since some of them are based on a completely different technology, the detail will be reviewed in Chap. 4 after some preliminaries on the Bergman kernel. SKODA once told the author that his method explained in 3.2.2 is quite natural and satisfying. It was in 1984 when the author was trying to find what happens in Corollary 3.3 as → 0, with a hope to get some idea of proving Theorem 2.86. SKODA was right because the main ingredient of his theory is Lemma 3.3 which remained untouchable in the L2 extension theory and around which a new development has been taking place. It was combined in [V’08] with (3.4) to yield another kind of refined L2 division theorems. The following is one of the recent results in this direction. It extends Skoda’s theory by sacrificing the estimate of the solutions. Theorem 3.34 (cf. [K’16]) Let X be a projective algebraic manifold, let L → X 0 be a holomorphic line bundle and let g1,...,gp ∈ H (X, L) be holomorphic sections (p ≥ 1). Let H → X be another holomorphic line bundle and let a and b be singular fiber metrics on L and H, respectively, both with semipositive curvature current. Let q ≥ min (n, p − 1) be an integer, where n = dim X. If a holomorphic 0 ⊗(q+2) section f ∈ H (X, O(KX ⊗ H ⊗ L )) satisfies | ∧ | | |−2(q+1) ∞ f f baq+2 g a < , X References 161

0 ⊗(q+1) then there exist h1,...,hp ∈ H (X, O(KX ⊗ H ⊗ L )) such that f = g1h1 +···+gphp. Theorem 3.34 has a background in complex geometry similar to [F-O’87], for instance, where the following was proved. Theorem 3.35 Let X ⊂ Cn be an algebraic submanifold of pure codimension 2 such that the projection to the ﬁrst n − 2 coordinates p : X → Cn−2 is proper. Suppose that the canonical line bundle of X is topologically trivial. Then the ideal of X can be generated by two functions F1,F2 ∈ Rn−2[zn−1,zn], where m Rm stands for the ring of holomorphic functions f = f(z1,...,zm) on C P(|z |+···+|z |) m satisfying |f(z1,...,zM )|≤e 1 m for all (z1,...,zm) ∈ C for some real polynomial P depending on f . Concerning the multiplier ideal sheaves, Nadel’s coherence theorem has been generalized to the coherence of those for singular Hermitian vector bundles in a generalized sense (see [DC’98, Rf’15, L’14] and [I’18]). It was shown in [L-L’07] and [K’14] that not all integrally closed coherent ideal sheaves are the multiplier ideal sheaves. Further, Theorem 3.21 was nicely supplemented by the following. Theorem 3.36 (cf. [K’16], Proposition 2.1) Let X be a complex manifold and let ϕ,ψ ∈ PSH(X). Suppose that the Lelong number of ψ is zero at every point of X. Then we have I (ϕ) = I (ϕ + ψ). Results on the strong openness conjecture has become sharper in [G’17], where an inequality of the form 2 −t |F | ≥ e cF,ϕ(D) {ϕ<−t} plays an essential role. Here D is a pseudoconvex domain in Cn, F ∈ O(D), ϕ ∈ PSH(D), ϕ < 0 and cF,ϕ(D) ∈[0, ∞]. Recently, Demailly [Dem’17] extended Theorem 3.5 to holomorphically convex Kähler manifolds by using a technique of approximate solutions of ∂¯ in [Dm-9] (see also [CJ]).

~~References~~

[B’85] Beatrous, F.: Lp-estimates for extensions of holomorphic functions. Michigan Math. J. 32(3), 361–380 (1985) [B-S’89] Berenstein, C., Struppa, D.: Complex analysis and convolution equations, current problems in mathematics. Fundamental directions, vol. 54, pp. 5–111, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. (Moscow, 1989). Encyclopedia of Mathematical Sciences, vol. 54 Khenkin, G.M. (ed.) Several Complex Vaiables , pp. 1–108. Springer, Berlin/Heidelberg (1990) 162 3 L2 Oka–Cartan Theory

[B-T’80] Berenstein, C., Taylor, B.: Interpolation problems in Cn with applications to harmonic analysis. J. Anal. Math. 38, 188–254 (1980) [Brd’13] Berndtsson, B.: The openness conjecture for plurisubharmonic functions. arXiv: 1305.5781r1 [math.cv] [B-C’00] Berndtsson, B., Charpentier, P.: A Sobolev mapping property of the Bergman Kernel. Math. Z. 235(1), 1–10 (2000) [B-L’16] Berndtsson, B., Lempert, L.: A proof of the Ohsawa-Takegoshi theorem with sharp estimates. J. Math. Soc. Jpn. 68(4), 1461–1472 (2016) [C-D-M’17] Cao, J.-Y., Demailly, J.-P., Matsumura, S.: A general extension theorem for cohomology classes on non reduced analytic subspaces. Sci. China Math. 60(6), 949–962 (2017) [C’58] Carleson, L.: An interpolation problem for bounded analytic functions. Am. J. Math. 80, 921–930 (1958) [C’62] Carleson, L.: Interpolations by bounded analytic functions and the corona problem. Ann. Math. 76(2), 547–559 (1962) [Cl’07] Claudon, B.: Invariance for multiples of the twisted canonical bundle. Ann. Inst. Fourier (Grenoble) 57, 289–300 (2007) [DC’98] de Cataldo, M.A.: Singular hermitian metrics on vector bundles. J. Reine Angew. Math. 502, 93–122 (1998) [Dem’81] Demailly, J.-P.: Scindage holomorphe d’un morphisme de ﬁbrés vectoriels semi- positifs avec estimations L2, Seminar Pierre Lelong-Henri Skoda (Analysis), 1980/1981, and Colloquium at Wimereux, May 1981. Lecture Notes in Mathematics, vol. 919, pp. 77–107. Springer, Berlin/New York (1982) [Dem’17] Demailly, J.-P.: Extension of holomorphic functions and cohomology classes from non reduced analytic subvarieties. In: The 12th Korean Conference on Several Complex Variables (KSCV12), July 2017, Gyeong-Ju, [D-M’00] Diederich, K., Mazzilli, E.: A remark on the theorem of Ohsawa-Takegosh. Nagoya Math. J. 158, 185–189 (2000) [D’70] Duren, P.: Theory of H p spaces. In: Pure and Applied Mathematics, vol. 38, pp. xii+258. Academic, New York/London (1970) [E-G’96] Earle, C., Gardiner, F.: Geometric isomorphisms between inﬁnite-dimensional Teichmüller spaces. Trans. Am. Math. Soc. 348(3), 1163–1190 (1996) [E’70] Ehrenpreis, L.: Fourier analysis in several complex variables. In: Pure and Applied Mathematics, vol. XVII, pp. xiii+506. Wiley-Interscience Publishers, New York/London/Sydney (1970) [F-O’87] Forster, O., Ohsawa, T.: Complete intersections with growth conditions. In: Alge- braic Geometry (Sendai, 1985). Advanced Studies in Pure Mathematics, vol. 10, pp. 91–104. North-Holland, Amsterdam (1987) [G’80] Gamelin, T.W.: Wolff’s proof of the corona theorem. Israel J. Math. 37(1–2), 113– 119 (1980) [G’17] Guan, Q.-A.: A sharp effectiveness result of Demailly’s strong openness conjecture. arXiv: 1709.05880v4 [math.CV] [G-L’18] Guan, Q.-A., Li, Z.-Q.: A characterization of regular points by Ohsawa-Takegoshi extension theorem. J. Math. Soc. Jpn. 70(1), 403–408 (2018) [G-Z’17] Guan, Q.-A., Zhou, X.-Y.: Strong openness of multiplier ideal sheaves and optimal L2 extension. Sci. China Math. 60(6), 967–976 (2017) [H’70] Henkin, G.M.: Integral representation of functions in strongly pseudoconvex regions, and applications to the ∂¯-problem. Mat. Sb. (N.S.) 82(124), 300–308 (1970) [H’17-1] Hosono, G.: The optimal jet L2 extension of Ohsawa-Takegoshi type. arXiv: 1706.08725v1 [math.CV] [H’17-2] Hosono, G.: On sharper estimates of Ohsawa-Takegoshi L2-extension theorem. arXiv: 1708.08269v2 [math.CV] [H-S’06] Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336, p. xiv+431. Cambridge Univer- sity Press, Cambridge (2006) References 163

[I’18] Iwai, M.: Nadel-Nakano vanishing theorems of vector bundles with singular Hermi- tian metrics. arXiv: 1802.01794v1 [math.CV] [J’78] Jennane, B.: Extension d’une fonction définie sur une sous-variété avec contrôle de la croissance. Séminaire Pierre Lelong-Henri Skoda (Analyse), Année 1976/1977. Lecture Notes in Mathematics, vol. 694, pp. 126–133. Springer, Berlin (1978) [K’14] Kim, D.: The exactness of a general Skoda complex. Michigan Math. J. 63(1), 3–18 (2014) [K’16] Kim, D.: Skoda division of line bundle sections and pseudo-division. Int. J. Math. 27(5), 1650042, 12 pp. (2016) [K’80] Koosis, P.: Introduction to H p-Spaces. With an Appendix on Wolff’s Proof of the Corona Theorem. London Mathematical Society Lecture Note Series, vol. 40, p. xv+376. Cambridge University Press, Cambridge/New York (1980) [L-L’07] Lazarsfeld, R., Lee, K.: Local syzygies of multiplier ideals. Invent. Math. 167, 409– 418 (2007) [L’14] Lempert, L.: Modules of square integrable holomorphic germs. arXiv: 1404.0407v2 [math.CV] [M’93] Manivel, L.: Un théorème de prolongement L2 de sections holomorphes d’un fibré hermitien. Math. Z. 212(1), 107–122 (1993) [M-V’07] McNeal, J., Varolin, D.: Analytic inversion of adjunction: L2 extension theorems with gain. Ann. Inst. Fourier (Grenoble) 57(3), 703–718 (2007) [M-V’17] McNeal, J., Varolin, D.: Extension of jets with L2 estimates, and an application. arXiv: 1707.04483v2 [math.CV] [Nk’72] Nakamura, I.: On complex parallelisable manifolds and their small deformations. Proc. Jpn. Acad. 48, 447–449 (1972) [N’80] Nishimura, Y.: Problème d’extension dans la théorie des fonctions entières d’ordre fini. J. Math. Kyoto Univ. 20(4), 635–650 (1980) [Oh’84] Ohsawa, T.: Boundary behavior of the Bergman kernel function on pseudoconvex domains. Publ. Res. Inst. Math. Sci. 20(5), 897–902 (1984) [Oh’94] Ohsawa, T.: On the extension of L2 holomorphic functions. IV. A new density concept. In: Mabuchi, T., et al. (eds.) Geometry and Analysis on Complex Manifolds, pp. 157–170. World Scientific Publishing, River Edge (1994) [Oh’96-1] Ohsawa, T.: On the analytic structure of certain infinite-dimensional Teichmüller spaces. Nagoya Math. J. 141, 143–156 (1996) [Oh’96-2] Ohsawa, T.: Applications of L2 estimates and some complex geometry. In: Noguchi, J. (ed.) Geometric Complex Analysis (Hayama, 1995), pp. 505–523. World Scientific Publishing, River Edge (1996) [Oh’03] Ohsawa, T.: On the extension of L2 holomorphic functions. VI. A limiting case. In: Explorations in Complex and Riemannian Geometry. Contemporary Mathematics, vol. 332, pp. 235–239. American Mathematical Society, Providence (2003) [Oh’05] Ohsawa, T.: On a curvature condition that implies a cohomology injectivity theorem of Kollár-Skoda type. Publ. Res. Inst. Math. Sci. 41(3), 565–577 (2005) [Oh’17] Ohsawa, T.: On the extension of L2 holomorphic functions VIII–a remark on a theorem of Guan and Zhou. Internat. J. Math. 28(9), 1740005, 12 (2017) [P’70] Palamodov, V.: Linear differential operators with constant coefficients. Translated from the Russian by Brown, A.A. Die Grundlehren der mathematischen Wis- senschaften, Band 168, p. viii+444. Springer, New York/Berlin (1970) [P’05] Popovici, D.: L2 extension for jets of holomorphic sections of a Hermitian line bundle. Nagoya Math. J. 180, 1–34 (2005) [R’70] Ramírez de, A.E.: Ein Divisionsproblem und Randintegraldarstellungen in der komplexen Analysis. Math. Ann. 184, 172–187 (1969/1970) [Rf’15] Raufi, H.: Singular hermitian metrics on holomorphic vector bundles. Ark. Mat. 53(2), 359–382 (2015) 164 3 L2 Oka–Cartan Theory

[Siu’96] Siu, Y.-T.: The Fujita Conjecture and the Extension Theorem of Ohsawa-Takegoshi. Geometric Complex Analysis (Hayama, 1995), pp. 577–592. World Scientiﬁc Publishing, River Edge (1996) [Siu’02] Siu, Y.-T.: Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type. In: Complex Geometry (Göttingen, 2000), pp. 223–277. Springer, Berlin (2002) [Ty’06] Takayama, S.: Pluricanonical systems on algebraic varieties of general type. Invent. Math. 165(3), 551–587 (2006) [T’02] Tsuji, H.: Deformation invariance of plurigenera. Nagoya Math. J. 166, 117–134 (2002) [V’08] Varolin, D.: Division theorems and twisted complexes. Math. Z. 259, 1–20 (2008) [Y-Z’15] Ye, F., Zhu, Z.-X.: On Fujita’s conjecture in dimension 5. arXiv: 1511.09154v1 [Y’81] Yoshioka, T.: Cohomologie à estimation L2 avec poids plurisousharmoniques et extension des fonctions holomorphes avec contrôle de la croissance. Osaka J. Math. 19(4), 787–813 (1982) Chapter 4 Bergman Kernels

Abstract Applications of the L2 method to the Bergman kernels will be discussed. Emphasis is put on the results obtained in recent decades. Among them, there are various estimates for the Bergman kernel from below on weakly pseudoconvex domains, including the solution of a long-standing conjecture of Suita by Błocki (Invent Math 193:149–158, 2013) and Guan and Zhou (Ann Math 181:1139–1208, 2015). Recently discovered variational properties due to Maitani and Yamaguchi (Math Ann 330:477–489, 2004) and Berndtsson (Ann Inst Fourier (Grenoble) 56(6):1633–1662, 2006; Ann Math 169:531–560, 2009) are also discussed. In a broader framework, they are describing the parameter dependence of the Bergman kernels associated to families or sequences of complex manifolds and vector bundles. Most of these new results are closely related to the L2 extension theorems in the previous chapter. Among them, a surprise is that a variational property of the relative canonical bundles generalizing that of the Bergman kernels, which originally belongs to the theory of variation of Hodge structures, happens to imply an optimal L2 extension theorem (cf. Berndtsson and Lempert, J Math Soc Jpn 68:1461–1472, 2016.

~~4.1 Bergman Kernel and Metric~~

The Bergman kernel, named after Stefan Bergman (1895–1977), is by deﬁnition the reproducing kernel of the space of L2 holomorphic n-forms on connected n- dimensional complex manifolds. Its signiﬁcance in complex geometry has been gradually understood through many spectacular works in the last century. For instance, C. Fefferman [F] analyzed the boundary behavior of the Bergman kernel on strongly pseudoconvex domains with C∞ boundary, and proved that any biholomorphic map between such bounded domains in Cn extends smoothly to a difffeomorphism between their closures. Recently, methods for analyzing certain generalized Bergman kernels have brought new insights into some aspects of algebraic geometry and differential geometry (cf. [Siu-7, Siu-9, Siu-10, Siu-11, Siu-12, B-Pa, Ds] and [Ma]). The purpose of this chapter is to review some of the results on the Bergman kernels related to such a new development.

© Springer Japan KK, part of Springer Nature 2018 165 T. Ohsawa, L2 Approaches in Several Complex Variables, Springer Monographs in Mathematics, https://doi.org/10.1007/978-4-431-56852-0_4 166 4 Bergman Kernels

Since it seems worthwhile to recognize the surroundings of the Bergman kernel in complex analysis and its connection to various concepts, let us briefly review some history here. The circle division theory of C.F. Gauss (1777–1855), which was discovered on 3/30/1796 is a giant leap in mathematics and the first step towards complex geometry. In the early nineteenth century, it brought new progress in the theory of elliptic integrals. It had been developed by L. Euler (1707–1783) and A.- M. Legendre (1752–1833) as an art of change of variables in their integration. Inspired by the work of Gauss, N.-H. Abel (1802–1829) was led at first to algebraic insolvability of equations of degree 5, and subsequently discovered that the inverse functions of elliptic integrals are nothing but doubly periodic analytic functions in one complex variable, i.e. elliptic functions. He eventually reached a remarkable characterization of principal divisors in the theory of algebraic functions of one variable (Abel’s theorem). The latter is now regarded as the starting point of algebraic geometry. As a generalization of Abel’s theory on elliptic functions, the theory of multiply periodic functions in several variables was developed by G. Jacobi (1804–1851), K. Weierstrass (1815–1897) and B. Riemann (1826–1866). On the other hand, in spite of an important contribution of H. Poincaré (1854–1912) on normal functions and a subsequent work of S. Lefschetz (1884–1972), it was not before the appearance of the theory of harmonic integrals on Kähler manifolds by W.V.D. Hodge (1903–1975) that Abel’s idea became really efficient in several variables (cf. [Ho]). This delay is, to the author’s opinion, mainly because of the lack of the viewpoint of orthogonal projection in Hilbert spaces. Recall that it was only in 1899 that D. Hilbert (1862–1943) awoke Riemann’s idea of Dirichlet’s principle from a deep sleep (cf. [R-2]) and that the basic representation theorem of F. Riesz (1880–1956), which is often crucial in the existence proofs under orthogonality conditions, was not available until 1907. History shows that such a systematic construction in abstract mathematics emerged only after the detailed studies of orthogonal polynomials in the nineteenth century. It culminated in a general method of orthogonal projection by H. Weyl (1885–1955). Weyl’s method (cf. [Wy-1]) became the analytic base of Hodge’s theory, which was later combined with analytic sheaf theory by K. Kodaira [K-2, K-3] and developed into the method presented in Chap. 2. That Weyl anticipated a lot in this method is modestly suggested in [Wy-2]. Anyway, the Bergman kernel came into the picture around 1922 (cf. [Be] and [Bo]) in such a circumstance.

~~4.1.1 Bergman Kernels~~

For any set S,thesetofC-valued functions on S, simply denoted by CS, is naturally equipped with the structure of a complex vector space. A subspace of CS,sayH equipped with an inner product is called a reproducing kernel Hilbert space if the following conditions are satisﬁed. 4.1 Bergman Kernel and Metric 167

(i) H is complete with respect to the associated norm. (ii) For any element x of S,themap[x] from H to C deﬁned by [x](f ) = f(x)is continuous. Given a reproducing kernel Hilbert space H, Riesz’s representation theorem implies that there exists uniquely a function on S × S say KH , satisfying the following:

~~(a) KH (∗,y)∈ H for any y. (b) (u, KH (∗,y))= u(y) for any u ∈ H and y ∈ S.~~

We shall say that KH is the reproducing kernel of H . Among such H and KH , some deserve special attention when S has certain structures as manifolds or groups. For our purpose, an important example is the case where S is a complex manifold M equipped with a Hermitian metric and H is 0,0 2 H(2) (M), the space of L holomorphic functions on M. 0,0 This way of regarding H(2) (M) as a reproducing kernel Hilbert space is naturally generalized for the space of L2 holomorphic sections of Hermitian holomorphic vector bundles over M, replacing [x] in (ii) by the evaluation of the sections of E at x. Accordingly, for any Hermitian holomorphic vector bundle E over a Hermitian manifold M, or more generally for any holomorphic vector bundle with a singular 0,0 ﬁber metric over a Hermitian manifold, H(2) (M, E) has a reproducing kernel as a ∗ ⊗ ∗ × section of p1E p2E.Herepj denotes the projection to the j-th factor of M M. 0,0 Note that the spaces H(2) (M, E) are separable, so that they are isomorphic to

~~∞ 2 := ∞ ; ∈ C | |2 ∞ C (cj )j=1 cj and cj < j=1~~

0,0 =∞ if dim H(2) (M, E) . An important special case is when E is the canonical K 0,0 = n,0 bundle M of M. In this situation H(2) (M, E) H(2) (M) if M is of pure dimension n, and the inner product of the space is independent of the choices of the metric on M as long as the ﬁber metric of the canonical bundle is induced from the n,0 metric on M. More explicitly, H(2) (M) is equipped with a canonical inner product − 2 2 nin u ∧ v. (4.1) M

n,0 The reproducing kernel of H(2) (M) with respect to this inner product is called the Bergman kernel of M, denoted by KM instead of K n,0 for simplicity. H(2) (M) A question of basic interest is how KM detects the geometry of M. Some of the properties of KM follow immediately from the deﬁnition. For instance, it is clear that 168 4 Bergman Kernels

KM×N ((z, s), (w, t)) = KM (z, w) × KN (s, t) (4.2) holds up to the order of the variables. n Example 4.1 Let D be a bounded domain in C ,letΩz = dz1 ∧ dz2 ∧···∧dzn, { ∈ N n,0 and let fj (z)Ωz ; j } be a complete orthonormal system of H(2) (D). Then

∞ KD(z, w) = fj (z)fj (w)Ωz ⊗ Ωw, (4.3) j=1 where the series converges locally uniformly on D × D. Note that the boundedness n,0 assumption on D was used only to ensure that the space H(2) (D) is inﬁnite dimensional. A straightforward calculation yields a formula

−n −n−1 KBn (z, w) = (2π) n!(1 − z · w) Ωz ⊗ Ωw, (4.4) · = n Bn ={ ∈ Cn; } 2 = ·¯ where z w j=1 zj wj and z z < 1 ( z z z). −n/2 In the situation of Example 4.1,theset{2 fj (z) ; j ∈ N} becomes a 0,0 complete orthonormal system of H(2) (D) with respect to the Lebesgue measure. The reproducing kernel K 0,0 is called the Bergman kernel function of D and H(2) (D) denoted by kD. By an abuse of language, kD(z, z) will also be called the Bergman kernel of D. kD(z, z) is denoted by kD(z) for simplicity. Note that kD(z, w) can be recovered from kD(z), because the Taylor coefﬁcients of kD(z, w) along the diagonal z = w are all recorded in kD(z) as the Taylor coefﬁcients in z and z¯. Similarly we put

~~KM (z) = KM (z, z) (z ∈ M). (4.5)~~

−1 Then KM (z) is naturally identiﬁed with a singular ﬁber metric of the canonical bundle KM of M whenever KM (z) does not vanish almost everywhere. Example 4.2 If M is a complex torus Cn/Γ equipped with a metric induced from 2 dsCn , then KM is related to the volume of M,sayVol(Γ ),by

~~−n −1 KM (z, w) = 2 Vol(Γ ) Ωz ⊗ Ωw. (4.6)~~

~~From this formula, it is expected that KM generally detects some geometric properties of M. This point will be further discussed in the following sections.~~

~~To evaluate KM (z) in more general situations, the following is most basic. 4.1 Bergman Kernel and Metric 169~~

~~Proposition 4.1~~

~~= { ⊗ ; 2 = ∈ n,0 } KM (z) sup σ(z) σ(z) σ 1,σ H(2) (M) (4.7) holds for any z ∈ M. Here the right-hand side of (4.7) is understood to be zero if n,0 ={ } H(2) (M) 0 .~~

Corollary 4.1 For any open set U ⊂ M, KU (z) ≥ KM (z) holds for any z ∈ U. {| |2; 2 = ∈ 0,0 } It is also of basic√ importance that sup f(z) f 1,f H(2) (D) is ∗ attained by kD( ,z)/ kD(z), and that KUj (z, w) converges to KM (z, w) locally uniformly on M × M if Uj increasingly converges to M.

~~4.1.2 The Bergman Metric~~

Let Mj (j = 1, 2) be two complex manifolds of pure dimension n. Suppose that there exists a biholomorphic map f : M1 → M2. Then f induces an isometry n,0 between H(2) (Mj ) by the pull-back of (n, 0)-forms. Accordingly,

~~= ∗ KM1 f KM2 . (4.8)~~

~~In terms of the local coordinates ζ1 around p ∈ M1 and ζ2 around q ∈ M2 such that f(p)= q, the relation (4.8) is explicitly written as~~

~~2 k1(ζ1) = k2(f (ζ1))| det(∂ζ2/∂ζ1)| , (4.9)~~

~~= ⊗ where KMj (z) kj (ζj )Ωζj Ωζj . Therefore, if KMj (z) are nowhere zero, the −1 curvature forms θj of KMj (z) also satisfy the relation~~

~~∗ θ1 = f θ2. (4.10) = n α ∧ β In particular, if θj α,β=1 θjαβ dζj dζj , and (θjαβ ) are everywhere positive deﬁnite, f is an isometry with respect to the metrics~~

n α ⊗ β θjαβ dζj dζj , (4.11) α,β=1 which are called the Bergman metrics of Mj . Here the notation for the metric is as a ﬁber metric of a holomorphic tangent bundle. The Bergman metric of a complex 2 manifold M will be denoted by dsM,b. Equation (4.10) means that biholomorphic maps preserve the Bergman metrics. 170 4 Bergman Kernels

~~Example 4.3~~

2 = − 2 −1 2 + − 2 −2 2 ⊗ ¯ 2 dsBn,b (1 z ) dsCn (1 z ) ∂ z ∂ z , (4.12) 2 = n ⊗ 2 where dsCn j=1 dzj dzj .(dsCn,b does not exist.) Cn 2 For a bounded domain D in , the Bergman metric dsD,b will be identiﬁed with ¯ ∂∂ log kD(z) by an abuse of notation. In terms of the notation (4.2),

2 = ¯ = −1 ¯ − −2 ¯ dsD,b ∂∂ log kD(z) kD ∂∂kD kD ∂kD∂kD 2 −2 ¯ ¯ = ( |fj | ) (fj ∂fk − fk∂fj )(fj ∂fk − fk∂fj ), (4.13) j =k where ∧ and ⊗ are omitted to avoid confusion. 2 Consequently, dsD,b is characterized as follows. 2 = ∗ 2 CP∞ 2 \{ } C∗ Proposition 4.2 dsD,b ι dsCP∞ , where is the quotient of C 0 by , 2 2 = : :··· dsCP∞ is induced from ds 2 , and ι(z) (f1(z) f2(z) ). C For any bounded domain D ⊂ Cn, z ∈ D and ξ ∈ T 1.0D, we put 0 z0

~~= {| |; ∈ 0,0 = = } b(ξ) sup ξf f H(2) (D), f (z0) 0, f 1 . (4.14)~~

~~Another immediate consequence of (4.13)is: 2 Proposition 4.3 In the above situation, the length of ξ with respect to dsD,b is & b(ξ)/ kD(z0). (4.15)~~

~~4.2 The Boundary Behavior~~

Among the analytic properties of the Bergman kernels which reﬂect the geometry of complex manifolds, the boundary behavior is studied from various viewpoints. Since it is often hard to calculate the Bergman kernels explicitly, description of principal terms of their singularities and their asymptotic expansions is aimed at. It is expected that this can be achieved in terms of geometric quantities such as the Levi form of the boundary and the curvature form of the bundles. The L2 method for the ∂¯ operator is available to localize the problem. It was ﬁrst applied in the case of strongly pseudoconvex domains by Hörmander [Hö-1]. To estimate the Bergman kernels from below on weakly pseudoconvex domains, the L2 extension theorem (Theorem 3.3)isuseful. 4.2 The Boundary Behavior 171

~~4.2.1 Localization Principle~~

Let the notations be as above. A basic question to be discussed here is the following: Given an open subset U ⊂ M,forwhichV ⊂ U, can one find a positive constant C such that KU (z) ≤ CKM (z) holds for any z ∈ V ? This is asked to understand the behavior of KM at infinity or the boundary behavior of kD by comparing them with those on local models. Results are described on complete Kähler manifolds in general. As we mentioned earlier, pseudoconvex domains in Cn are complete Kähler manifolds (cf. Corollary 2.15). ∞ For any point z in a complex manifold M, we denote by Pz the set of C negative functions φ on M \{z} such that e−φ is not integrable on any neighborhood of z.LetU ⊂ M be an open set, let χ : M →[0, 1] be a C∞ function such that χ|M\U = 0, and let V ⊂{z ∈ M; χ(z) = 1}. Theorem 4.1 In the above situation, assume that M admits a complete Kähler ∞ metric and there exist C0 > 0 and a bounded C function ψ on M such that one can find for every z ∈ V a function φz ∈ Pz satisfying C0 +φz > 0 on supp dχ and ¯ ¯ ∂∂(φz + ψ) ≥ ∂χ∂χ (4.16) holds on M \{z}. Then there exists a constant C depending only on ψ and C0 such that & KU (z) ≤ (1 + CKU (∗,z)/ KU (z)supp dχ)KM (z) (4.17)

2 holds for any z ∈ V .Here∗K denotes the L norm over K. Proof Since M \{z} admits a complete Kähler metric for any ﬁxed z, one may apply − − Theorem 2.14 for the trivial line bundle equipped with the ﬁber metric e φz ψ to solve the ∂¯-equation & ¯ ¯ ∂u = ∂(χ · KU (∗,z)/ KU (z))

− with L2 norm estimate. Since u(z) = 0 by the nonintegrability of e φz , one has a holomorphic n form & χ · KU (∗,z)/ KU (z) − u √ 2 on M \{z} whose value coincides with that of KU (∗,z)/ KU (z) at z.BytheL 2 ¯ property, it√ extends holomorphically to M. Thus, evaluation of the L norm of ∂(χ · KU (∗,z)/ KU (z)) deduced from (4.16) yields (4.17). 172 4 Bergman Kernels

~~n Corollary 4.2 Let D be a bounded pseudoconvex domain in C ,letz0 ∈ ∂D and n let Uj (j = 1, 2) be neighborhoods of z0 in C such that U1 U2. Then there exists C>0 such that~~

−1 C kD(z) < kU2∩D(z) < CkD(z) (4.18) holds for any z ∈ U1 ∩ D. n Corollary 4.3 Let D be a pseudoconvex domain in C ,letz0 ∈ ∂D, and let U be a neighborhood of z0. Suppose that ∂D is strongly pseudoconvex at z0. Then

~~lim kU∩D(z)/kD(z) = 1. (4.19) z→z0~~

~~In view of the proof of Theorem 4.1, it is easy to see that a similar localization principle holds for the Bergman metric. Proposition 4.4 In the situation of Corollary 4.2, there exists C>0 such that~~

~~−1 C bD(ξ) < bU2∩D(ξ) < CbD(ξ) (4.20)~~

~~1,0 n holds for any ξ ∈ Tz C with z ∈ U1 ∩ D. Combining Propositions 4.3 and 4.4 with Corollary 4.2, we obtain: Theorem 4.2 In the situation of Corollary 4.2, there exists C>0 such that~~

~~− C 1 ds2~~

~~4.2.2 Bergman’s Conjecture and Hörmander’s Theorem~~

Let us start from a naïve observation. Let D be a bounded domain in Cn and let n z0 ∈ ∂D. Assume that there exist domains Dj ⊂ C (j = 1, 2) with D1 ⊂ D ⊂ D2 n such that there exist biholomorphic authomorphisms αj (j = 1, 2) of C satisfying n n n n−1 α1(D1) = B , α2(D2) = D , α1(z0) ∈ ∂B and α2(z0) ∈ ∂D × D . Then one can ﬁnd C>0 such that

~~−1 −2 −n−1 C δD(z)~~

~~Here ρ(z) = δD(z) if z ∈ D and ρ(z) =−δD(z) if z/∈ D. Then~~

~~n+1 −n lim kD(z)δD(z) = n!π (z0). (4.24) z→z0~~

~~Proof A direct combination of Example 4.1, Corollary 4.1 and Corollary 4.3. We note that Theorem 3.4 is also available to prove Theorem 4.3 (cf. [Oh-33]).~~

~~4.2.3 Miscellanea on the Boundary Behavior~~

Again, let D be a pseudoconvex domain in Cn. There are at least two types of questions related to Theorem 4.2. One is in the direction of deeper analysis on the asymptotics of KD(z) near z0 under the strong pseudoconvexity assumption. A decisive result of this kind is the following (cf. Kerzman [Kzm] and Fefferman [F]). Theorem 4.4 Let D ⊂ Cn be a strongly pseudoconvex domain with C∞-smooth ∞ boundary. Then kD(z, w) ∈ C (D × D \{(z, z); z ∈ ∂D}) and

−n−1 kD(z) = φ(z)δ (z) + ψ(z)log δ(z) (4.25) holds as z → ∂D.HereD denotes the closure of D in Cn and φ,ψ ∈ C∞(D). Another direction which we are going to describe below is less quantitative 2 and concerns with weaker divergence properties of kD(z) and dsD,b on weakly pseudoconvex domains. One of the results motivating such studies is the following 2 criterion for the completeness of dsD,b due to S. Kobayashi [Kb-2].

~~Proposition 4.5 Suppose that limz→∂D kD(z) =∞and the set of bounded 0,0 2 holomorphic functions on D is dense in H(2) (D). Then dsD,b is complete. 174 4 Bergman Kernels~~

∞ Proof Let z0 ∈ D and let γ :[0, 1) → D be a C curve with γ(0) = z0 and → → 0,0 γ(t) ∂D as t 1. Then, by the assumption on H(2) (D), one can ﬁnd for any >0 a bounded holomorphic function f on D such that ' ' 2 ' kD(z, z0)' |f(z0)| = kD(z0) and 'f − √ ' <. kD(z0)

Hence, since kD(z) explodes at the boundary, one can ﬁnd t0 and an isometric ∞ N ∗ embedding ι : D → PC (:= C \{0}/C ) such that ι(z0) = (1 : 0 : 0 ···) = : :··· 2 and ι(γ (t0)) (0 1 ). Hence dsD,b is complete. Kobayashi also proved that any bounded analytic polyhedron satisﬁes the assumptions of Proposition 4.5.

Theorem 4.5 (cf. [Kb-1]) Let P1,...,Pm be polynomials in z = (z1,...,zn) and n let D ⊂ C be a bounded connected component of {z;|Pj (z)| < 1, 1 ≤ j ≤ m}. =∞ 2 Then limz→∂D kD(z) and dsD,b is complete. Applying Skoda’s L2 division theorem, P. Pﬂug [Pf] obtained the following. n Theorem 4.6 Let D ⊂ C be a bounded pseudoconvex domain and let z0 ∈ ∂D. { }∞ ⊂ Cn \ Assume that there exist α>0 and a sequence pμ μ=1 D converging α n to z0 such that {z;z − pμ < z0 − pμ }⊂C \ D for all μ ∈ N. Then =∞ limz→z0 kD(z) .

Pflug’s theorem suggests that kD(z) will explode along ∂D under some weak regularity assumption on ∂D. A natural class to be studied has existed for a long time in potential theory (cf. [Wn, Bou]). As a class of complex manifolds it is defined as follows. Definition 4.1 A complex manifold M is said to be hyperconvex if there exists a bounded strictly plurisubharmonic exhaustion function on M. Diederich and Fornaess [D-F-1] proved that any bounded pseudoconvex domain in Cn with C2 smooth boundary is hyperconvex. Kerzman and Rosay [K-R] generalized the result to the C1 smooth case. The following simple observation is useful. Proposition 4.6 M is hyperconvex if and only if there exists a strictly plurisubharmonic exhaustion function ϕ on M satisfying ∂∂ϕ¯ ≥ c∂ϕ∂ϕ¯ for some positive constant c. Proof Let φ be a bounded strictly plurisubharmonic exhaustion function on M such = that supM φ 0. Then

~~∂∂φ¯ ∂φ∂φ¯ ∂∂(¯ − log (−φ)) = + ≥ ∂(log (−φ))∂(¯ log (−φ)). −φ φ2 4.2 The Boundary Behavior 175~~

Conversely, if ∂∂ϕ¯ ≥ c∂ϕ∂ϕ¯ for some positive constant c, one can find a bounded increasing function λ such that λ(ϕ) is a strictly plurisubharmonic exhaustion function on M. By virtue of the detailed study of homogeneous domains (cf. [PS]), homogeneous bounded domains are known to be hyperconvex (cf. [K-Oh]). Based on Bers’s realization of Teichmüller spaces as bounded domains in Cn, Krushkal’ [Kr]showed that any finite-dimensional Teichmüller space is hyperconvex. When dim M = 1, hyperconvexity of M is equivalent to the exhaustiveness of the Green function of M (cf. Proposition 3 in [Oh-16]), which can be seen easily from the definition of the Green function. Recall that the Green function of a Riemann surface M is by definition the maximal element of the set of continuous functions g : M × M → [−∞, 0) such that, for each point w ∈ M, g(z,w) is subharmonic in z and, for any local coordinate ζ around w, g(z,w) − log |ζ | is bounded on {z; 0 < |ζ(z)| < 1}. The Green function of M will be denoted by gM if it exists. Otherwise we put gM ≡−∞for the convenience of the notation. Example 4.4 z − w gD(z, w) = log . 1 − zw

2 Combining the properties of gM with the L extension theorem, one can show the following. Theorem 4.7 (cf. [Oh-16]) Let D ⊂ Cn be a bounded hyperconvex domain. Then limz→∂D kD(z) =∞. For the proof, the following elementary and obvious fact is useful. Lemma 4.1 Let D be a bounded domain in Cn and let u be a bounded continuous = exhaustion function on D with supD u 0. Then for any δ>0, lim sup inf dλ = 0. (4.26) →0 ζ∈D ζ, ∩{u<−δ} =∅ ∩{u>−}

n Here denotes the complex lines in C and dλ the Lebesgue measure on . Proof of Theorem 4.7.Letφ be a bounded strictly plurisubharmonic function on D with sup φ = 0. Then (4.26) holds for u = φ. Moreover, as is easily seen from the deﬁnition of the Green function, for any δ>0 one can ﬁnd k>0 such that

{z; kφ(z) > −}∩ ⊂{z; g ∩D(z, w) > −} (4.27) holds for a complex line if >0 and φ(w) < −δ. Hence, combining (4.26) and (4.27) with a well-known symmetry property g ∩D(z, w) = g ∩D(w, z),wehave

~~lim sup inf dλ = 0 (4.28) → w, ∩{φ<−δ} =∅ 0 w∈{φ>−} ∩g ∩D(z,w)<−1 176 4 Bergman Kernels for any δ>0. By (4.28) and the localization principle for the Bergman kernel, one has~~

~~lim inf sup k ∩D(z) =∞. →0 φ(z)>− z~~

2 Hence, by the L extension theorem we conclude that limz→∂D kD(z) =∞holds. 2 In view of Theorem 4.7, it is natural to ask whether dsM,b is complete if M is hyperconvex. Błocki and Pﬂug [B-P] and Herbort [Hb] independently proved the following. Theorem 4.8 The Bergman metric of a bounded hyperconvex domain in Cn is complete. This was generalized by B.-Y. Chen [Ch-1]: Theorem 4.9 The Bergman metric of a hyperconvex manifold is complete. The proofs of Theorems 4.8 and 4.9 are based on Bedford and Taylor’s theory of the complex Monge–Ampère operator [B-T-1, B-T-2], which is, however, beyond the scope of the present monograph. Manipulation of the distance function with respect to the Fubini–Study metric leads to the following (cf. [Oh-S]). Theorem 4.10 Let D ⊂ Pn be a pseudoconvex domain. Assume that ∂D is nonempty and C2-smooth. Then D is hyperconvex. In fact, in the situation of the above theorem, the distance from z ∈ D to ∂D with respect to the Fubini–Study metric, say r(z), turns out to have a property that −r(z) is strictly plurisubharmonic near ∂D for sufﬁciently small >0. Such a special bounded exhaustion function can be used to obtain a quantitative result. Theorem 4.11 (cf. [D-Oh-4]) Let D Cn be a pseudoconvex domain, on which there is a bounded plurisubharmonic C∞ exhaustion function ρ : D →[−1, 0) satisfying the following estimate with suitable positive constants C1,C2 > 0:

~~−1 C2 − 1/C2 C1 δD (z) < ρ(z) < C1δD (z). (4.29)~~

~~Then there are, for any z0 ∈ D, positive constants c3,c4 > 0 such that~~

distD(z0,z1)>c3 log | log (c4δD(z1))|−1 (4.30) holds for all z1 ∈ D.HeredistD(z0,z1) denotes the distance between z0 and z1 with 2 respect to dsD,b. The proof is an application of a slight reﬁnement of the localization principle in Theorem 4.1. Błocki [Bł-1] has improved the estimate (4.30)to 4.2 The Boundary Behavior 177

~~log 1/δD(z1) distD(z0,z1)> ,C>0. (4.31) C log | log (c4δD(z1))|~~

~~The proof relies on the pluripotential theory. Whether or not~~

~~log 1/δ (z ) dist (z ,z )> D 1 ,C>0 (4.32) D 0 1 C holds remains an open question.~~

~~4.2.4 Comparison with a Capacity Function~~

Let D be a domain in C. Then, because of the transformation formula (4.9), kD is closely related to the theory of conformal mappings and related quantities such as capacity functions (cf. [A, Ca, S-O] and [Su-2]). In view of the L2 method in Chap. 2, it is easy to see that kD ≡ 0 if there exists a bounded nonconstant subharmonic function on D, or equivalently, there exists a continuous function g : D × D →[−∞, 0) such that the following hold for any w ∈ D: (i) ∂∂g(¯ ∗,w)= 0onD −{w}. (ii) g(z,w) − log |z − w| is bounded on a neighborhood of w.

~~The maximum element, say gD, of the set of such g is nothing but the Green function of D. Accordingly, kD(z) ≡ 0 if the Green function exists on D. We put~~

~~γ(z)(= γD(z)) = lim (gD(z, w) − log |z − w|) (4.33) w→z and~~

~~γ(z) cβ (z)(= cβ,D(z)) = e . (4.34)~~

γ and cβ are called the Robin function and the logarithmic capacity on D, 2 −1 respectively. It is straightforward that cβ (z) = (1−|z| ) if D = D. Hence πkD = 2 cβ . Example 4.5 Let D(r) ={z ∈ C;|z|

~~1 r2 kD (z) = . (4.38) (r) π (r2 −|z|2)2~~

Letting γ ≡−∞and cβ ≡ 0ifgD does not exist, one can say that kD ≡ 0 if and only if cβ ≡ 0, as was observed by Oikawa and Sario in [S-O]. In fact, this elementary but nontrivial remark is an interpretation of Carleson’s theorem on the negligible singularities of Lp holomorphic functions (cf. [Ca, §VI. Theorem 1]). The main ingredient of [Ca] is a systematic study of “thin sets” by means of capacities, Hausdorff measures, arithmetical conditions etc., dealing with the signiﬁcance of these concepts to existence problems for harmonic and holomorphic functions, boundary behavior, convergence of expansions and to harmonic analysis. Based on this, Oikawa and Sario suggested comparing kD and cβ for any domain D.The question makes sense for Riemann surfaces. Namely, using the local coordinates z 2 ⊗ ¯ 1,0 ⊗ 0,1 and w in (4.33) and (4.34), we regard cβ,M(z) dz dz as a section of TM TM , 2 ⊗ ¯ so that the question is to compare KM (z) and cβ,M(z) dz dz.Bytheway,themain theme of [S-O] is the study of the boundary behavior of conformal mappings aiming at applications to the classiﬁcation of open (= noncompact ) Riemann surfaces. At that time, a general question which attracted attention was the relation between the function spaces on a Riemann surface M and the magnitude of its boundary.A typical approach was to consider an extremal problem in such a way that triviality of the solution implies degeneration of certain function spaces (cf. [A-Bl]). KM and cβ,M are certainly solutions of extremal problems on M. In this context, Oikawa and Sario also asked for comparison of KM with the Ahlfors constant

:= | | cB (z) sup|f |≤1 f (z) , where f ∈ O(M). N. Suita (1933–2002) ﬁrst considered this latter question and solved it completely in [Su-1] with a sharp bound by generalizing Hejhal’s result in [Hj’72] for smoothly bounded domains. After that, he proceeded to study the relation between KM and cβ . As for the annuli Ar := {r<|z| < 1},0≤ r<1, he proved the following in [Su-1]. Theorem 4.12 πk (z) ≥ c2 (z) holds for all z ∈ A . The equality holds if and Ar β,Ar r only if r = 0. Suita proved this by exploiting a formula of Zarankiewicz [Za] which expresses kAr in terms of the Weierstrass functions. ≥ 2 ⊗ ¯ Suita’s conjecture. πKM (z) cβ,M(z) dz dz holds for any Riemann surface M. Moreover, the equality holds if and only if M is conformally (=biholomorphically) equivalent to D \ E for some E satisfying cβ,C\E ≡ 0. 4.2 The Boundary Behavior 179

In [Oh-16, Addendum], the L2 extension theorem was applied to Suita’s ≥ 2 conjecture, and 750πKM cβ,M was obtained for any Riemann surface M.In 2012, Z. Błocki [Bł-2] proved: ≥ 2 Theorem 4.13 πkD cβ holds for any plane domain D. Błocki’s proof is a reﬁnement of a simpliﬁed variant of [Oh-T-1] by B.-Y. Chen [Ch-2]. For that, Błocki had to solve an ODE problem for two unknown functions. The following sharpened version of Theorem 4.13 is in [Bł-3]. Theorem 4.14 Let D be a pseudoconvex domain in Cn and let

~~GD,w = sup {u ∈ PSH(D); u<0 and lim sup (u(z) − log |z − w|)<∞}. z→w~~

~~Then~~

~~≥ 1 kD(w) 2na . (4.39) e Vol ({GD,w < −a})~~

~~Here Vol (·) denotes the Euclidean volume.~~

~~Proof Let G = GD,w and ⎧ ≥− ⎨ 0 t a −t χ(t) = − − ⎩ s 1e ns ds t < −a. a~~

¯ ¯ = ¯ ◦ ∈ 0,0 −2nG Then, solving the ∂-equation ∂α ∂(χ G) with α L(2) (D, e ) in such a way that |α|2 ≤ C Vol ({G<−a}) D where C is an absolute constant, one has a holomorphic function f := χ ◦ G − α satisfying

~~∞ − − f(w)= χ(−∞) = s 1e s ds =: E (na). na~~

~~Since √ & f ≤χ ◦ G+α≤(χ(−∞) + C) Vol ({G<−a}), the estimate~~

~~|f(w)|2 c(n, a) k (w) ≥ ≥ D f 2 Vol ({G<−a}) 180 4 Bergman Kernels holds, where~~

~~E (na)2 c(n, a) = √ . (E (na) + C)2~~

~~∈ N ˜ = m ⊂ Cnm ˜ = ∈ ˜ ˜ = For any m ,letD D and w (w,...,w) D. Then kD˜ (w) m 1 m = j (kD(w)) . Since GD˜ ,w˜ (z ,...,z ) max1≤j≤m G(z ), one has~~

~~{ − } = { − } m Vol ( GD˜ ,w˜ < a ) Vol ( G< a ) .~~

~~Hence, by the above estimate applied for D˜ ,~~

~~c(nm, a) (k (w))m ≥ . D (Vol ({G<−a}))m~~

~~The desired inequality follows from the fact that~~

~~− lim c(nm, a)1/m = e 2na. m→∞~~

In 2013, Q. Guan and X.-Y. Zhou [G-Z-1] proved the following, also by exploiting the solutions of an ODE problem. Theorem 4.15 Suita’s conjecture is true for any Riemann surface. Theorem 4.15 is a corollary of Theorem 3.4 except for the equality criterion. There exists an intimate relation between kD and gD besides the above inequality:

~~∂2 πk (z, w) = 2 g (z, w) (Bergman–Schiffer formula) (4.40) D ∂z∂w D and~~

~~∂2 πk (z) = γ (z) (Suita’s formula) (4.41) D ∂z∂z D~~

F. Maitani and H. Yamaguchi [M-Y] have exploited (4.40) to obtain an interesting variational property of kD, which was generalized by Berndtsson [Brd’06] and eventually led Berndtsson and Lempert [Brd-L] to discover a completely new proof of Theorem 3.3. These materials will be discussed in Sects. 4.4.2, 4.4.3, 4.4.4 and 4.4.5. According to Lempert, the ﬁnal blow was inspired by Theorem 4.14. 4.3 Sequences of Bergman Kernels 181

~~4.3 Sequences of Bergman Kernels~~

Let Mμ (μ ∈ N) be a sequence of Hermitian manifolds, let Eμ be holomorphic vector bundles over Mμ, and let hμ be singular ﬁber metrics of Eμ. Then, the behavior of the associated sequence of reproducing kernels K 0,0 is H(2) (Mμ,Eμ) expected to reﬂect that of (Mμ,Eμ,hμ). Some instances of results in this direction will be presented below. K 0,0 and its restriction to the diagonal will also be H(2) (Mμ,Eμ) called the Bergman kernels.

~~4.3.1 Weighted Sequences of Bergman Kernels~~

Let (M, ω) be a complete Kähler manifold, let φ be a nonnegative C∞ plurisubharmonic function on M and let M0 be the interior of {z ∈ M; φ(z) = 0}.For ∈ N 2 the sequence K n,0 −mφ ,(m ), the following is straightforward by the L H(2) (M,e ) method. × Proposition 4.7 On M0 M0, K n,0 −mφ locally uniformly converges to KM0 H(2) (M,e ) as m →∞. Let (L, b) be a positive line bundle over a connected compact Kähler manifold

(M, ω) of dimension n. The behavior of the sequence K 0,0 μ μ is related H(2) ((M,ω),(L ,b )) to the existence of certain extremal metrics on M as was suggested by S.-T. Yau in [Yau-2]. The ﬁrst result indicating this relationship was shown by G. Tian [Ti]. = = For simplicity we put KM,μ(z, w) K 0,0 μ μ (z, w) and KM,μ(z) H(2) ((M,ω),(L ,b )) KM,μ(z, z). Tian proved: 1/μ −1 Theorem 4.16 limμ→∞ KM,μ(z) = b(z) holds for any z ∈ M. ∈ Proof Let z0 M be any point. Let z be a local coordinate around z0 such that = n ∧ + 2 ω i j=1 dzj dzj O( z ), and let ζ be a ﬁber coordinate of L over a 2 2 ∞ neighborhood U of z0 such that b(z, ζ) =|ζ | + O(z ). Let s be a C section of L which is identically equal√ to 1 with respect to ζ on a neighborhood of z0 and ≡ 0 outside U. Put v = ∂(χ(¯ μz)sμ), where χ is a C∞ real-valued function on R such that supp χ ⊂[−2, 2] and χ ≡ 1on[−1, 1]. Then, by Theorem 2.14, ¯ ¯ one can solve the ∂-equation ∂u = v with a side condition u(z0) = 0 and with an 2 2 ≤ L estimate√ u C, where C is a constant independent of μ. Hence one has an μ − 0,0 μ μ 1/μ element χ( μ z )s u of H(2) ((M, ω), (L ,b )) approximating KM,μ(z0) in the desired way. 1/μ 2 Tian proved moreover that KM,μ(z) converges to b(z) in the C -topology. As −1 a result, 1/μ times the curvature form of KM,μ(z) converges to Θb. (Recall that Θb denotes the curvature form of b.) Later, D. Catlin [Ct] and S. Zelditch [Ze] independently proved the following. 182 4 Bergman Kernels

~~Theorem 4.17 In the above situation, assume moreover that ω = iΘb. Then there ∞ exist C functions am(m = 0, 1, 2,...)on M such that the asymptotic expansion~~

~~n n−1 n−2 KM,μ(z) ∼ a0(z)μ + a1(z)μ + a2(z)μ +··· (4.42)~~

μ μ holds with a0(z) = 1.HereL ⊗ L is identified with the trivial bundle by the fiber metric b. In [Ct] and [Ze], an asymptotic formula of Boutet de Monvel and Sjöstrand for the boundary behavior of the Bergman kernel, which is similar to (4.41), was used. It may be worthwhile to note that the above proof of Tian’s theorem can be refined to give an elementary proof of Theorem 4.17. (See [B-B-S].) Apparently there exists a parallelism between Theorems 4.3, 4.7 and Theo- rems 4.16, 4.17, the counterpart of 4.3 (resp. 4.7) being 4.16 (resp. 4.17). Strong pseudoconvexity of ∂D corresponds to the (strict) positivity of (L, b). Accordingly, it is natural to expect that Theorem 4.16 can be extended as a convergence 1/μ theorem for KM,μ(z) under weaker positivity assumptions. Such an instance is an approximation theorem of Demailly to be explained below.

~~4.3.2 Demailly’s Approximation Theorem~~

Let D be a domain in Cn and let φ(z) be a plurisubharmonic function on D (φ ∈ PSH(D)). Recall that φ(z) can be locally approximated from above by C∞ plurisubharmonic functions (cf. 1.2.5). It was shown by Bremermann [Brm] that any φ ∈ PSH(D) can be approximated on compact subsets of D by linear combinations of log |f | for f ∈ O(D) as long as D is pseudoconvex. This is because the domain {(z, w); z ∈ D and |w| 0 such that

~~− C1 ≤ 1 ≤ + 1 C2 φ(z) log kD,m(z) sup φ(ζ) log n (4.43) m m |ζ−z|~~

~~Ec(φ) := {z ∈ D; ν(φ,z) ≥ c} (cf. Deﬁnition 3.5) is an analytic set in D. Proof From Theorem 4.18, ( E (φ) = E − n (log k (z)). c c m D,m m≥1~~

~~Since~~

~~(α) E − n (log k (z)) ={z;|α|~~

~~4.3.3 Towering Bergman Kernels~~

Up to Theorem 4.18, Bergman kernels on a ﬁxed manifold were considered. The method can be naturally extended to study the Bergman kernels on a family of different base manifolds. A particularly interesting situation arises when the group actions are involved. By a tower of complex manifolds, we shall mean a sequence of complex manifolds Mj (j = 1, 2,...)such that Mj = M/Γj for some decreasing sequence of discrete subgroups Γj of the group Aut M of biholomorphic automorphisms of ∞ = M, acting on M properly discontinuously without ﬁxed points such that j=1 Γj { } [ : ] ∞ = { }∞ idM and Γ1 Γj < for all j. We put M∞ M.Atower Mj j=1 is said to be normal if Γj is a normal subgroup of Γ1 for all j. Given a tower {Mj }, one has natural projections πk,j : Mk → Mj (j < k) which are Galois coverings if {Mj } is normal. 184 4 Bergman Kernels

~~∗ In this setting, a natural question is whether or not the sequence π∞,j KMj (z) converges to KM∞ (z). J.A. Rhodes [Rh] proved the following.~~

Theorem 4.19 If {Mj } is a normal tower with M∞ = D such that M1 is compact, ∗ π∞,j KMj (z) converges to KM∞ (z) locally uniformly. 2 Sketch of proof. By the Aut D invariance of dsD, it follows from Proposition 4.1 and Cauchy’s estimate that the sequence π ∗ K (z) is equicontinuous with respect to ∞,j Mj 2 ∞ ={ } dsD. By the assumption j=1 Γj idM , the inequality

∗ ≤ lim sup π∞,j KMj (z) KM∞ (z) (4.46) j→∞ is obvious. On the other hand, since Mj are compact for all j =∞,bythe Gauss–Bonnet formula and (4.4) one knows = KMj (z) KM∞ (z). (4.47) Mj Mj

Here KM∞ (z) is identiﬁed with a form on Mj by the invariance under Γj . Since ∗ π∞,j KMj (z) are equicontinuous, combining (4.44) and (4.46) with the normality of the tower, the desired convergence is obtained. In [Oh-25], an example of a non-normal tower of compact Riemann surfaces is ∗ given for which π∞,j KMj (z) does not converge to KM∞ (z).

~~4.4 Parameter Dependence~~

~~Given a continuous family of n-dimensional complex manifolds Mt and Hermitian holomorphic vector bundles Et over Mt , the dependence of K n,0 on t will H(2) (Mt ,Et ) be discussed.~~

~~4.4.1 Stability Theorems~~

Let M be a complex manifold, let π : M → D be a surjective holomorphic map with smooth ﬁbers, and let (E, h) be a holomorphic Hermitian vector bundle over M.We = −1 = | = = − put Mt π (t), Et E Mt and KM,E,t K n,0 , where n dim M 1. H(2) (Mt ,Et ) By Proposition 4.1 and Cauchy’s estimate, it is easy to see that KM,E,t(z) is upper semicontinuous on D × M.Ifπ is proper, it is also immediate that KM,E,t(z) n,0 is continuous if and only if dim H (Mt ,Et ) is constant. It is the case if Et are 4.4 Parameter Dependence 185

Nakano positive as is easily seen from Theorem 2.20.IfMt are Kählerian, that n,0 dim H (Mt ) does not depend on t follows from the Hodge decomposition (cf. p,q Theorem 2.33) and the upper semicontinuity of dim H (Mt ) in t. Moreover, as n,0 we have noted in Sect. 3.1.4, the invariance of dim H (Mt ,Et ) is also true if M is holomorphically embeddable into some CPN and E ( Km for some m ∈ N. t Mt For the case of Nakano positive bundles, one can also deduce from Theorem 2.14 a continuity result for nonproper π. Theorem 4.20 Let π : M → D be a surjective holomorphic map with smooth fibers, and let (E, h) be a Nakano positive vector bundle over M. Assume that there exists a complete Kähler metric on Mt for every t depending continuously on t and there exists a Kähler metric ds2 on M and a constant c>0 such that the least eigenvalue of the curvature form of (E, h) with respect to ds2 is everywhere ≥ c. Then KM,E,t(z) is continuous. Concerning the proof of Theorem 4.20, the reader is referred to [D-Oh-3]forthe argument of solving the ∂¯-equation continuously in t by applying Theorem 2.14. Remark 4.1 It might be interesting if one can find a natural condition for the m continuity of KM,ωm , where the fiber metric of K is induced from the Bergman Mt Mt kernel on Mt . As for the Bergman metrics ds2 , a criterion for the continuity can be described Mt similarly to that above. 2 Theorem 4.21 Let M,π,dst be as in Theorem 4.19. Assume moreover that there exists a bounded strictly plurisubharmonic function on M. Then the family of Bergman metrics ds2 is continuous on M × D. Mt Continuity of the derivatives of the Bergman kernels can be analyzed similarly as long as the directions of derivatives are tangent to Mt (cf. [G-K]). As for the smoothness with respect to t, not many facts seem to be known.

~~4.4.2 Maitani–Yamaguchi Theorem~~

~~F. Maitani and H. Yamaguchi [M-Y] observed from~~

∞ (eφ(t)|z|)2j log k −φ(t)D(z) =−log π + 2φ(t) + 2 , (4.48) e j j=1 which is known to hold for any upper semicontinuous function φ = φ(t) on D by (4.38), that log ke−φ(t)D(z) is plurisubharmonic in (t, z) if and only if φ(t) is subharmonic in t, and generalized the result to Stein families of Riemann surfaces. As an exercise, let us verify by a straightforward computation that log ke−φ(t)D(z) is plurisubharmonic if φ(t) is subharmonic and in C2(D). 186 4 Bergman Kernels

~~Since~~

~~e−φ(t) k −φ(t)D(z, w) = , e π(e−φ − wz)2 =− − − −φ(t) −| |2 2 log ke−φ(t)D(z) φ(t) log π log (e z ) .~~

~~Hence we have~~

~~2 −φ(t) ∂φ ∂ 2e ∂t z log k −φ(t)D(z) = , ∂z∂t e (e−φ(t) −|z|2)2~~

~~2 ∂ − (−φ(t)− log (e φ(t) −|z|2)2) ∂t∂t 2 2e−φ(t) ∂φ =−∂ φ(t) + ∂ ∂t ∂t∂t ∂t e−φ(t) −|z|2~~

~~2 2 2e−φ(t)(| ∂φ|2 + ∂ φ(t)) 2e−2φ(t)| ∂φ|2 =−∂ φ(t) + ∂t ∂t∂t + ∂t ∂t∂t e−φ(t) −|z|2 (e−φ(t) −|z|2)2 and~~

~~2 ∂ − (− log (e φ(t) −|z|2)2) ∂z∂z | |2 = 2 + 2 z e−φ(t) −|z|2 (e−φ(t) −|z|2)2 2e−φ(t) = . (e−φ(t) −|z|2)2~~

2 2 ∂ − ≥ ∂ − ≥ Hence it is easy to see that ∂t∂t log ke φ(t)D(z) 0, ∂z∂z log ke φ(t)D(z) 0 and ∂2 ∂2 ∂2 2 log k −φ(t)D(z) log k −φ(t)D(z) − log k −φ(t)D(z) ∂t∂t e ∂z∂z e ∂z∂t e 4e−2φ(t)| ∂φ|2 ≥ ∂t ≥ 0. (e−φ(t) −|z|2)3

~~The generalized result is as follows. 4.4 Parameter Dependence 187~~

Theorem 4.22 Let p : D → D × C be a Riemann domain, let p1 : D × C → D −1 be the projection to the ﬁrst factor, and let Dt = p ({t}×C). Assume that D is ∈ D D pseudoconvex. Then log kDt (z) (z t ) is plurisubharmonic on . Let us present below only a sketchy account on the proof of Theorem 4.22, since the result was further generalized by Berndtsson [Brd’06, Brd] and Guan and Zhou [G-Z-1] by completely different methods. Their proofs will be discussed later in detail. Sketch of proof Since the assertion is local in t, one may assume that ∂D is smooth and Dt are domains with real analytic smooth boundary. Let Φ be a deﬁning D = = function of ∂ . We put g(t,z,w) gDt (z, w) and γ(t,z) γDt (z). Since

g(t,z,w) = log |z − w|+γ(t,z)+ h(t, z, w), (4.49) where h is harmonic in z in a neighborhood of w ∈ Dt and h(t, w, w) = 0for 2 all t, ∂γ (t, z)/∂t and ∂ γ(t,z)/∂t∂t are harmonic on Dt . Hence, by a generalized Poisson formula on Dt , one obtains ) ∂γ(t,w) 1 ∂Φ ∂Φ ∂g(t,z,w)2 = ds (4.50) ∂t π ∂Dt ∂t ∂z ∂z

Here ds denotes the arc length element of ∂Dt . Similarly, modifying the contour integral by Stokes’ formula, one has ∂2γ(t,w) 1 ∂g(t,z,w)2 4 ∂2g(t,z,w)2 = L(t, z) ds + dλ, ∂t∂t π ∂Dt ∂z π Dt ∂t∂z (4.51) where ) ∂2Φ ∂Φ2 ∂2Φ ∂Φ ∂Φ ∂Φ2 ∂2Φ ∂Φ3 L(t, z) = − 2Re + . ∂t∂t ∂z ∂t∂z ∂t ∂z ∂t ∂z∂z ∂z (4.52)

~~Let 2 ∂g(t,z,w) L (t,z,w)= π ∂z∂w and 2 ∂g(t,z,w) K (t,z,w)= . π ∂z∂w 188 4 Bergman Kernels~~

Then, combining (4.49) and (4.50) with the Bergman–Schiffer and Suita formulas (cf. () and (4.40)), one has ∂2kD (w) 1 t = L(t, z)(|L (t,z,w)|2 +|K (t,z,w)|2)ds ∂t∂t 4 ∂Dt ∂L (t,z,w)2 ∂K (t,z,w)2 + + dλ. (4.53) Dt ∂t ∂t

~~(The computation is actually quite involved.)~~

Hence, since L(t, z) is nonnegative by the pseudoconvexity assumption, kDt (z) is subharmonic in t for any fixed z. By a holomorphic coordinate transformation of the form (t, z) → (t, w + f (t)(z − w)) and by using the subharmonicity in t of the transformed Bergman kernels for fixed z, one deduces that log kDt (z) is subharmonic in t for any z. Similarly one sees the subharmonicity of log kDt (z) along any local D → D holomorphic sections of . Therefore log kDt (z) is plurisubharmonic. Formulas (4.49) and (4.50) can be regarded as variants of the corresponding formulas in [Y-3] and [L-Y-1] for the Robin functions on higher-dimensional ∂γ(t,w) ∂2γ(t,w) domains. Although ∂t and ∂t∂t had been studied in [Y-1, Y-2], motivated by a classification theory of entire functions (cf. [Ni-1]), their relation to the Bergman kernel was made explicit only in [M-Y] as an application of formulas of Schiffer and Suita. So, [M-Y] has the nature of a side remark in the theory of Robin functions. As a further development of Yamaguchi’s theory, see [Ha] for instance. Unfortunately, the higher-dimensional variational formulas in [L-Y-1]arediffi- cult to apply to study the Bergman kernels since there is no precise analogue of () and (4.40)forn>1. In this situation, B. Berndtsson came up with a completely new method of generalizing Theorem 4.22 to higher dimensions by making an observation that the plurisubharmonicity of log kDt (z) in (t, z) is a consequence of Nakano semipositivity of a Hilbertian bundle over D. The purpose of Sect. 4.4.3 is to review this work after [Brd’06].

~~4.4.3 Berndtsson’s Method~~

Let π : M → D be as in Sect. 4.4.1. A natural question after Theorem 4.22 is −1 K := K ⊗ whether or not KMt (z) is a singular ﬁber metric of the bundle M/D M ∗ −1 π KD with positive curvature current if M is a Stein manifold. This is indeed the case as explained below after [Brd’06]. Let Ω be a bounded domain in a complex manifold M0. We assume that Ω admits a complete Kähler metric and a bounded plurisubharmonic function which is strictly plurisubharmonic at some point. Let φ be any C∞ plurisubharmonic t function on a neighborhood of D × Ω in D × M0. For each t ∈ D, put φ (∗) = ∗ 2 n,0 2 = φ(t, ) and denote by At the space H(2) (Ω) with respect to the norm h 4.4 Parameter Dependence 189

~~ 2 = n2 ∧ −φt D h t i Ω h he .LetE be the vector bundle over of inﬁnite rank with = 2 ﬁber Et At .~~

Theorem 4.23 If φ is plurisubharmonic, then the Hermitian bundle (E, ∗t ) is semipositive. If φ is strictly plurisubharmonic, then (E, ∗t ) is positive. 2 n,0 −φt 2 Proof Let Lt denote the space L(2) (Ω, e ) consisting of L (n, 0)-forms on Ω ∗ 2 equipped with the norm t .ByF we denote the vector bundle with fiber Lt , a trivial bundle with a possibly nontrivial metric. Clearly, just as in the finite rank case, the curvature form of this Hermitian bundle F is given by the operator of multiplication by (∂2φ/∂t∂t)dt ∧ dt. To get an expression of the curvature form ΘE dt ∧ dt of (E, ∗t ), we apply the Griffiths formula (3.16) and obtain ∂2φ ∂φ ∂φ u, v = π⊥ u ,π⊥ v + (Θ u, v) (4.54) ∂t∂t ∂t ∂t E

∈ 2 for any u and v Lt lying in the domain of ΘE.Hereπ⊥ denotes the orthogonal projection to the orthogonal complement of E.(u and v do not depend on t.) Hence, in order to show the semipositivity of E, it sufﬁces to show that ' ' ' ∂φ '2 ∂2φ 'π⊥ u ' ≤ u, u (4.55) ∂t ∂t∂t

~~∈ 2 holds for any u At . However, noticing that~~

~~n 2 ¯ ∂φ ∂ φ ∂z u = dzj ∧ u, ∂t ∂t∂zj j=1~~

~~∂φ 2 so that π⊥( ∂t u) is the L minimal solution of~~

~~n 2 ¯ ∂ φ ∂zw = dzj ∧ u, ∂t∂zj j=1~~

~~∈ 2 one can easily see from (2.17) that (4.55)isvalidforanyu At as long as φ is plurisubharmonic. Positivity of E for strictly plurisubharmonic φ is similar. ~~

Let KΩ,φ (z) = K 2 (z, z). t At −1 Corollary 4.5 In the above situation, KΩ,φt (z) is a singular ﬁber metric of K(D×Ω)/D with positive curvature current. ∞ Let ψ be a C strictly plurisubharmonic function on D × Ω and let Dc = {(t, z) ∈ D × Ω; ψ(t,z) < c}. By Proposition 4.7, Corollary 4.5 is immediately extended to the following more general positivity assertion. 190 4 Bergman Kernels

−1 Proposition 4.8 The curvature form of KDc,t (z) is positive on Dc. Furthermore, by a continuity argument, one can immediately conclude from Corollary 4.5 that the following generalization of Theorem 4.22 holds. Theorem 4.24 Let M be a connected Stein manifold and let π : M → D be a ≡ −1 ∈ surjective holomorphic map with smooth fibers. If KMt 0, then KMt (z) (z Mt ) is a singular fiber metric of the bundle KM/D with positive curvature current. As for the positivity of the curvature on higher-dimensional parameter spaces, n,0 the above proof of Theorem 4.23 can be naturally extended to the H(2) (Ω)-bundles m ∞ m (E, ∗t ) over the polydiscs D for a C function φ on a neighborhood of D × Ω to obtain the following. Theorem 4.25 If φ is plurisubharmonic (resp. strictly plurisubharmonic), then (E, ∗t ) is Nakano semipositive (resp. Nakano positive). Theorem 4.25 is naturally generalized to the following (see also [Brd-1]). Theorem 4.26 Let f : M → U be a Kähler fibration with compact fibers over an open set U in Cm. Let L be a (semi-)positive line bundle over M and let E → U be O = O ⊗ K ⊗ ∗K−1 the vector bundle with (E) f∗ (L M f U ) such that E is equipped with the canonical fiber metric induced from the L2 inner product on the fibers of f . Then E is Nakano (semi-)positive. Here, “f is a Kähler fibration” means that f is differentiably a locally trivial fibration and every fiber of f admits a neighborhood carrying a Kähler metric. Berndtsson’s proof is simple enough and theoretically important because it clarifies a link between a variational problem and the optimality of the L2-estimate (2.16) for the ∂¯-operator originally due to Kodaira and Hörmander. It must be noted that Theorem 4.26 generalizes also Fujita’s semipositivity theorem which was discovered in a context of the classification theory of projective algebraic varieties. In this context, Theorem 4.26 has been generalized in [M-T’08] and [P-T’18] (see also [P’17]).Yet there is still another approach to Theorem 4.22 which was recently discovered by Q. Guan and X.-Y. Zhou [G-Z-1]. This method is even simpler than Berndtsson’s, once the L2 extension theorem with optimal constant is admitted. For the detail see the next subsection.

~~4.4.4 Guan–Zhou Method~~

Let the notation be as above. Guan and Zhou’s approach to Berndtsson’s theorem (Theorem 4.24) starts from an observation that the L2 extension theorem can be understood as a sub-mean-value property of the ﬁberwise Bergman kernels, as one can see from Proposition 4.1. Actually, to prove the following, it sufﬁces to make this remark more precise. 4.4 Parameter Dependence 191

Theorem 4.27 (cf. [G-Z-1]) Let M be a connected complex manifold and let p : M → D be a holomorphic map with smooth fibers. Assume that there exists an analytic set X ⊂ M such that M \ X is Stein and p−1(0) ∩ X is nowhere dense −1 ≡ −1 in p (0). Then, either KMt (z) 0 or KMt (z) is a singular fiber metric of the bundle KM/D with positive curvature current. Proof By Theorem 1.8 it suffices to prove the assertion by assuming that X =∅. Let z0 be any point of M0 and let (t, z) be a local coordinate of M around (0,z0).

~~In terms of (t, z), the Bergman kernels KMt (z) are expressed as~~

~~= −n ∧ ∧···∧ ⊗ ∧ ∧···∧ KMt (z) 2 k(t,z)dz1 dz2 dzn dz1 dz2 dzn. (4.56)~~

~~To prove the assertion, it sufﬁces to show that the average of log k(t,z0) over |t|~~

~~KM0 (z, z0) u0 = √ . (4.57) k(0,z0)dz1 ∧ dz2 ∧···∧dzn~~

~~2 By the L extension theorem, if 0~~

~~=˜| ˜2 ≤ 2 u0 ur M0 and ur πr . (4.58)~~

~~On the other hand, by Proposition 4.1,~~

~~u˜ (z) ⊗ u˜ (z) K (z) ≥ r r (4.59) Mt ˜2 ur t if |t|~~

~~Hence, in view of (4.55) and (4.58) with z = z0,(4.59) implies ≥ 1 |˜ ∧ ∧···∧ |− 1 exp 2 (2log ur (t, z0)/(dz1 dz2 dzn) log k(t,z0)) dλt πr |t|~~

x by the convexity of the function y = e .Butlog|˜ur (t, z0)/(dz1 ∧ dz2 ∧···∧dzn)| is harmonic, so that, in view of (4.56) and the ﬁrst half of (4.57), the desired sub- mean-value property 192 4 Bergman Kernels ≤ 1 log k(0,z0) 2 log k(t,z0)dλt (4.62) πr |t|

~~4.4.5 Berndtsson–Lempert Theory and Beyond~~

In 2014, Berndtsson and Lempert [Brd-L] found a remarkable connection between the plurisubharmonicity of log kD and Theorem 3.3. To illustrate the idea of [Brd-L], let us begin with an alternate proof of Theorem 3.2 along with Lempert’s proof of Theorem 4.13 which was presented in [Bł’15, Introduction].

Proof of Theorem 3.2 It sufﬁces to show that πkD(0) ≥ 1, or equivalently that log(πkD(0)) ≥ 0. By the Maitani–Yamaguchi theorem (Theorem 4.22), log k|t|D(0) is subharmonic in log t. Since log k|t|D(0) depends only on log |t|, it is convex in log |t|. Therefore, the function

~~2 − log πkD(0) + log πk|t|D(0) + log |t| turns out to be increasing in log |t|, since it is convex and bounded. Hence~~

~~2 lim (− log πkD(0) + log πk|t|D(0) + log |t| ) t→0 2 ≤ lim (− log πkD(0) + log πk|t|D(0) + log |t| ) = 0. t→1~~

2 Admitting that Theorem 3.2 is inﬁnitesimally true, i.e. that limt→0 πk|t|D(0)|t| = 1, we obtain the desired conclusion. Keeping this argument in mind, it may not be too hard to generalize it to prove Theorem 3.26 for any pseudoconvex domain Ω ⊂ Cn−1 × D ⊂ Cn. Namely, it g (t,0) sufﬁces to consider the domain {(z, t) ∈ Ω × D;|zn|

2 2 Proof of Theorem 3.26. Let Aϕ(Ω) (resp. Aϕ(Ω )) denote the space of holomorphic functions on Ω (resp. on Ω ) with respect to the Lebesgue measure. To prove the assertion, it sufﬁces to show that

~~− { 2 | ∈ 2 | = }≤ 2 sup inf F F Aϕ(Ω), F Ω f πcD(0) . ∈ 2 = f Aϕ (Ω ), f 1 4.4 Parameter Dependence 193~~

~~For that we look at the function~~

~~{ 2 | ∈ 2 | = } sup inf F F Aϕ(Ωt ), F Ω f , ∈ 2 = f Aϕ (Ω ), f 1~~

~~g (t,0) where Ωt ={z ∈ Ω ||z1|~~

2 = 2 ⊕ 2 ∩ I Aϕ(Ωt ) Aϕ(Ω ) (Aϕ(Ωt ) (Ωt )), where I (Ωt ) denotes the subspace of O(Ωt ) consisting of the functions vanishing 2 on Ω and the norm in Aϕ(Ω ) on the right hand side is that of the quotient 2 2 ∩ I Aϕ(Ωt )/(Aϕ(Ωt ) (Ωt )), i.e.

~~ 2 := 2 f t min F . ∈ 2 | = F Aϕ (Ωt ), F Ω f~~

~~−g (t,0) In this situation, the proof will be over if the function gG(t, 0) →f t e D ∈ 2 \{ } turns out to be decreasing for every f Aϕ(Ω ) 0 , because it will then follow that~~

~~− { 2 | ∈ 2 | = }= 2 2gD(t,0) inf F F Aϕ(Ω), F Ω f lim f t e t→1 − − ≤ 2 2gD(t,0) = 2 lim f t e πcD(0) f . t→0~~

~~−g (t,0) To evaluate f t e D we apply the equality~~

f t = sup |ξ(f)|/ξ(t), ∈ 2 ∗ ξ Aϕ (Ω ) where ·(t) denotes the dual norm of ·t . Since the monotonicity of functions g (t,0) is preserved by taking the supremum, it sufﬁces to show that ξ(t)e D is increasing for every ξ = 0. But this follows similarly to the above proof of πkD(0) ≥ 1 from the subharmonicity of log ξ(t), which is guaranteed by Theorem 4.23. By this argument, an L2 extension theorem was proved in [Brd-L] for any pseudoconvex domain D ⊂ Cn, for any φ ∈ PSH(D) and for any complex submanifold V ⊂ D of codimension k with respect to − A2(D) = f ∈ O(D); |f |2e φ < ∞ D 194 4 Bergman Kernels and − A2(D) = f ∈ O(V ); |f |2e φ < ∞ . V

Here the integration is with respect to the Lebesgue measure and the volume element on V induced by the euclidean metric, respectively. Let dV (z) be the distance from z to V and let G be a negative plurisubharmonic function on D which satisﬁes

~~≤ 2 + G(z) log dV (z) A and~~

≥ 2 − G(z) log dV (z) B(z) as z goes to V .HereA is some constant and B is a continuous function on D. Then the precise estimate of Berndtsson–Lempert is the following. Theorem 4.28 Let f ∈ A2(V ). Then there is a function F in A2(D) whose restriction to V euqals f , which satisﬁes 2 −φ 2 −φ+kB |F | e ≤ σk |f | e , D V

k where σk is the volume of the unit ball in C . It is easy to see that Theorem 3.26 is a corollary of Theorem 4.28. It turned out recently that the method of Berndtsson and Lempert can be applied to prove an L2 extension theorem for jets of holomorphic functions (cf. [H’17-1] and [M-V’17]). Hosono [H’17-2] also obtained the following highly nontrivial reﬁnement of Theorem 3.2 by this method. Theorem 4.29 In the above situation, assume that n = 1 and φ(0) = 0. Let D˜ ={(z, w) ∈ C2; z ∈ D,|w|2

~~4.5 Notes and Remarks~~

~~Since the Bergman kernel has many aspects of independent interest, some of them that have not been mentioned above will be reviewed below. 4.5 Notes and Remarks 195~~

The transformation formula (4.8) for the Bergman kernel can be used to explore the properties of biholomorphic maps. For instance, given a simply connected domain D Cand p ∈ D, the biholomorphic map f : D → D with f(p)= 0 and f (p) > 0 can be expressed in terms of kD by using

~~1 kD(z, F (z))F (w) = f (z)kD(f (z), w) = f (z) , (4.63) π(1 − wf (z))2 where F = f −1. Indeed, by letting w = 0 one has~~

~~−1 kD(z, p)F (0) = π f (z), (4.64) which immediately yields a well-known formula * π z f(z)= kD(ζ, p) dζ . kD(p, p) p~~

~~On the other hand, by differentiating (4.64) with respect to w one has~~

~~2 1 k (z, F (w))F (w) + kD(z, F (w))F (w) = 2f (z)f (z) , D π(1 − wf (z))3 (4.65)~~

~~= ∂ = where kD(z, w) ∂w kD(z, w). Letting w 0in(4.65) one has~~

~~2 + = −1 kD(z, p)F (0) kD(z, p)F (0) 2π f (z)f (z). (4.66)~~

~~Dividing (4.66)by(4.64) one has~~

~~kD(z, p) 2F(z) = c1 + c2, kD(z, p)~~

~~where c1 = F (0) and c2 = F (0)/F (0). This shows that the mapping~~

kD(z, p) kD(z, p) is a biholomorphic map onto a disc. For multiply connected domains in C, Suita and Yamada [S-Y’76] showed that the Bergman kernel must vanish at some point, so that the above expression loses some utility. On the other hand, for any bounded domain Ω in Cn, there is a k (z,p) generalization of D which has signiﬁcance as a global canonical coordinate kD(z,p) in some cases. An explicit expression of this coordinate will be described below. 196 4 Bergman Kernels

~~Let 2 = ⊗ dsΩ,b gj,k dzj dzk j,k~~

~~kj ∈ and let g be the (k, j)-th entry of the inverse matrix of (gjk). Then, for any p Ω, we put~~

~~= repp(z) (ζ1(z), . . . , ζn(z)), where kj ∂ log kΩ (z, w) ∂ log kΩ (w, w) ζj (z) = g (p) − . ∂w ∂w w=p k k k~~

∂ζ Since k | = = δ , the map rep is a holomorphic local coordinate around p. ∂z z p k p Bergman has found the following. Theorem 4.30 (cf. [B’50]) If f : Ω → Ω˜ is a biholomorphic mapping of bounded ◦ ◦ −1 C domains, the composite repf(p) f repp is -linear. The proof of Bergman is a direct computation using (4.8). Recently it was observed by S. Yoo [Y’16] that Theorem 4.28 is essentially a consequence of an obvious fact that the Bergman metric has locally a potential whose Taylor expansion has no holomorohic or anti-holomorphic part. It is well-known that the idea of linearizing biholomorphic maps was carried over by Webster [W’79-1] and Bell [B’79], and culminated in [B-L’80] and [L’81] as the great simpliﬁcation of the proof of Fefferman’s theorem for the C∞-smooth extendability of biholomorphic maps between C∞ strongly pseudoconvex domains.

~~The map kD/kD is related to Suita’s conjecture in an intriguing manner. Let us follow this passage for a while starting from~~

Theorem 4.31 (cf. [B’06, Theorem 2.2]) Let D1 and D2 be n-connected bounded domains in C, each bounded by n non-intersecting C∞ smooth simple closed curves, and let f : D1 → D2 be a biholomorphic map. Given pj ∈ ∂Dj for j = 1, 2, there j j j j exist a1 and a2 in Dj such that kDj (z, a1 )/kDj (z, a0 ) map the points of ∂Dj near pj to smooth real-algebraic sets, and f becomes linear after these transformations. The same procedure as in Theorem 4.29 can be carried out for Riemann surfaces. If Ω is a smoothly bounded domain in a compact Riemann surface M with an anti- holomorphic involution ι such that ∂Ω ={x ∈ M; ι(x) = x} (M is the double of Ω), it is known that the Bergman kernel KΩ extends to M meromorphically. ˆ More explicitly, letting fp : M → C be an n-to-one holomorphic map such that fp(p) = 0 and fp(∂Ω) = ∂D, which exists and unique up to multiplication by eiθ ∈ ∂D,itisshownin[B’99] that the function 4.5 Notes and Remarks 197

~~fp(z)KΩ (z, w)fp(w)~~

dfp(z) ⊗ dfp(w) extends as a single-valued function to M × M which is meromorphic in w and anti- meromorphic in z. In this situation, it was pointed out by Yamada [Y’98] that the validity of Suita’s conjecture implies that the function L(z, w) := L (0,z,w) for ˆ Ω = D0 in 4.4.2 is zero-free if M = C.L(z,w)is known as the adjoint L-kernel of Ω, which is characterized, among the class of meromorphic function on Ω with the same singularity as L(z, w) and with ﬁnite values lim L(z, w)ω ∧ ω →0 Ω\{|z−w|<}

~~= ∈ 1,0 for any ω( ω(z)) H(2) (Ω), by the property that lim L(z, w)ω ∧ ω = 0 →0 Ω\{|z−w|<}~~

∈ 1,0 holds for all ω H(2) (Ω) (see [S’97]). The localization principle is the key idea in the proof of Theorem 4.3, as Hörman- der remarks in [Hö-3]. Theorem 4.4 is a far-reaching reﬁnement of Theorem 4.3, involving the idea that geometric invariants of ∂D can be differentiated out from the coefﬁcients φ(z) and ψ(z) in (4.25). Equation (4.24) says that the principal −n −n−1 term of the singularity of kD(z) at z0 is n!π (z0)δD(z) . For a class of weakly pseudoconvex domains, the boundary behavior of kD(z) has been analyzed in the same spirit (cf. [H’83, D-H’93, D-H’94, B-S-Y’95, K’04, Ch-K-Oh’04]). An illustrative result is in [H’83], where the Bergman kernel of the domain

~~3 6 2 2 6 Ω ={z ∈ C ; Re z1 +|z2| +|z2| |z3| +|z3| < 0} satisﬁes −c −c 1~~

~~1Graham [G’87] is based on an unpublished note of Burns. In [H-K-N’93] a computation needed in Graham’s proof is written. 198 4 Bergman Kernels~~

[BM’88](seealso[W’79-2]). The condition ψ = O(ρ∞) is satisﬁed if the Bergman metric is Kähler–Einstein (cf. [F-W’97]), so that combining a theorem of Lu Qi- Keng [L’66], asserting that a bounded domain with complete Bergman metric of constant holomorphic sectional curvature is biholomorphic to the ball, with Cheng and Yau’s theorem on the uniqueness of complete Kähler–Einstein metrics (up to constant) (cf. [C-Y’80]), one knows that the Ramadanov conjecture implies that D is biholomorphic to the ball if and only if the Bergman metric is Kähler–Einstein. The latter is known as Cheng’s conjecture (cf. [F-W’97]). Recently, it turned out that the Ramadanov conjecture is true if D is sufﬁciently close to the ball. Curry and Ebenfelt proved it for the case n = 3 and Hirachi proved it for the general case (cf. [C-E’18]). An effective version of Theorem 4.9 was obtained in [Ch’17] by restricting a class of hyperconvex domains. Let D ⊂ Cn be a domain with a negative α plurisubharmonic exhaustion function ρ.If−ρ(z) ≤ CδD(z) for some constants α, C > 0, let α(D) be the supremum of such α. If there exists η>0 such that −1 η − ≤ η C δD < ρ CδD for some C>0, let η(D) be the supremum of such η. α(D) and η(D) are called the hyperconvexity index and the Diederich–Fornaess index, respectively (cf. [D-F-1, A-B, F-S’16, H’08, H’17]). It is known that, if D is a bounded psuedoconvex domain in Cn with Lipschitz boundary, then D is hyperconvex (cf. [Dem’87]) and η(D) > 0(cf.[H’08, H’17]). Theorem 4.32 (cf. [Ch’17, Theorem 1.7]) Let D ⊂ Cn be a bounded hyperconvex domain with α(D) > 0. Then, for every 0 0 such that |k (z, w)|2 ν(z) ν(z) r D ≤ C min , ,z,w∈ D, kD(z)kD(w) μ(z) μ(z) where μ =|ρ|/(1 +|log (−ρ)|) and ν =|ρ|(1 +|log (−ρ)|)n. Combining Theorem 4.30 with an estimate of the form + | |2 kD(z, w) 2 1 − ≤ 2(distD(z, w)) , kD(z)kD(w) the following generalization of (4.31) has been obtained. Theorem 4.33 ([Ch’17, Corollary 1.8]) If D is a bounded hyperconvex domain with α(D) > 0, there exists for any point z0 ∈ D a constant C>0 such that

~~| log δD(z)| distD(z0,z)≥ C log | log δD(z)| holds for any z sufﬁciently close to ∂D. 4.5 Notes and Remarks 199~~

Concerning Theorem 4.17,see[M-M’07] for an extensive account coupled with Demailly’s complex Morse inequality and studies on the asymptotics of the analytic torsion. In the case where the curvature form of (L, h) degenerates at a point z0 ∈ M,an expansion of the form β a1 aN lim KM,μ(z) − μ a0 + +···+ = 0 μ→∞ μα μNα was shown for some α, β > 0 under some homogeneity condition on h near z0 (cf. [Ch-K-N’11]). Demailly [Dm-6] applied Theorem 4.18 to extend the deﬁnition of a positive closed current (i∂∂u)¯ k, which was originally deﬁned in Bedford–Taylor theory [B-T-1, B-T-2] for a bounded plurisubharmonic function u recursively as

~~− (∂∂u)¯ k = ∂∂(u(∂¯ ∂u)¯ k 1).~~

2 For instance, the product turned out to make sense for u = c log(|f1| + ··· + 2 |fm| ) + b, where c>0, fj are holomorphic and b is bounded, if the unbounded locus of u is small enough compared to k. This definition of the product has a continuity property which was exploited in [A-B-W’18]togiveanalternate ¯ k = | |2 +···+| |2 + definition of (∂∂u) for u c log ( f1 fm ) b. { } = [ : ] ∞ ∞ A tower of complex manifolds Mj (Mj M/Γj , γj Γj+1 < , j=1 Γj ={idM })issaidtobesupported on M1 (cf. [Y’17]). M1 is called the base and M is called the top (cf. [Ch-F’16]). A typical tower with a simply connected top is supported on a manifold whose fundamental group is a finitely generated subgroup of SL(n,C) [B’63]. Arithmetic quotients of bounded symmetric domains are of such type. Limiting behavior of the spectrum of the Laplacian on Mj as j →∞has been studied by De George and Wallach [DG-W’78] for a tower of coverings that are topped by symmetric spaces of noncompact type. Kazhdan [K’70, K’83] proved that, for a normal tower {Mj } with simply connected top M, = 0,0 K [ : ] KM 0 if lim supj→∞ (dim H (Mj , Mj )/ Γ1 Γj )>0. This was applied to show that the arithmeticity of projective varieties is invariant under the base change (transformation of the field of definition under an isomorphism over Q), which is a phenomenon discovered in a special case by Doi and Naganuma in [D-N’67]. Kazhdan suggested that, for a tower of coverings supported on a Riemann surface, the pull-back of the Bergman metric on Mj converges to that of the top (cf. [M’75, Yau-2]), We shall say that a tower is Bergman stable if the pull-back of the Bergman kernels to the top converges to the Bergman kernel locally uniformly. Donnely [Dn’96] proved results analogous to Theorem 4.19 for a tower supported on a Riemannian manifold under the condition that the top has bounded sectional curvature and that the smallest nonzero eigenvalues of the Laplacian on Mj is uniformly bounded from below by a positive constant. His method is based on estimates by Cheeger, Gromov and Taylor for the heat kernel (cf. [C-G-T’82]) and Atiyah’s L2-index theorem [A’76]. As a continuation of Donnelly’s work, Yeung 200 4 Bergman Kernels

K [Y’00, Y’01] showed that Mj is very ample if the injectivity radius of Mj is greater than some constant depending on the top and the base (see also [W’17]). In [Y’17] it is shown that a class of compact locally symmetric space of noncompact type { } − 1 K = supports a tower Mj such that (1 2n−2 ) Mj (n dim M) is very ample for sufﬁciently large j. This direction is apparently parallel to that of Fujita’s conjecture. Some works on the asymptotics of the Bergman kernel for (Lμ,hμ)(μ→∞) are related to the towers (see [M-M’15] and [W’16]). On the other hand, an alternate proof of Theorem 4.19 was given by [Ch-F’16] based on the following. ˜ Theorem 4.34 Let (M, ω) and (M,ω)˜ be complete Kähler manifolds and let {Mj } ˜ be a normal tower with Mj = M/Γj . Then {Mj } is Bergman stable if the following two conditions are satisﬁed: (1) There exist a compact set K ⊂ M and a C2 function ψ of SBG on M \ K such that C−1ω ≤ i∂∂ψ¯ ≤ Cω holds for some constant C>0. (2) There exists a C2 function ψ˜ of SBG on M˜ such that C˜ −1ω˜ ≤ i∂∂¯ψ˜ ≤ C˜ ω˜ holds for some constant C>˜ 0.

Theorem 4.20 is far from optimal with respect to the continuity of KM,E,t.This is partly because the need for Theorem 4.20 came from a work of Bonneau and Diederich [B-D’90], where local integral solution operators for ∂¯ were constructed on certain weakly pseudoconvex domains in Cn as an extension of the works of Henkin [H’70] and Ramirez [R’70] on strongly pseudoconvex domains. Range [R’78] had already put forth the study of integral kernels on weakly pseudoconvex domains. A novelty in [B-D’90] was the use of Theorem 2.85, but Range [R’91] made an objection that the measurability of the kernel in [B-D’90] is not obvious. Theorem 4.20 was designed to clarify this point. Maitani–Yamaguchi’s formula (4.53) in the proof of Theorem 4.22 is a generalization of “Formula 9” in [M’84] which corresponds to the case L(t, z) = 0. Theorem 4.23 was preceded by [Brd’06] where Theorem 4.22 was ﬁrst generalized to the following. Theorem 4.35 Let D be a pseudoconvex domain in Cn+k,letφ ∈ PSH(D),let ={ ∈ Cn; ∈ } t = | ∈ Ck = ∧ Dt z (z, t) D ,letφ φ Dt for t and let KDt ,φt kt dz1 ···∧dzn ⊗ dz¯1 ∧···∧dz¯n. Then log kt (z) ∈ PSH(D). According to [Brd’06], the idea of generalizing Theorems 4.22 and 4.23 came from Prékopa’s theorem in [P’73], which says that for any convex function φ(x,y) on Rm × Rn the function φ˜ on Rm deﬁned by − ˜ − e φ(x) = e φ(x,y) dy Rn is also convex. Yamaguchi’s variational formula (4.51) for the Robin function γD(z) was also generalized to higher dimension in [Brd’06]. References 201

~~Harrington [H’17] and McNeal and Varolin [M-V’17] improve the estimates obtained earlier by Popovici in [P’05]. There are other non-reduced variants of Theorem 4.28 (cf. [Dem’17]).~~

~~References~~

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[Y’17] Yeung, S.-K.: On the canonical line bundle of a locally Hermitian symmetric space. J. Geom. Anal. 27, 3240–3253 (2017) [Y’16] Yoo, S.-M.: The Bergman representative map via a holomorphic connection. arXiv:1509.01668v2 [math.CV] Chapter 5 L2 Approaches to Holomorphic Foliations

Abstract Results on the L2 ∂¯-cohomology groups are applied to holomorphic foliations with an emphasis on the cases with Levi ﬂat hypersurfaces as stable sets. Nonexistence theorems are discussed for holomorphic foliations of codimension one on compact Kähler manifolds under some assumptions on geometric properties of the complement of stable sets. For the special cases such as CPn, complex tori and Hopf surfaces, nonexistence, reduction and classiﬁcation theorems will be proved. Closely related materials have been already discussed in Sect. 2.4., e.g. Theorem 2.79.

~~5.1 Holomorphic Foliation~~

Geometric structures of holomorphic foliations on complex manifolds are reﬂected in the curvature properties of the normal bundle, as in the case of submanifolds. Some results on the curvature of holomorphic foliations of codimension one are discussed. Ghys’s turbulent foliations on complex tori and Nemirovski’s example on torus bundles are described in terms of meromorphic connections.

~~5.1.1 Foliation and Its Normal Bundle~~

By deﬁnition, a foliation on a differentiable manifold M is a (possibly disconnected) : → = ∗ manifold F with a bijective embedding ι F M such that TF ι TF˜ holds for r some differentiable subbundle TF˜ of T. F is called a foliation of class C if TF˜ is of class Cr . Connected components of F are called the leaves of F .IfM is a complex manifold and TF˜ is a holomorphic subbundle of TM , F will be called a holomorphic foliation on M. For simplicity we shall not distinguish TF˜ from TF . Let F be a holomorphic foliation of codimension r on a complex manifold M of dimension n. Then, for any point x ∈ M, one can ﬁnd a neighborhood U x and holomorphic 1-forms ω1,...,ωr on U which are pointwise linearly independent ≤ ≤ O 1,0 ∗ and annihilated by TF . ωj (1 j r) locally generates a subsheaf of ((TM ) )

© Springer Japan KK, part of Springer Nature 2018 205 T. Ohsawa, L2 Approaches in Several Complex Variables, Springer Monographs in Mathematics, https://doi.org/10.1007/978-4-431-56852-0_5 206 5 L2 Approaches to Holomorphic Foliations which we shall denote by ΩF for simplicity. The collection of local generators ωj ∧···∧ of ΩF defines by ω1 ωr a global holomorphic section of the projectivization r 1,0 ∗ −{ } C∗ r 1,0 ∗ ∧ ( (TM ) 0 )/ of (TM ) . The system of locally defined r-forms ω1 ···∧ ωr is naturally identified with a globally defined nowhere-vanishing r-form r with values in the bundle det NF (= NF ), where NF denotes the normal bundle | =∼ 1,0| =∼ of F (in M). Recall that NF is defined as the quotient of TM F ( TM F ) by TF ( 1,0 1,0 O ∗ =∼ TF ).(NF will be also denoted by NF .) Clearly, (NF ) ΩF .Itiseasyto see that one may take as ωj 1-forms of the form dfj for some fj ∈ O(U) by shrinking U if necessary. Hence F is locally a collection of the level sets of Cr - valued holomorphic functions. Concerning the normal bundle of compact leaves, the following is basic. Proposition 5.1 Let F be a holomorphic foliation of codimension r

~~As for NF , let us mention some curvature properties. Proposition 5.2 Let M be a compact complex manifold of dimension n and let F be a holomorphic foliation of codimension r~~

Similarly, one has the following. = − 1,0 Proposition 5.3 Let (M, F ) be as above. If r 1 or n 1, and TM is trivial, then det NF is not positive. 0,0 n Proof If det NF were positive, no element of H (M, ⊕ det NF ) would be nowhere zero, because positive dimensional analytic sets must intersect with the zeros of holomorphic sections of positive line bundles on compact complex manifolds. 5.1 Holomorphic Foliation 207

~~Corollary 5.1 The normal bundle of a holomorphic foliation of codimension one on a complex torus is not positive.~~

The reader may suspect that NF is more or less ﬂat. However, the following phenomenon must not be neglected. Theorem 5.1 There exist a two-dimensional compact complex manifold M and a holomorphic foliation F of codimension one on M such that NF has a ﬁber metric whose curvature form is positive along F . Proof Let R be a compact Riemann surface of genus ≥ 2, let D be the open unit disc and let Γ be a discrete subgroup of Aut D such that D/Γ =∼ R. Since Aut D ⊂ Aut Cˆ , Cˆ being the Riemann sphere, Γ acts also on D × Cˆ by

~~(z, w) −→ (γ (z), γ (w)) (γ ∈ Γ).~~

Let M = (D × Cˆ )/Γ ,letπ : D × Cˆ → M be the natural projection, and let F be the collection of the images of D ×{w} in M by π. Then M is a compact complex manifold of dimension 2 and F is a holomorphic foliation of codimension one on M. To define a fiber metric of NF , first note that the Bergman metric (1 − 2 −2 ˆ ¯ |w| ) dw⊗ dw¯ on D ∪ (C − D) is a fiber metric of NF |M−π(D×∂D), because of its invariance under Γ . Hence, by multiplying (1−|w|2)−2 dw⊗dw¯ by a C∞ function ρ defined by ⎧ ⎪ z − w 2 2 ⎨ 1 − if z, w ∈ D = wz¯ − 1 ρ(z,w) ⎪ wz¯ − 12 2 ⎩ 1 − if z ∈ D,w∈ Cˆ − D, z − w

∞ one has a C fiber metric of NF ,sayb. The curvature form of b is positive along the leaves of F since it is twice the Bergman metric along them. Conjecture Let M be a compact complex manifold of dimension ≥ 3 and let F be a holomorphic foliation on M of codimension one. Then NF does not admit a fiber metric whose curvature form is positive along F . The reader will find several pieces of supporting evidence for it in subsequent sections. A holomorphic foliation F on a dense open subset U of M is called a O 1,0 ∗ singular holomorphic foliation on M if the subsheaf of ((TM ) ) generated by holomorphic 1-forms annihilated by T 1,0F , to be called the defining sheaf of F , is locally finitely generated over the structure sheaf of M. For instance, the holomorphic foliation on C2 −{0} whose tangent bundle is Ker (w dz − zdw) is a singular holomorphic foliation on C2.Sing(F ) will denote the set of points for which F is not extendible to a holomorphic foliation on their neighborhoods. Sing (F ) is called the singular set of F . In contrast to the case of holomorphic foliation, not every singular holomorphic foliation is locally expressed as the level set of a vector-valued holomorphic function. Concerning singular holomorphic 208 5 L2 Approaches to Holomorphic Foliations foliations of codimension one, it is easy to see that the defining sheaf is invertible, so that the normal bundle NF is well defined also for a singular holomorphic foliation F . A long-standing open question is whether there exists a singular holomorphic foliation F on CP2 with a leaf which does not accumulate to any point in Sing (F ) (cf. [C-LN-S] and [C]). A singular holomorphic foliation F on M is said to be a foliation by rational curves if for every x ∈ M there exists a rational curve (a complex space bimeromorphically equivalent to Cˆ ) through x and tangent to F . The following was obtained by M. Brunella [Br-1] in the context of bimeromorphic classification theory of algebraic varieties. (See also [Br-4].) Theorem 5.2 Let F be a singular holomorphic foliation of dimension one on a compact Kähler manifold M. Suppose that F is not a foliation by rational curves. Then its canonical bundle KF is pseudoeffective (i.e. KF admits a singular fiber metric whose curvature current is semipositive). Definition 5.1 AclosedsetS ⊂ M is called a stable set of a singular foliation F if S is the closure (in M) of the union of some leaves. Minimal stable sets are particularly of interest. Given a singular holomorphic foliation F of codimension one on a compact complex manifold, the complement of a stable set of F is locally pseudoconvex. Hence, minimal stable sets can arise as the boundary of a locally pseudoconvex domain. Therefore, the L2 method is naturally expected to be useful to study such foliations. In the general theory of several complex variables, holomorphic foliations of codimension one first arose in a paper of Grauert [Gra-6], where a locally pseudoconvex domain without nonconstant holomorphic functions was presented as a counterexample to a Levi problem on complex manifolds. In the next subsection, we shall collect some examples of holomorphic foliations of codimension one arising as variants of Grauert’s example.

~~5.1.2 Holomorphic Foliations of Codimension One~~

Let M be a compact complex manifold of dimension n. A complex manifold of dimension n + 1 with a holomorphic foliation of codimension one arises as a relatively compact locally pseudoconvex domain in a differentiable disc bundle over M as follows. Let {Uα} be a locally ﬁnite open covering of M by open sets and let Φ ={φαβ } be a system of injective holomorphic maps from D to C ﬁxing the origin, such that φαβ ◦ φβγ = φαγ holds on a neighborhood of 0 as long as Uα ∩ Uβ ∩ Uγ = ∅. Then, by gluing a neighborhood of Uα ×{0} in Uα × D and that of Uβ ×{0} in Uβ × D by

(z, ζα) ∼ (z, ζβ ) ⇐⇒ ζα = φαβ (ζβ ) 5.1 Holomorphic Foliation 209 for z ∈ Uα ∩ Uβ , one obtains a complex manifold, say Ω, containing M as a closed submanifold ζα = 0. On Ω one has a holomorphic foliation, say FΦ locally defined by the level sets of the coordinates on D.IfM is a compact Riemann surface of genus ≥ 1, some of such FΦ can contain infinitely many compact leaves which are mutually homotopically nonequivalent (cf. [U-1]). If φαβ are all rotations around the origin, Ω is a tubular neighborhood of the zero section of the holomorphic line ˆ bundle NM over M.IfM is the Riemann sphere C, it is easy to see that FΦ are all equivalent to the fibers of the projection π : Cˆ × D → D on a neighborhood of M, so that they are holomorphically convex, but they need not be so if the genus of M is ≥ 1. Indeed, if the rotations are given in such a way that the tensor powers k ∈ N NM are not trivial for any k , then Ω is never holomorphically convex, since the leaves of FΦ are then dense in the level sets of |ζα| by Kronecker’s theorem (or by Dirichlet’s pigeon hole principle). There exist holomorphic foliations of a similar kind on complex tori, i.e. holomorphic foliations induced from mutually parallel affine subspaces in Cn. Such foliations will be called linear foliations. Leaves are dense in most cases, but there exist cases where the foliation admits a real hypersurface as a stable set. If a leaf is dense in such a hypersurface X, then the complement of X is not holomorphically convex, because it is the union of parallel translates of X. Grauert’s observation in [Gra-6] is essentially up to this point. Let D → M be a holomorphic disc bundle. Then, as a domain in the associated Cˆ -bundle, (the total space of) D is locally pseudoconvex and bounded by a real- analytic hypersurface which is a stable set of the foliation locally consisting of the constant sections of D → M, which shall be denoted by FD . It was proved in [D-Oh-2] that D is pseudoconvex whenever M is Kählerian (cf. Theorem 2.79 in Chap. 2). We shall see later, independently from [D-Oh-2], that D is not “too pseudoconvex” if M is Kählerian. Similarly, let A → M be a holomorphic affine line bundle, i.e. a holomorphic fiber bundle over M with typical fiber C. Since Aut C consists of holomorphic affine transformations, A has its associated line bundle, say A0 → M.IfM is Kählerian and the first Chern class of A0 is zero, then A is equivalent to the bundle whose transition functions are locally constant as maps to Aut C. Therefore, A is equipped with a holomorphic foliation of codimension one whose leaves are locally the constant sections of A . The following is essentially contained in [U-2]. Proposition 5.4 If M is a compact Kähler manifold, then the total space of topologically trivial holomorphic affine line bundles over M are pseudoconvex. Proof Since M is Kählerian, the transition functions of A can be given by

~~iθαβ ζα = ζβ e + cαβ ,θαβ ∈ R,cαβ ∈ C.~~

Then, by using the Kählerianity again, one has a system of pluriharmonic functions iθ 2 hα satisfying cαβ = hα − e αβ hβ . Then |ζα − hα| is a well-deﬁned plurisubhar- ¯ monic exhaustion function on A . In fact, since ∂∂hα = 0 one has 210 5 L2 Approaches to Holomorphic Foliations

~~¯ 2 i∂∂|ζα − hα| ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ = i(dζα ∧ dζα − dζα ∧ ∂hα − ∂hα ∧ dζα + ∂hα ∧ ∂hα + ∂hα ∧ ∂hα) ¯ ¯ ≥ i∂hα ∧ ∂hα ≥ 0. ~~

As in the case of disc bundles, A are not “too pseudoconvex”. This point will also be discussed later. Disc bundles and afﬁne bundles are variants of tubular neighborhoods of submanifolds, although they are not necessarily their deformations. As a variant of foliations on complex tori consisting of mutually parallel leaves, there exists a distinguished class of holomorphic foliations of codimension one, which will be described below. Let R be a compact Riemann surface (of any genus), let T be a complex torus and let π : P → R be a principal T -bundle. Let g be the Lie algebra pf T . The kernel of the exponential map exp : g → T will be denoted by g0. For simplicity, we put exp ζ =[ζ ] and do not distinguish T from g/g0.Byameromorphic connection on T → R,wemeanasystemofg-valued meromorphic 1-forms, say {ωα}, associated to an open covering {Uα} of R with local trivializations

~~−1 φα : π (Uα) −→ Uα × T, such that ωα are deﬁned on Uα and mutually related on Uα ∩ Uβ by~~

~~ωα − ωβ = dcαβ .~~

~~Here cαβ are deﬁned by~~

~~◦ −1 [ ] = [ + ] φα φβ (z, ζ ) (z, ζ cαβ (z) ).~~

Existence of nontrivial meromorphic connections is a consequence of the classical theory of Riemann surfaces, or by Kodaira’s vanishing theorem more directly speaking. Since the difference of adjacent ωα’s are d-exact, the parallel transports of the points in P along the paths in R \{poles of ωα} are well defined, depending only on the homotopy class of the paths. Let S be the set of poles of ωα.Bythis parallel transport, any linear foliation on a fiber outside π −1(S) yields a foliation on P \ π −1(S). If its codimension is one, by adding the fibers of π over S, one has an extension of the foliation to that on P . A holomorphic foliation on P arising by such a construction will be called a turbulent foliation. Theorem 5.3 (cf. [Gh]) Any holomorhic foliation of codimension one on a complex torus is either linear or turbulent. Proof Let F be a holomorphic foliation of codimension one on a complex torus T . Since the assertion is trivially true if dim T = 1, we may assume that 5.1 Holomorphic Foliation 211 dim T = n ≥ 2. Then F yields a holomorphic map from T to CPn−1 by associating TxF to x ∈ T , since the tangent bundle of T is trivial. Let L be the connected component of any smooth fiber of this map. Then L must be a complex torus, since its normal bundle is trivial and hence so is TL. Clearly, F is linear if L = T . That F is turbulent otherwise can be seen by induction by considering the factor space of T by L. As for the results on singular holomorphic foliations on T ,see[Br-3] and [C-LN]. As we have seen above, some disc-bundles and affine line bundles are naturally equipped with holomorphic foliations of codimension one which are extendable to the associated Cˆ -bundles. They admit stable real hypersurfaces and/or complex submanifolds which are minimally stable and with pseudoconvex complements. In the turbulent foliations, the preimages of the poles of the meromorphic connection are minimal stable sets. In [Nm], S. Nemirovski discovered that some turbulent foliations can contain real hypersurfaces as stable sets which are not minimal. Let us describe his construction below. Let N be a compact complex manifold of dimension m ≥ 1 and let p : E → N be a holomorphic line bundle. Let s be a meromorphic section of E whose zeros and poles are all of order one along a nonempty smooth submanifold say B of N. Then we put

~~S ={c · s(x); x/∈ B and c>0}.~~

Taking the quotient of S by the action of the infinite cyclic group Z on E∗ = E \{zero section} by fiberwise multiplication by 2a for a ∈ Z, we obtain a real hypersurface S/Z in E∗/Z. Since the order of zeros and poles of s is one, the closure of S/Z in E∗/Z becomes a smooth real hypersurface. The union of (the images of) the sections ζ · s(ζ∈ C \{0}) over N \ B and the preimage of B is a holomorphic foliation of codimension one on E∗ which induces a foliation on E∗/Z, and S/Z is a stable set of this foliation. A remarkable feature of Nemirovski’s hypersurface S/Z is that its complement is Stein if E is positive and s is everywhere holomorphic. In fact, N \ B is then an affine algebraic manifold by Kodaira’s embedding theorem, hence it is Stein, and (E∗/Z) \ (S/Z) is Stein because it is a holomorphic fiber bundle over a Stein manifold with one-dimensional Stein fibers (cf. [Mk]). In [Oh-24], a generalization of Nemirovski’s construction is given. It turned out that, for a turbulent foliation to admit a stable real hypersurface, the meromorphic connection must satisfy a period condition. As for the analytic continuation of holomorphic foliations, a positive result was obtained by T. Nishino [Ni-2, Ni-3] when the leaves are compact and of codimension one. Theorem 5.4 (cf. [Ni-3]) Let F be a holomorphic foliation of codimension one on a nonempty open subset of a complex manifold M. Suppose that the leaves of F are compact and M is an increasing union of relatively compact locally pseudoconvex domains. Then there exists a holomorphic foliation on M extending F . 212 5 L2 Approaches to Holomorphic Foliations

~~5.2 Applications of the L2 Method~~

As was indicated in the preceding section, the method of L2 estimates can be applied to prove that the Kähler condition imposes certain restrictions on holomorphic foliations. To see this, the structure of L2 ∂¯-chohomology on the locally pseudoconvex domains in Kähler manifolds has to be observed more closely, extending what has been seen in Chap. 2.

~~5.2.1 Applications to Stable Sets~~

Let D be a locally pseudoconvex relatively compact domain in a complex manifold. First we shall study the case where ∂D is a smooth real hypersurface of (real) codimension one. Let us first observe that the classical theory of ∂¯-cohomology groups on Stein manifolds and compact Kähler manifolds already yields a prototypical result. Definition 5.2 A closed submanifold of real codimension one in a complex manifold is said to be a Levi flat hypersurface if its complement is locally pseudoconvex. If ∂D is C2-smooth with a defining function ρ, it is clear that ∂D is Levi flat if ¯ and only if ∂∂ρ|(Ker ∂ρ)∩(T ⊗C) ≡ 0. By an abuse of language, we shall say that ∂D ∂D¯ is Levi flat at x ∈ ∂D if ∂∂ρ|Ker ∂ρ = 0atx. Proposition 5.5 A Levi flat hypersurface of class Cr (r ≥ 2) in a complex manifold M admits a foliation of class Cr of real codimension one whose leaves are complex submanifolds in M. Proof Let X ⊂ M be a Levi flat hypersurface. Then the analytic tangent bundle 1,0 1,0 1,0 1,0 T X = TX⊗ C ∩ T M|X has a property that T X ⊕ T X is involutive, i.e. it is closed under the Lie bracket, since

∂ρ([ξ,η]) = ∂ρ([ξ,η]) − ξ∂ρ(η) + η∂ρ(ξ) = ∂∂ρ(ξ,¯ η) = 0 if X ={ρ = 0} (ρ ∈ Cr ) and ξ and η are local Cr−1 sections of T 1,0X. Therefore, by the Frobenius theorem one has the desired foliation. 1,0 = 1,0 We shall call the foliation F on X satisfying TF TX the Levi foliation on X. The Levi foliation on X will be denoted by LX. Clearly, NLX is equivalent to 1,0 := 1,0| 1,0 NX (TM X)/TX , which is defined for any real hypersurface X and called the analytic normal bundle of X. Proposition 5.6 A real analytic Levi flat hypersurface locally admits a pluriharmonic defining function. 5.2 Applications of the L2 Method 213

Proof Given a real analytic Levi ﬂat hypersurface X in a complex manifold M of dimension n,letx ∈ X,letU be a neighborhood of x in X, and let f : U → R be a real analytic function with df (x) = 0 whose level sets are contained in the levels of LX. Then, by shrinking U if necessary, one can ﬁnd a real analytic equivalence from (0, 1)×Dn−1 to U,sayα(t,z), which is holomorphic in the variable z ∈ Dn−1. Then the conclusion is obvious because α can be extended to a biholomorphic equivalence between some neighborhoods of (0, 1) × Dn−1 and U,inCn and M, respectively.

Let us prove a nonexistence result for Levi flat hypersurfaces which are stable sets of holomorphic foliations. Theorem 5.5 Let (M, ω) be a compact Kähler manifold of dimension n whose holomorphic bisectional curvature (see Sect. 2.4.4) is positive. If n ≥ 3, singular holomorphic foliations of codimension one on M do not admit C∞ hypersurfaces as stable sets. Proof Suppose that there existed a C∞ Levi flat hypersurface X in M. Then, by taking the double cover of M if necessary, we may assume that M \ X = D+ ∪ D−, where D± are mutually disjoint locally pseudoconvex domains. Then, by a theorem of Takeuchi, Elencwajc and Suzuki (cf. Theorems 2.73 and 2.74), the curvature 0,2 = condition on M implies that D± are Stein domains. Therefore H0 (D±) 0if n ≥ 3. Now, suppose moreover that X is a stable set of some holomorphic foliation of codimension one, say F , on a neighborhood of X. Then the normal bundle NF is topologically trivial on a neighborhood of X because so is NX ⊗ C which is topologically equivalent to NF |X. The curvature form, say θ, of the fiber metric of 1,0 NF induced from that of TM is positive along F by virtue of the Gauss–Codazzi– Griffiths formula, since the holomorphic bisectional curvature of M is positive by assumption. Since NF |X is topologically trivial, there exists a neighborhood U ⊃ X and a 1-form η satisfying θ = dη on U. Splitting η into the sum of the (1, 0)- component η1,0 and the (0, 1)-component η0,1, one has ∂η¯ 0,1 = 0, because θ is of 0,2 = 0,1 type (1, 1). Since H0 (D±) 0, η can be extended from a neighborhood of X to M as a ∂¯-closed (0, 1)-form, say η˜. Since M is a compact Kähler manifold, the harmonic representative of η˜ is ∂-closed. This means that θ = ∂∂φ¯ holds for some C∞ function φ, which is absurd because of the compactness of X and the maximum principle for the plurisubharmonic functions on F . Remark 5.1 The idea of the above proof for the C∞ case is taken from Siu [Siu-8]. Nonexistence of Levi flat hypersurfaces in CPn for n ≥ 3 was first proved by Lins Neto in [LN] for the real analytic case by a method independently from the L2 method, and by Siu [Siu-8], Cao and Shaw [C-S] and Brunella [Br-2]forless regular cases. The latter works are based on the L2 method. It was asked in [C] whether or not CP2 contains a Levi flat hypersurface. It is known that if it does, then the hypersurface has to satisfy a seemingly very ristrictive curvature condition (cf. [A-B]). Let D be a locally pseudoconvex relatively compact domain in a Kähler manifold (M, ω) of dimension n. Although it is not known whether or not D 214 5 L2 Approaches to Holomorphic Foliations carries a plurisubharmonic exhaustion function, definite results still hold for the ∂¯-cohomology of such D. Some of them can be applied to study holomorphic foliations. Let ρ be a defining function of D, i.e. ρ is a C∞ function defined on a neighborhood U of D such that D ={x ∈ U; ρ(x) < 0} and dρ vanishes nowhere on ∂D. We shall analyze the ∂¯-cohomology of D by the L2 method under some restrictions on the eigenvalues of the Levi form ∂∂(¯ − log (−ρ)) of − log (−ρ). These conditions implicitly appeared above when ∂D is a stable set of a holomorphic foliation on complex manifolds with certain curvature properties n ¯ such as CP .Letλ1 ≥ λ2 ≥ ··· ≥ λn be the eigenvalues of i∂∂(− log (−ρ)) with respect to ω. Since ∂∂(¯ − log (−ρ)) =−∂∂ρ/ρ¯ + ∂ρ ∧ ∂ρ/ρ¯ 2,

~~2 lim inf ρ(x) λ1 > 0. (5.1) x→∂D~~

~~Proposition 5.7 Suppose that~~

~~2 lim inf ρ(x) λp > 0 for 2 ≤ p ≤ k, (5.2) x→∂D and~~

2 lim inf ρ(x) λp ≥ 0 for k + 1 ≤ p ≤ n (5.3) x→∂D hold for some k. Then, for any Nakano semipositive vector bundle E over M, n,p = ≥ − + 0,p ∗ = ≤ − H (D, E) 0 for p n k 1 and H0 (D, E ) 0 for p k 1. Proof By assumption, ω + i∂∂(¯ − log (−ρ)) is a complete Kähler metric on D for sufﬁciently small >0. By (5.1), (5.2) and (5.3), λk +λk+1 +···+λn > 0 outside a compact subset of D. Therefore, one can ﬁnd c ∈ R and a C∞ function μ : R → R satisfying

μ|(−∞,c] ≡ 0,μ|(c,∞) > 0,μ|(c,∞) > 0 such that the sum of n − k + 1 eigenvalues of i∂∂μ(¯ − log (−ρ)) is nonnegative everywhere and positive on {x ∈ D; ρ(x) > −e−c}. Hence, as in the proof of Theorem 2.42 obtained from Corollaries 2.10 and 2.11, recalling (2.17) one has n,p = ≥ − + 0,p ∗ = similarly H (D, E) 0 for any p n k 1. That H0 (D, E ) 0for p ≤ k − 1 follows from this by Serre’s duality theorem. Theorem 5.6 (cf. [Oh-29]) Let (M, ω) be a compact Kähler manifold of dimension ⊂ 1,0 n and let X M be a real analytic Levi ﬂat hypersurface. Then NX does not admit a ﬁber metric whose curvature form is semipositive of rank ≥ 2 everywhere along LX.

Proof Let us take a locally finite open covering {Uj } of X and real analytic functions : → R fj Uj such that TLX is locally equal to Ker dfj . We may assume that 5.2 Applications of the L2 Method 215 fj = Re hj for some holomorphic function hj on a neighborhood of Uj . Note that 1,0 Im hj is then a local defining function of X. Suppose that NX had admitted a fiber metric with semipositive curvature of rank ≥ 2. Then one would have a system of ∞ =|dfj |2 ∩ C positive functions aj on Uj such that ak df aj holds on Uj Uk,by k ¯ taking a refinement of {Uj } if necessary, such that −i∂∂ log aj is a positive (1, 1)- 2 2 form on LX ∩ Uj for each j. Since ak|Im hk| − aj |Im hj | vanishes along Uj ∩ Uk with order at least 3, one can find defining functions of the components of M \ X satisfying the conditions (5.1), (5.2) and (5.3)fork = 3. On the other hand, it follows also from the real analyticity of X that X is a stable set of a holomorphic foliation of codimension one on a neighborhood of X. Hence, similarly to the proof of Theorem 5.5, we obtain the conclusion. 1,0 In view of Theorem 5.1, the condition on the rank of the curvature form of NX is optimal. In particular, the boundary of holomorphic disc bundles over compact Kähler manifolds cannot carry positive analytic normal bundles. Thus, Theorem 5.6 may well be regarded as supporting evidence for Conjecture 5.1.1. Now, turning our attention from the geometry of Levi flat hypersurfaces to that of the domains they bound, it is natural to ask for their q-convexity properties. The following answer may be regarded as a pseudoconvex counterpart of Theorem 5.6. Theorem 5.7 (cf. [Oh-22]) Let (M, ω) be a compact Kähler manifold and let X ⊂ M be a real analytic Levi flat hypersurface. Then there exist no plurisubharmonic exhaustion functions on M \ X whose Levi form has everywhere at least three positive eigenvalues outside a compact subset of M \ X. Proof By assumption, one can find a neighborhood U ⊃ X and a holomorphic foliation F on U of codimension one extending LX. Then, let {dfα} be a system { } 1,0 of holomorphic 1-forms associated to an open covering Uα of U such that TF is locally equal to Ker dfα.Letdfα = eαβ dfβ hold on Uα ∩ Uβ . {eαβ } is a system of transition functions of NF , and {dfα} is naturally identified with an NF -valued 1-form. By shrinking U if necessary, we may assume that NF is topologically u trivial, so that one may assume that eαβ = e αβ for some additive cocycle {uαβ } 0,2 \ = of holomorphic functions. Since H0 (M X) 0 by assumption, the proof being similar to that in Proposition 5.7, NF is extendible as a topologically trivial ˜ ˜ ¯ holomorphic line bundle over M,sayNF . Now, concerning the NF -valued ∂- \ \ cohomology on M X, the pseudoconvexity assumption on M X allows us to n−2 : 1,1 \ ˜ ∗ → extend Theorem 2.49 to conclude that the map ω H0 (M X, (NF ) ) n−1,n−1 ˜ ∗ H (M \ X, (NF ) ) is an isomorphism, so that the natural homomorphism

1,1 \ ˜ −→ 1,1 \ ˜ H0 (M X, NF ) H (M X, NF ) ˜ is injective. Therefore, {ωα} is extendible as an NF -valued d-closed holomorphic 1-form on M. Therefore, we may assume in advance that {dfα} are related by

~~iθαβ dfα = e dfβ ,θαβ ∈ R. 216 5 L2 Approaches to Holomorphic Foliations~~

Hence one can measure the distance d(x) from a point x ∈ U to X with respect to ¯ dfα ⊗dfα. More explicitly, one may put d(x) = infα,c |fα(x) + c|.Hereα is chosen so that x ∈ Uα and, for each α, c runs through the complex numbers satisfying | + |= = iθαβ + ∩ infy∈X∩Uα fα(y) c 0. (Note that fα e fβ ηαβ on Uα Uβ for some ηαβ ∈ C , where Uα ∩ Uβ are implicitly chosen to be connected.) By shrinking −1 ∈ U if necessary, we may assume that d is constant on fα (ζ ) for any ζ fα(Uα), so that the levels of d are compact and foliated by complex submanifolds of M of codimension one near X, which obviously contradicts the maximum principle for the assumed exhaustion function on M \ X. Remark 5.2 From the last part of the proof, the reader may well have an impression that the assumption on the number of positive eigenvalues of the exhaustion function might be superﬂuous. However, the number 3 is optimal, as the example in Theorem 5.1 shows. There is a straightforward extension of Theorem 5.7 for the stable sets of certain singular holomorphic foliations. The following is essentially a repetition of what Theorem 5.7 says. Theorem 5.8 Let (M, ω) be a compact Kähler manifold of dimension n and let X ⊂ M be a closed set. Suppose that there exist a neighborhood U ⊃ X and a singular holomorphic foliation F of codimension one on U having X as a stable set, such that the deﬁning sheaf of F is locally generated by a d-closed form. Then, M \ X does not admit a plurisubharmonic exhaustion function whose Levi form has at most n − 3 nonpositive eigenvalues everywhere on U \ X. When X is a divisor, a somewhat stronger theorem holds. For the proof, which reduces the situation to that of the above theorem, one needs a property of a subsheaf of the germs of holomorphic 1-forms consisting of d-closed ones. By a standard argument of algebraic geometry, it is shown that X would extend to a singular foliation on M such that X does not intersect any other leaf, if M \ X is too pseudoconvex. Similarly to Theorems 5.7 and 5.8, one has the following. Theorem 5.9 (cf. [Oh-23]) Let M be a compact Kähler manifold and let D ⊂ M be a domain. Suppose that B := M \ D is a complex analytic set of codimension one such that there exists an effective divisor A with support B for which the line bundle [A]|B is topologically trivial. Then D does not admit a plurisubharmonic exhaustion function whose Levi form has at most n − 3 nonpositive eigenvalues everywhere outside a compact subset of D. For the detail of the proof, see [Oh-23]. Theorem 5.9 is also a supporting evidence of Conjecture 5.1.1. In the above study of stable sets of holomorphic foliations, the L2 method played a role in extending the ∂¯-cohomology classes and ∂¯-closed forms. Therefore, in order to put the argument into a wider scope, we shall make a digression in the next subsection to prove several Hartogs-type extension results on complex manifolds by the L2 method. The author is inclined to believe that these general results can 5.2 Applications of the L2 Method 217 be applied not only to foliations but also to other questions in several complex variables, dynamical systems for instance.

~~5.2.2 Hartogs-Type Extensions by L2 Method~~

In this subsection, we shall restrict ourselves to the study of extension phenomena for holomorphic functions on complex manifolds. The following classical theorem attributed to Bochner and Hartogs is our prototype. Theorem 5.10 Let M be a Stein manifold of dimension ≥2, let K ⊂ M be a compact set with connected complement. Then every holomorphic function on M\K has a holomorphic extension to M. ≥ 0,1 = Proof Since M is Stein and dim M 2, H0 (M) 0 by Theorem 2.22. By this proof it is clear that Theorem 5.10 is also true for any (n − 1)-complete n-dimensional complex manifold M (cf. Corollary 2.14). We note that the result was extended to (n − 1)-complete spaces by a different method (cf. [M-P]). Now, let D be a relatively compact locally pseudoconvex domain in a complex manifold M of dimension n. We ask for a condition on ∂D for a similar extendibility result to hold on D. Theorem 5.11 (cf. [Oh-26]) In the above situation, assume that M admits a 0,1 = Kähler metric. Then H0 (D) 0 in the following cases. Case I. ∂D is a C2-smooth real hypersurface and not everywhere Levi ﬂat. Case II. There exists an effective divisor E on M with |E|=∂D such that the line bundle [E] admits a ﬁber metric whose curvature form restricted to the Zariski tangent spaces of ∂D is semipositive everywhere but not identically 0. For the proof of Case I, let us prepare the following elementary lemma. Lemma 5.1 Let ρ be a real-valued C2-function on the closed unit disc {z ∈ C;|z|≤1} such that ρ(0) = 0 and dρ vanishes nowhere, let U ={z;|z| < 1 and ρ(z) > 0}, and let f be a holomorphic function on U. Suppose that |f(z)|2/ρ(z) dλ < ∞. U

~~Here dλ denotes the Lebesgue measure. Then there exists r>0 such that f(z)= 0 on U ∩{|z|~~

~~{z ∈ D; y<−2x2}⊂U ⊂{y<−x2} (x = Re zandy= Im z). 218 5 L2 Approaches to Holomorphic Foliations~~

~~For 0 0weput~~

~~−Ai(z+ia)/(1−iaz) fa,A(z) = e f(z).~~

~~Note that~~

~~fa,A(−ia) = f(−ia) if − ia ∈ U. (5.4)~~

~~Since z + ia −i(x + iy + ia) (y + a) − ix −i = = , 1 − iaz 1 − ia(x + iy) 1 − iax + ay one has z + ia a + (a2 + 1)y + a(x2 + y2) Re − i = . 1 − iaz (1 + ay)2 + a2x2~~

~~Therefore one can ﬁnd >0 and a0 > 0 so that −ia ∈ U and z + ia Re − i < − (5.5) 1 − iaz 2 holds if 0~~

~~Hence, given 0 0, one can choose a δ>0 in such a way that ρ(−ia) > δ and 1 |f(z)||dz| < . ρ=δ N~~

~~Thus, for any a ∈ (0,a0), one can ﬁnd sequences δμ → 0 and Aμ →∞such that~~

~~lim |fa,A (z)||dz|=0. (5.6) μ→∞ μ ρ=δμ 5.2 Applications of the L2 Method 219~~

~~On the other hand, since Aμ →∞, one has~~

~~lim inf |fa,A (z)||dz|=0 (5.7) μ→∞ μ {|z|=1−δμ}∩U − = − = by (5.5). Consequently one has f( ia) limμ→∞ fa,Aμ ( ia) 0if0~~

~~∂∂ρ¯ 1 2 ∂∂(¯ 1/ log (−ρ)) = − + ∂ρ ∧ ∂ρ.¯ ρ(log (−ρ))2 ρ2(log (−ρ))2 ρ2(log (−ρ))3 (5.8)~~

0,1 = By the above lemma, it suffices to prove that H(2) (D) 0 with respect to ωA. Since 1/ log (−ρ) is bounded, ωA is complete, and the sum of n eigenvalues of ¯ i∂∂(1/ log (−ρ)) with respect to ωA is bounded from below by a positive constant 0,1 near ∂D, we know already that H(2) (D) is Hausdorff (cf. Theorems 2.13 and 2.4). Therefore, by virtue of Aronszajn’s unique continuation theorem, it suffices to show 2 that there exist no nonzero L harmonic forms of type (0, 1) with respect to ωA and the fiber metric eτ of the trivial bundle, for some C∞ bounded function τ on D.To find such a weight function τ, let us first take a Levi non-flat point x ∈ ∂D and a compactly supported nonnegative C∞ function χ on M such that χ(x) = 1 and ∂D is nowhere Levi flat on ∂D ∩ supp χ. Then we put τ = λ(χ + 1/ log (−ρ)) for >0 and a C∞ weakly convex increasing function λ such that λ(t) = 0fort<0 and λ (t) > 0fort>0. Then, in view of (5.8), it is easy to see that for sufficiently − ¯ small, for any choice of (n 1) eigenvalues τ1,...,τn−1 of i∂∂τ with respect to n−1 ∩ ωA, j=1 τj is nonnegative everywhere and positive on supp χ D. Therefore, by ¯ 2 +¯ ∗ 2 ≥ ¯ ∧ ∈ 0,1 (2.19) and the basic inequality ∂u ∂ u (iΛ∂∂τ u, u) for u C0 (D) (and by recalling Gaffney’s theorem again), we obtain the conclusion. The proof of Case II is similar. Since the construction of the metric and the weight function on D is more involved, the reader is referred to [Oh-26] for the detail. Remark 5.4 It is likely that the Kähler assumption is superfluous in Theorem 5.11. As supporting evidence, let us mention a result in [D-Oh-1] which asserts that two- dimensional locally pseudoconvex bounded domains with real analytic, Levi non- flat and connected boundary are holomorphically convex. 220 5 L2 Approaches to Holomorphic Foliations

~~5.3 A History of Levi Flat Hypersurfaces~~

The history is brief, not only because it starts from Grauert’s paper [Gra-6] in 1963 but also because the classification in CP2 is not yet complete. Anyway, let us begin with the character of [Gra-6] as a counterexample to the Levi problem. By virtue of the celebrated work of K. Oka [O-1, O-4], it is known that unramified domains over Cn are pseudoconvex if and only if they are holomorphically convex. The situation becomes subtler for the domains on complex manifolds. Namely, Grauert [Gra-3] proved that bounded domains in complex manifolds with strictly pseudoconvex boundaries are holomorphically convex, with a remark that not all pseudoconvex domains are so. Narasimhan described Grauert’s counterexample in [N’63], by showing that a generically chosen complex torus of dimension ≥2 contains a pseudoconvex domain which contains real hypersurfaces foliated by dense complex leaves of dimension one. By the maximum principle, such pseudoconvex domains do not admit nonconstant holomorphic functions. These hypersurfaces are the first examples of Levi flat hypersurfaces. Grauert [Gra-6] showed that a tubular neighborhood of the zero section of a generically chosen line bundle over a non-rational Riemann surface is also such an example. On the contrary, it was noticed in [Oh’82] and [D-Oh-2] that generic disc bundles over non-rational compact Riemann surfaces are Stein manifolds although their boundaries are foliated by complex leaves as well. These examples naturally raised a question of classifying Levi flat hypersurfaces. Let us recall some examples and describe their basic properties, before proceeding to the specific problem of classification in tori and Hopf surfaces. Simple closed curves in Riemann surfaces are Levi flat hypersurfaces on which nothing is left to say. Preimages of such curves by proper and smooth holomorphic maps are more general but still trivial. However, they can be deformed to Levi flat hypersurfaces sometimes in a nontrivial way as the reader can see from the examples below. So a question of general interest is how far does the theory of compact complex manifolds extend to Levi flat hypersurfaces. (i) Boundary of pseudoconvex domains without nonconstant holomorphic func- tions1 First we shall recall Grauert’s example described by Narasimhan [N’63]. = 1 n ∈ Cn ≤ ≤ Let n>1 and let wj (wj ,...,wj ) be chosen for 1 j 2n in such a way that

~~1. w1 = (1, 0,...,0), 2. w1,...,w2n are linearly independent overR and 1 ≤ ≤ Q 3. Im wj (2 j 2n) are linearly independent over .~~

1These example can be counted also as trivial ones. BEDFORD once warned the author not to talk about them anymore. 5.3 A History of Levi Flat Hypersurfaces 221 = 2n Z ⊂ Cn Cn : Cn → Let Γ j=1 wj , let Tn be the torus /Γ and let π Tn be ⊂ Cn 1 = the natural map. Let U be defined by 0 < Re z1 < 2 and let D π(U). = = 1 Then the components of ∂D are defined by Re z1 0 and Re z1 2 , so that ∂D is a Levi flat hypersurface in T n. The domain D is pseudoconvex because it admits a plurisubharmonic exhaustion function − log (1 − 2Re z1) − log Re z1.But every holomorphic function f on D is constant: in fact |f | has a maximum at a = { = 1 } − point x0 on K π( Re z1 4 ) and there is a connected n 1 dimensional analytic set A through x0 in K which is dense in K (i.e. π({z1 = c1})) where c is such that π(c) = x0, by Kronecker’s theorem. Hence f is constant on A and so on K. Since K has real dimension 2n − 1, f is constant on D.In[N’63]itwas asked if a bounded domain D in a complex manifold is holomorphically convex when it is assumed that ∂D is smooth and contains at least one point where ∂D is strictly pseudoconvex. Grauert [Gra-6] immediately constructed a counterexample to Narasimhan’s conjecture, based on a yet another class of Levi flat hypersurfaces bounding nonconstant holomorphic functions, which we shall describe below. Let M be a compact complex manifold of dimension ≥1 and let E → M be a holomorphic vector bundle whose transition matrices are all unitary. Then the zero section of E admits a pseudoconvex neighborhood system of the form {w 0}. Then X is Levi flat, ∂Ω+ = X and Ω+ is equivalent to (C \{0}) × ({Im ξ>0}/Z) by the map (ζ, [z]) → (ζ, [ζz]), where the action of Z on {Im ξ>0} is generated by ξ → 2ξ. Thus Ω+ is Stein because it is 2 the product of C \{0} and the annulus {e−2π / log 2 < |w| < 1}. This example is from [Oh’82], which was inspired by the foregoing works of Huckleberry and Omsby [H-O’79] and Diederich and Fornaess [D-F’77] as well as by the following observation attributed to J.-P. Serre (cf. [H’77] and [Nm’88]). Proposition 5.8 Let T = (C \{0})/Z and let F = (C × (C \{0}))/Z, where the actions of Z are defined by z → ez and (ζ, z) → (ζ + 2πi,ez), respectively. Then i ζ F is equivalent to (C \{0})2 by the map induced by (ζ, z) → (eζ ,e2π z). Recall that Serre’s example shows the distinction of analytic equivalence and algebraic equivalence on Stein manifolds, in contrast to Oka’s principle which says that topological equivalence implies analytic equivalence there. Based on the example Ω+, Barrett [B’86] has observed a similar phenomenon for Stein domains with smooth boundary. Namely, equivalence of the domains does not imply the equivalence of their boundaries. This observation was later carried over to Kiselman’s work [K’91] on the irregularity of the Bergman projection and culminated in Christ’s analysis [Ch’96] of the Bergman projection on the worm domains of Diederich and Fornaess [D-F’77](seealso[B’12]). Note that the fibers of the projection from Ω+ to Y are equivalent to the upper ∼ half plane H ={Im ζ>0} and that Ω+ = (H × (C \{0}))/Γ,where Γ is the subgroup of Aut (H × (C \{0})) generated by (ζ, z) → (2ζ,2z). That Ω+ is equivalent to the product of C \{0} and an annulus can also be seen from this. Generalizing this, Diederich and Fornaess [D-F-5] gave an answer to Grauer’s question whether or not every smoothly bounded pseudoconvex domain in a compact complex manifold admits a plurisubharmonic exhaustion function by showing Ωn in Example 2.10 in Chap. 2. This counterexample suggests an interesting relationship between pseudoconvexity and Kählerianity. Since dim Ωn ≥ 3, Grauert’s question remains open for two-dimensional domains. It is also open for the domains in Kähler manifolds. (iii) Boundary of disc bundles In [D-Oh-2], the above-mentioned examples have been studied further as disc bundles over compact manifolds, since H =∼ D. Theorem 2.79 is the main result of [D-Oh-2], so that let us give a short account on its proof. Proof of Theorem 2.79 (an outline) First observe that the transition maps of analytic D-bundles are locally constant, i.e. they do not depend on the base variables, because otherwise Aut D would contain a complex 1-parameter subgroup which would yield a nonconstant holomorphic map from C to D, contradicting Liouville’s theorem. Then, one appeals to a fact that D-bundles with locally constant transition functions over compact Riemannian manifolds admit either harmonic sections with respect to the Poincaré metric on the fibers, or locally constant sections on the boundary, i.e. locally constant sections of the associated Cˆ -bundles whose images 5.3 A History of Levi Flat Hypersurfaces 223 are contained in the boundary of the D-bundles. This follows from the energy- decreasing property of the solution of a heat equation which is associated to the Euler–Lagrange equation for the energy functional of the sections. This property of the energy functional was proved by Eells and Sampson [E-S’64] for the maps to Riemannian manifolds with negative sectional curvature, whose generalization for the sections is straightforward. It is also immediate from the argument of [E-S’64] that the solution of the heat equation either converges to a harmonic section, or converges to a locally constant section on the boundary. Once this existence result is available, the next step is to exploit Siu’s variant of the Bochner trick of integrating by parts in [Siu’80] which shows the pluriharmonicity of harmonic maps from compact Kähler manifolds to locally symmetric spaces of negative curvature (see also [C-Td’89] and [J-Y’83]). Here a map is called pluriharmonic if its restriction to complex curves are harmonic. Consequently, the harmonic sections of D-bundles over a compact Kähler manifold M turn out to be pluriharmonic, so that plurisubharmonic exhaustion functions are obtained in this case as the logarithm of the fiberwise (diagonalized) Bergman kernel functions with respect to the fiber coordinates centered at the images of harmonic sections. When there exist no harmonic sections, a plurisubharmonic exhaustion function is obtained as the squared length of the fiber vectors of a line bundle with constant transition functions of modulus one which is pluriharmonically equivalent to transition functions to the affine line bundle whose section at infinity is naturally identified with the locally constant section of the Cˆ -bundle associated to the D-bundle. The Hodge theory on M is applied here. We note that the logarithm of the Bergman kernel on D is log 1 , so that π(1−|z|2)2 1+|z| ∈ D it is asymptotically (quasi-)equivalent to the distance log 1−|z| from 0 with respect to the Poincaré metric. Based on this construction, it is easy to produce a strictly plurisubharmonic exhaustion function on the D-bundle when the D- bundle is over a compact Riemann surface and admits no holomorphic sections, so that a function theoretic consequence of Theorem 2.79 can be summarized as Theorem 2.80. Corollary of Theorem 2.80 1 For any D-bundle over a compact Riemann surface, being Stein is stable under small deformations. Somewhat more is proved in [Oh’15] answering a question whether or not D-bundles over analytic families of compact Riemann surfaces are locally pseudoconvex over the parameter spaces. Note that the set of equivalence classes of D-bundles over M is naturally identified with the set of equivalence classes of Aut D-representations of the fundamental groups of M, so that it carries a natural topology. This space contains the Teichmüller space of M when M is a compact Riemann surface, so that it seems to be the case that Theorem 2.80 is related to the convexity properties of the Teichmüller spaces that have been known by other methods. On the other hand, there is some independent interest in the properties of functions living in the special D-bundle D over a compact Riemann surface of 224 5 L2 Approaches to Holomorphic Foliations genus ≥2, say Σ = D/Γ for Γ ⊂ Aut D, such that D = D2/Γ for the action of Γ on D2 defined by

γ · (z, w) = (γ (z), γ (w)) (γ ∈ Γ,z,w ∈ D), and D → Σ induced by the projection (z, w) → w. D contains a holomorphic section of D2/Γ → D/Γ which is the image of the diagonal in D2 by the projection D2 → D2/Γ , so that D is not Stein. But D is a modiﬁcation of a Stein space because w − z − log 1 − 1 − zw is a plurisubharmonic exhaustion function on D which is strictly plurisubharmonic on D \ Σ. A recent study of Adachi [A’17] explores an explicit system of functions that generates O(D). Let us have a look at the main result of [A’17] because it is about the L2 space of holomorphic functions. For that, consider the inner products −| |2 α −| |2 α = 1 (1 z ) (1 w ) ∧ ∧ ∧ (f, g)α f g + (i dz dz) (i dw dw) Γ(α+ 1) D |1 − zw|4 2α for measurable functions f, g on D and α>−1, the weighted L2 spaces

~~2 D ={ ; 2 := ∞} Lα( ) f f α (f, f )α < and the weighted Bergman space~~

~~2 D = 2 D ∩ O D Aα( ) Lα( ) ( ).~~

~~Theorem 5.12 There exists an injective linear map~~

∞ ( : 0 O K⊗N −→ 2 D ⊂ O D I H (Σ, ( Σ )) Aα( ) ( ) N=0 α>−1 having dense image in O(D) such that I (ψ)(z, w) = ψ if N = 0, 1 w ψ(τ)(dτ)⊗N I (ψ)(z, w) = ⊗(N−1) B(N,N) z [w, τ, z]

~~≥ = ⊗N ∈ 0 O K⊗N if N 1 for ψ ψ(τ)(dτ) H (Σ, ( Σ )),~~

~~(w − z) dτ [w, τ, z]:= (w − τ)(τ − z) and B(p,q) is the beta function. 5.3 A History of Levi Flat Hypersurfaces 225~~

~~2 D =∞ − Corollary 5.2 dim Aα( ) if α> 1. Theorem 5.12 also entails a formula ∗ ⊗ I γ dτ N (z, w) = (γ (z) − γ(w))N ,N≥ 2. γ ∈Γ γ ∈Γ~~

Theorem 2.80 also synchlonizes with Hartshorne’s question on the complement of a compact complex curve C in a projective algebraic surface S. The question is if S \ C is Stein whenever C intersects with all the compact complex curves in S.It is natural to ask this in view of the properties of ample divisors and Proposition 5.8. Ueda’s work [U-2] on the neighborhood of complex curves of self-intersection zero gives a partial but deep answer to this question. See [B’90, B-I’92] and [K-O’17] for applications of Ueda’s theory to Levi flat hypersurfaces. Pluriharmonic sections are applied also to study the fundamental groups of compact Kähler manifolds (cf. [Td’99]). This direction is closely related to a conjecture by Shafarevich [Sh’74] asserting that the universal covering space of every compact Kähler manifold is holomorphically convex. In [E-K-P-R], the Shafarevich conjecture was solved when the fundamental group is residually finite. (iv) Levi flat hypersurfaces in torus bundles. In (ii), the projection from the ˆ domain Ω+ to the first factor C induces another fiber structure of Ω+; a bundle over C \{0} whose fibers are annuli. Nemirovski’s hypersurface is a Levi flat hypersurface of this type, generalizing this understanding of ∂Ω+. In particular, if S = CPn, B = Cn+1 \{0} and the fibration B → S is given by the natural projection, B/Z is a Hopf manifold, say H (n) := k k k n+1 (n) {{(2 z0, 2 z1,...,2 zn); k ∈ Z}; (z0,z1,...,zn) ∈ C \{0}}. Hence H (n) contains a Levi flat hypersurface, say X , defined by {Im z0 = 0} corresponding z1 zn (2) to s((z : z : ··· : zn)) = (1, ,..., ). X has an intriguing property: The 0 1 z0 z0 (2) line bundle associated to the divisor z1 = 0 is positive on X , i.e. it admits a fiber metric whose curvature form is positive along the leaves of LX(2) . However there exist no nonconstant real analytic map f : X(2) → CPn for any n ≥ 1 such that the restriction of f to the leaves of LX(2) are holomorphic, because otherwise f would extend to a meromorphic map on H (n) by Ivashkovitch’s theorem (cf. [I’92]), so that H (2) would be of algebraic dimension two, which is an absurdity. For the C∞ maps to CPn the situation becomes more delicate (see [A’14]). As for the related embeddability results for abstract Levi flat CR manifolds, see [Oh-S’00, Oh-28] and [H-M’17]. The above construction of Levi flat hypersurfaces can be generalized for other torus bundles (cf. [Oh’06],2[Oh-24]), and for the quotients of B by a more general action of Z (cf. [Oh-30]). Real analytic Levi flat hypersurfaces in Hopf surfaces will be classified from this viewpoint (see Theorem 5.3).

~~2The proof of Theorem 0.2 in [Oh’06] is incomplete. 226 5 L2 Approaches to Holomorphic Foliations~~

~~5.4 Levi Flat Hypersurfaces in Tori and Hopf Surfaces~~

Although it is still a big open problem whether or not CP2 contains no Levi ﬂat hypersurfaces, the nonexistence result on CPn for n ≥ 3 due to Lins Neto [LN]was generalized in several ways, as we have seen in Sect. 5.2.1. Therefore it is natural to proceed this way to see what can be said for Levi ﬂat hypersurfaces in other typical complex manifolds. We shall present such results in the case of several complex homogeneous manifolds and their deformations.

~~5.4.1 Lemmas on Distance Functions~~

Let Ω be a domain of holomorphy in Cn with ∂Ω =∅and let δ : Ω →]0, ∞[ be the Euclidean distance to ∂Ω. Recall that − log δ is plurisubharmonic on Ω.This follows from a more basic fact that, for each v ∈ Cn −{0}, the Euclidean distance from z ∈ Ω to ∂Ω ∩{z + ζv; ζ ∈ C},sayδv, has a property that − log δv is, as a function on Ω with values in [−∞, ∞[, plurisubharmonic. The functions δ and δv are naturally generalized on the domains in complex manifolds; δ of course makes sense on Hermitian manifolds and so does δv on holomorphically foliated Hermitian manifolds, while v has to be replaced by the foliation. In the latter case, δv is generalized as the distance along the leaves with respect to a semipositive (1, 1)-form inducing a metric on each leaf of the foliation. Let us consider a speciﬁc situation where Ω is a pseudoconvex domain in the ball Bn ={z ∈ Cn;z < 1} deﬁned by an inequality Re f(z)>0, where f(z)is a holomorphic function on Bn such that f(z)− zn = O(2) at the origin z = 0. Here O(ν) is Laudau’s symbol for

~~ν ∈ N.Letδ0(z) denote the distance from z = (z ,zn) ∈ Ω to ∂Ω∩{(z ,ζ); ζ ∈ C} with respect to the Euclidean metric. Lemma 5.2 (cf. [Oh-31]) There exists ε>0 such that~~

~~¯ − = rank(∂∂( log δ0)|Ker dzn )|z=(0,t) rank(df|Ker dzn )|z=0 holds if (0,t)∈ Ω and 0~~

~~= iL(z) + 2 + f(z) z2e cz1 O(3), where L(z) = az1 + bz2. Then, for sufﬁciently small t>0, the Taylor expansion of δ0 at (0,t)is calculated as~~

~~= + − + 2 + δ0(z1,z2) t cos(Re L(z)) Re(z2 t cz1) O(3) = − 2 + − + 2 + t t(Re L(z)) /2 Re(z2 t cz1) O(3). 5.4 Levi Flat Hypersurfaces in Tori and Hopf Surfaces 227~~

~~Hence~~

2 ¯ ¯ ¯ 4t (−∂∂ log δ0)|z=(0,t) = (t∂L∂L + 2 dz2 dz¯2)|z=(0,t). = ¯ − = Therefore a 0 if and only if rank(∂∂( log δ0)|Ker dz2 ) 0at(0,t). As an immediate consequence one has the following.

Proposition 5.9 The Levi form of the level set of δ0(z) at (0,ζ) ∈ Ω is zero for 2 sufﬁciently small ζ if and only if ∂ f/∂zj ∂zn(0) = 0 for j = 1, 2,...,n− 1. Putting Proposition 5.8 in another way, we obtain: Theorem 5.13 Let Ω and f be as above. If there exists ε>0 such that ¯ rank ∂∂(− log δ0) = 1

~~on Ω ∩{(0,ζ); 0 < |ζ | <ε}, then there exists ε > 0 such that ∂f = dzn holds at (0,ζ)if |ζ | <ε .~~

Thus we know how − log δ0 detects the tilt of the leaves of L∂Ω. It is expected that − log δ also has such a property. K. Matsumoto [M] made it explicit in the case where ∂Ω is the graph of a holomorphic function, i.e. when f is of the form zn − g(z1,...,zn−1). In this situation, let Ω ={f = 0},let ∂2g G = (0) ∂zj ∂zk 1≤j,k≤n−1 and let ∂2(− log δ) Φ(ζ) = (0,ζ) . ∂zj ∂z¯k 1≤j,k≤n−1

~~Lemma 5.3 There exists ε>0 such that~~

~~1 − Φ(ζ) = GG(I¯ − GG)¯ 1 2 holds for 0 < |ζ | <ε.HereI denotes the identity matrix.~~

~~Proof Let w = (w1,...,wn−1) and put~~

~~n−1 2 2 α(z,w) = |zj − wj | +|zn − g(w)| . j=1~~

~~Then, on a neighborhood of 0, δ(z)2 is characterized as α(z, w(z)), where w(z) is the solution to the functional equation~~

~~∂α ∂α (z, w(z)) = 0 and (z, w(z)) = 0 (1 ≤ j ≤ n − 1). (5.9) ∂wj ∂w¯j 228 5 L2 Approaches to Holomorphic Foliations~~

~~Therefore~~

~~∂δ2 ∂α = = zj − wj (1 ≤ j ≤ n − 1) ∂zj ∂zj and~~

~~2 2 ∂ δ ∂w¯ j = δjk − (1 ≤ j ≤ n − 1). ∂zj ∂zk ∂z¯k~~

~~By differentiating (5.9) one has~~

~~− ∂2α n 1 ∂2α ∂w ∂2α ∂w¯ + + = 0 ∂wj ∂zk ∂wj ∂w ∂zk ∂wj ∂w¯ ∂zk =1 and~~

~~− ∂2α n 1 ∂2α ∂w ∂2α ∂w¯ + + = 0. ∂w¯j ∂zk ∂w¯j ∂w ∂zk ∂w¯j ∂w¯ ∂zk =1~~

~~On the other hand, from the deﬁnition of α, one has~~

~~∂2α ∂α = 0, =−δjk, ∂wj ∂zk ∂w¯ j ∂zk ∂2α ∂2g = (g(w) − zn) ∂wj ∂wk ∂wj ∂wk and~~

~~∂2α ∂g ∂g¯ = δjk + . ∂wj ∂w¯k ∂wj ∂w¯ k~~

~~Hence it is easy to see that~~

~~n−1 2 ∂w¯ j ∂ g ∂w (0,ζ)= ζ¯ (0,ζ) ∂zk ∂wj ∂w ∂zk =1 and~~

~~n−1 2 ∂wj ∂ g¯ ∂w¯ (0,ζ)− δjk = ζ (0,ζ). ∂zk ∂w¯ j ∂w¯ ∂zk =1 5.4 Levi Flat Hypersurfaces in Tori and Hopf Surfaces 229~~

~~Hence ∂wj ∂w (0,ζ)− δ =|ζ |2 G G m (0,ζ). ∂z jk j m ∂z k ,m k~~

~~Therefore~~

~~2 2 ∂ δ − − (0,ζ) = I − (I − GG)¯ 1 =−GG(I¯ − GG)¯ 1. (5.10) ∂zj ∂z¯k 1≤j,k≤n−1~~

~~Since~~

2 2 − 2 2 2 2 =| | ∂δ = ∂ ( log δ) = 1 − 1 ∂ δ + 1 ∂δ ∂δ δ(0, 0) ζ , (0,ζ) 0 and 2 4 , ∂zj ∂zj ∂z¯k 2 δ ∂wj ∂w¯k δ ∂zj ∂z¯k the desired formula follows from (5.10).

~~5.4.2 A Reduction Theorem in Tori~~

Let T be a complex torus of dimension n and let X be a connected Levi flat ω 1,0 ⊂ hypersurface of class C in T . We shall say X is holomorphically flat if TX Ker σ for some nonzero holomorphic 1-form σ on T . Since T is the quotient of Cn n by the action of a lattice Γ ⊂ C , the leaves of LX for any holomorphically flat X are images of complex affine hyperplanes of Cn by the projection Cn → T . X is said to be flat if it is the image of an affine real hyperplane. We shall call a Kähler metric ω onT a flat metric if there exist holomorphic 1-forms σ1,...,σn on T such = n ∧ ∈ that ω i j=1 σj σ j .Aflat coordinate around x T is by definition a local coordinate z = (z1,...,zn) around x such that dzj are extendible holomorphically to T as holomorphic 1-forms. Let us denote by δω(z) the distance from z ∈ T to X with respect to ω. Since δω(z) is the infimum of the distances from z to the leaves of LX, Lemma 5.3 implies the following: ¯ Proposition 5.10 X is holomorphically flat if and only if ∂∂ log δω has rank one near X. If X is holomorphically flat, it is clear that either X is flat or there exists a surjective holomorphic map from T to a complex torus of dimension one, say p, such that X is the preimage of some simple closed curve by p. Therefore, the remaining interest is to classify the rest. What is known at present is the following. Theorem 5.14 If X is not holomorphically flat, there exist a complex torus T of dimension two, a surjective holomorphic map π : T → T , and a Levi flat hypersurface X ⊂ T such that π −1(X ) = X. 230 5 L2 Approaches to Holomorphic Foliations

~~Proof Let x be any point of X and let L be the leaf of LX containing x.Letz be a~~

ﬂat coordinate of T around x such that L ={z ; zn = f(z)} on a neighborhood = = of x for some holomorphic function f in z (z1,...,z n−1) satisfying df 0at = = n ∧ ¯ z 0. Then, with respect to the ﬂat metric ωx i j=1 dzj dzj ,

¯ ≤ rank ∂∂ log δωx 2 ¯ near x. In fact, if rank ∂∂ log δωx were strictly greater than two at points arbitrarily close to x, then by Lemma 5.3 one would have ∂2f rank ≥ 2 ∂zj ∂zk 1≤j,k≤n−1 at z = 0, which means by the real analyticity of X that the same would be true at almost all points of X with respect to some flat coordinates. This implies, by the real analyticity and linear algebra, that there exist flat metrics ω1,...,ω2n−1 on ⊃ 2n−1 − T and a neighborhood U X such that j=1 ( log δωj ) is a plurisubharmonic function whose Levi form has everywhere at least three positive eigenvalues on U \ X. But this is impossible by Theorem 5.7. Similarly, Lemma 5.3 implies that ¯ ∂∂(log δω + log δω ) has at most two nonzero eigenvalues near X, for any choice 1 2 ¯ of flat metrics ω1 and ω2, and that the zero-eigenspace of ∂∂ log δωx is equal to (the parallel translates of) Ker F at (0,ζ) for 0 < |ζ |1. Therefore Ker F must be actually all parallel as long as they are of dimension n − 2. Since they are contained in the tangent spaces of the foliation extending LX and X is not holomorphically flat, they have to be tangent to complex subtori of codimension two. They are the fibers of the desired fibration over T . Corollary 5.3 Real analytic Levi flat hypersurfaces in a complex torus without nonconstant meromorphic functions are flat. Proof Clearly, it suffices to prove that they are holomorphically flat. If they are not holomorphically flat, then take a reduction X as above, which is also real analytic. If X were not holomorphically flat, the Gauss map from X associating to x the tangent space of LX at x is a nonconstant map to the Riemann sphere, which extends holomorphically to a neighborhood of X and hence meromorphically to T because T \ X turns out to be Stein by Lemma 5.3. (As for the Hartogs-type extension theorem for meromorphic functions, see [Siu-2] and [M-P] for instance.)

The argument in the proof of Theorem 5.13 was originally used to prove the following more general assertion. Theorem 5.15 (cf. [Oh-24]) Let T be a complex torus, let A ⊂ T be a closed set, and let F be a singular holomorphic foliation of codimension one on a neighborhood U of A such that A is a stable set of F . Suppose that the deﬁning sheaf F of F is locally generated by a closed 1-form and topologically trivial on 5.4 Levi Flat Hypersurfaces in Tori and Hopf Surfaces 231

U. Then, either F is generated by a holomorphic 1-form on T or there exist a two- dimensional complex torus T and a surjective holomorphic map π : T → T such that A = π −1(π(A)). Moreover, if A is not a complex analytic subset of T , F = π −1(F ) holds on a neighborhood of A for some singular holomorphic foliation F on a neighborhood of π(A). Concerning singular holomorphic foliations of codimension one on complex tori, M. Brunella [Br-3] proved the following. Theorem 5.16 Let F be a singular holomorphic foliation of codimension one on a complex torus T such that Sing F =∅. Then there exist a complex torus T ,a surjective holomorphic map π : T → T , and a singular holomorphic foliation F −1 on T such that π (F ) = F and NF is positive. Remark 5.5 There exist plenty of singular holomorphic foliations of codimension one on any algebraic complex torus whose normal bundle is positive, for instance those induced from that on CPn by a branched covering map. It is easy to see that the normal bundle of a (nonsingular) holomorphic foliation of codimension one on a complex torus is never positive. Although our knowledge on Levi flat hypersurfaces in complex tori is generally incomplete because of the lack of classification in dimension two, we have still a complete classification at least when the torus has no nonconstant meromorphic functions (cf. Corollary 5.2). Therefore it is natural to expect that Levi flat hypersurfaces in non-algebraic surfaces can be classified similarly. In the case of Hopf surfaces, all the real analytic Levi flat hypersurfaces will be described below.

~~5.4.3 Classiﬁcation in Hopf Surfaces~~

The Hopf manifolds, introduced by H. Hopf [Hf] are most typical non-Kähler manifolds. By definition, they are compact complex manifolds of dimension n ≥ 2 whose universal covering space is Cn \{0}.WeshallclassifyCω Levi flat hypersurfaces in Hopf surfaces based on the classification of Hopf surfaces (n = 2) by Kodaira [K-4]. Let H be a Hopf surface and let π : C2 \{0}→H be the universal covering. By analyzing the group Gal (H ,π) := {σ ∈ Aut (C2 \{0}); π ◦ σ = π}, Kodaira proved that every Hopf surface is isomorphic to the quotient of H with Gal (H ,π)=∼ Z by a fixed point free action of a finite group. H is called primary if Gal (H ,π) =∼ Z, and of diagonal type if one can find τ ∈ Aut (C2 \{0}) such that Gal (H ,π◦τ)is generated by the transformation (z, w) → (αz, βw) for some α, β ∈ D\{0}. It is known that a Hopf surface of diagonal type admits a nonconstant meromorphic function if and only if αj = βk holds for some j,k ∈ N. In general, for a primary Hopf surface H , Kodaira proved, applying a normalization due to Lattes [L], that one can find τ such that a generator of Gal (H ,π ◦ τ) is given by

(z, w) −→ χ(z, w) = (αz + λwm,βw), 232 5 L2 Approaches to Holomorphic Foliations where 0 < |α|≤|β| < 1 and either λ = 0orα = βm. Although the choice of (α,β,λ,m)is not unique for one H , we shall denote the primary Hopf surfaces by H (α,β,λ,m), and the associated covering map C2 \{0}→H (α,β,λ,m)by π. Kodaira [K-4] proved that every primary Hopf surface is diffeomorphic to S1 × S3. Diffeomorphism types of general Hopf surfaces were classified by M. Kato [Ka]. As a basic complex analytic property of Hopf surfaces one has the following. Proposition 5.11 A Hopf surface with a nonconstant meromorphic function is holomorphically mapped onto CP1(= Cˆ ). Proof Let f be a nonconstant meromorphic function on H and let C = f −1(0). Then C does not intersect with f −1(1). In fact, had it not been the case, then the self-intersection number of C would be positive, so that the transcendence degree of the field of meromorphic functions over C would be two (cf. Theorem 2.45 for instance), which is absurd because it would imply that H is projective algebraic by the classical Chow–Kodaira theorem (cf. [C-K]). Therefore H is holomorphically mapped onto CP1 by f . Remark 5.6 It is easy to see that a primary Hopf surface with a nonconstant meromorphic function is of diagonal type. Let X be a real analytic Levi flat hypersurface in a primary Hopf surface H .We shall give explicit descriptions for X by assuming that H is primary, putting aside a question which X is invariant under finite group actions. Let us first assume that 1,0 H \ X is Stein. In this situation, the section of the projectivization P(TH ) over X induced by T 1,0 is extended to H as a meromorphic section, say h. It was observed LX in [Oh-30] that X can be described by studying the intersection of h(H ) and the levels of a nonconstant meromorphic function on P(T 1,0H ). For that, the first step is of course the following. 1,0 Proposition 5.12 P(TH ) admits a nonconstant meromorphic function. 1,0 2 Proof Let χ(z, w) be as above. Then TH is equivalent to the quotient of (C \ {0}) × C2 by the action of the group generated by (χ, dχ) . Letting ((z, w), (ξ, η)) C2 ×C2 ∂ + ∂ be the coordinate of so that (ξ, η) represents the vector ξ ∂z η ∂w, one has

~~− ∂ ∂ dχ(ξ,η) = (αξ + mλwm 1η) + βη . ∂z ∂w~~

~~ξw = Therefore ηz is invariant under (χ, dχ) if λ 0, and so is the function~~

~~ξ − mz ηwm−1 wm otherwise. 5.4 Levi Flat Hypersurfaces in Tori and Hopf Surfaces 233~~

Applying this, Cω Levi ﬂat hypersurfaces in primary Hopf surfaces can be speciﬁed. In order to give the description of these, let us recall the construction of Nemirovski in [Nm] in this situation. Let p : C2 \{0}→CP1 be the natural projection. Let ζ = z/w be the inhomogeneous coordinate of CP1 and let

~~U+ ={ζ ; 0 ≤|ζ | < ∞},~~

~~U− ={ζ ; 0 < |ζ |≤∞}.~~

~~Let ω+ and ω− respectively be meromorphic 1-forms on U+ and U− satisfying~~

~~ω+ − ω− = d log ζ on U+ ∩ U−.~~

Then, parallel transports of the points in C2 \{0} are deﬁned over the paths avoiding − − the poles of ω±. In terms of the ﬁber coordinates w, z of p 1(U+), p 1(U−),the parallel transport along γ :[0, 1]→U± is given by ω+ −1 w −→ we γ on p (U+) and ω− −1 z −→ ze γ on p (U−).

Let P∞ be the union of the sets of poles of ω+ and ω−. Then, for any point ζ0 ∈ 1 −1 CP \ P∞, a closed real analytic curve C in p (ζ0) yields a Levi flat hypersurface − in (C2 \{0}) \ p 1(P∞) as long as the parallel transport along a curve γ as above with γ(0) = γ(1) = ζ0 with respect to ω± leave C invariant. In such a case, if moreover the closure of the union of the parallel transports of C in C2 \{0} is a smooth hypersurface, we shall call it a Levi flat hypersurface of Nemirovski type since it is an analogue of S¯ ⊂ E∗ in Sect. 5.1.2. Theorem 5.17 Let X be a real analytic Levi flat hypersurface with Stein complement in a Hopf surface H (α,β,λ,m).Ifλ = 0 or m = 1, then the preimage of X by the covering map π is of Nemirovski type. 1,0 Proof Let h be a meromorphic section of the bundle q : P(TH ) → H induced by T 1,0.Ifλ = 0, then ξ/η is constant on h(H ), because otherwise there would be LX nonconstant meromorphic functions on H and h(H ) ∩ (ξw/ηz)−1(c) are mapped by q to complex curves in H which intersect with the images of z = 0 and w = 0. This contradicts the Chow–Kodaira theorem. Therefore, there exists a meromorphic ∂ + ∂ ∂ + 1 ∂ function c in z/w such that the vector field c ∂z ∂w or ∂z c ∂w is everywhere tangent to π −1(X). Hence π −1(X) is of Nemirovski type. If m = 1, a similar argument applies to the function (ξ/η − z/w)/(z/w − λ) instead of ξw/ηz. 234 5 L2 Approaches to Holomorphic Foliations

~~A similar method works for the case λ = 0 and m ≥ 2 to describe a Cω Levi ﬂat hypersurface X with Stein complement in terms of the holomorphic map~~

p˜ : C2 \{0}−→CP1, where awm p(z,˜ w) = z + : w m and a is a constant such that the image of h is contained in the preimage of a by ξ/ηwm−1 −mz/wm. Namely, the preimage of X by p˜ is “of generalized Nemirovski type” (cf. [Oh-30]). The following is due to Kim, Levenberg and Yamaguchi [K-L-Y] and Levenberg and Yamaguchi [L-Y-2]. (See also Miebach [Mb].) Theorem 5.18 Let X be a real analytic Levi ﬂat hypersurface in a Hopf surface of diagonal type H = H (α, β, 0, 0).IfH \ X is not Stein, then X is either of the form

log |α| k|z| log |β| =|w| (k > 0) or the preimage of a Jordan curve in CP1 by a surjective holomorphic map. Recently, Theorem 5.17 was complemented by the following. Theorem 5.19 (cf. [Oh-31]) A primary Hopf surface is of diagonal type if and only if it contains a real analytic Levi ﬂat hypersurface whose complement is not Stein. For the proof, Lemma 5.2 is crucial.

~~5.5 Notes and Remarks~~

Theorem 4.22 has a counterpart in foliation theory. Brunella [Br’03-1, Br’03-2, Br’05] extended a variational calculus for the Green functions of Riemann surfaces for Stein families, due to Yamaguchi [Y’81] and Kizuka [Kz’95], to that of leafwise Poincaré metrics of foliations. Although it is hard to imagine any counterpart of the L2 extension theorem, it tuned out in [Oh’18] that Theorem 2.89 is related to some stability property of affine line bundles. Proposition 5.4 was extended there as follows. Theorem 5.20 Let T be a complex manifold, let p : S → T be a proper holomorphic map with smooth one-dimensional fibers, and let q : L → S be an analytic affine line bundle. Then p ◦ q : L → T is locally pseudoconvex if one of the following conditions is satisfied. References 235

(i) Fibers Lt (t ∈ T)of p ◦ q are of negative degrees over the fibers St of p. (ii) Lt are topologically trivial over St and not analytically equivalent to line bundles. (iii) L → S is a U(1)-flat line bundle. It also turned out that not all naturally arising analytic families of affine line bundles are locally pseudoconvex. Example 5.2 Let A be a complex torus of dimension one (i.e. an elliptic curve), say A = (C \{0}/Z), where the action of Z on C \{0} is given by z → emz for m ∈ Z. Over the product A × C as an analytic family of compact Riemann surfaces over C, we define an affine line bundle F : A × C the quotient of the trivial bundle ((C \{0}×C) × C) × C → (C \{0}×C) by the action of Z defined by (z,t,ζ) → (emz, t, ζ + mt). Suppose that F is locally pseudoconvex with respect to the map π : F → C induced by the projection to the second factor of A × C. Then there will exist a neighborhood V 0 such that π −1(V ) is weakly 1-complete. Then, since the canonical bundle of F is obviously trivial, holomorphic functions on π −1(t) must be holomorphically extendable by Theorem 2.89. But this will mean that π −1(0) can be blown down to C in F because the other fibers of π are equivalent to (C \{0})2. This contradicts that the normal bundle of the divisor π −1(0) is trivial. As for Conjecture 5.1.1, Brinkschulte [Brs’18] has given a partial answer, which is decisive for the foliations with Levi flat stable sets. Theorem 5.21 Let M be a complex manifold of dimension n ≥ 3. Then there does not exist a smooth compact Levi-flat real hypersurface X in M such that the normal bundle to the Levi foliation admits a Hermitian metric with positive curvature along the leaves.

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A Cohomology group of X supported in Φ, 37 Addition formula, 143 Cokernel sheaf, 22 Adjunction formula, 33 1-complete, 16, 66 Akizuki–Nakano vanishing theorem, 61 Complete, 54 Ample, 60 Complex, 23 Analytic set, 26, 27 Complex curve, 28 Analytic sheaves, 22 Complex hyperplane, 11 Atlas, 10 Complex hypersurface(s), 6, 11 Complex Lie group, 17 Complex manifold, 10 B Complex projective space, 10 Basic inequality, 58 Complex semitorus, 12 Bergman stable, 199 Complex singularity exponent, 144 Biholomorphic, 11 Complex space, 27 Briançon–Skoda theorem, 141 Complex submanifold, 11 Bundle homomorphism, 32 Complex torus, 12 Constant sheaf, 20 Contractible, 89 C Converge V -weakly, 79 Canonical bundle, 33 Convex increasing, 17 Canonical flabby extension, 20 Coordinate transformations, 10 Canonical flabby resolution, 23 CR manifold, 105 Cartan’s coherence theorem, 26 Cr -smooth, 18 Cartan–Serre finiteness theorem, 30 Current of type (p, q), 38 Cartan’s theorem A, 28 Cartan’s theorem B, 29 Cauchy’s formula, 2 D Chart, 10 Defining function, 18 Cheng’s conjecture, 198 Defining sheaf, 207 Chern connection, 41 Degree, 41, 55 Closed complex submanifolds, 11 Demailly’s approximation theorem, 182 Codimension, 11 Diederich–Fornaess index, 198 Coherent, 25 Dimension, 26, 28 Coherent analytic sheaves, 27 Direct image sheaf, 20 ∂¯-Cohomology, 35 Direct sum, 20

~~© Springer Japan KK, part of Springer Nature 2018 255 T. Ohsawa, L2 Approaches in Several Complex Variables, Springer Monographs in Mathematics, https://doi.org/10.1007/978-4-431-56852-0 256 Index~~

Distinguished polynomials, 5 Hodge’s star operator, 52 Divisor, 34 Holomorphic, 7 Dolbeault complex, 35 Holomorphic affine line bundle, 209 Dolbeault’s isomorphism theorem, 36 Holomorphically convex, 13 Dolbeault’s lemma, 35 Holomorphically convex space, 28 Domain, 47 Holomorphically flat, 229 Domain of holomorphy, 8 Holomorphic bisectional curvature, 96 Domain over a topological space, 7 Holomorphic disc, 11 Domains of meromorphy, 21 Holomorphic foliation, 205 Donnely–Fefferman vanishing theorem, 65 Holomorphic function(s), 2, 27 Holomorphic line bundles, 33 Holomorphic map, 11, 27 E Holomorphic motion, 110 Effective divisor, 34 Holomorphic sectional curvature, 107 Embedding, 11 Holomorphic tangent bundle, 15 E-valued Dolbeault cohomology group, 35 Holomorphic vector bundle, 32 Exact, 32 Homogeneous coordinate, 11 Exact sequence, 23 Homogeneous lexicographical order, 145 Exhaustion function, 16 Homomorphism, 21 Extension, 7 Hopf manifolds, 231 Hyperconvexity index, 198 Hyperplane section bundle, 33, 34 F Factor of automorphy, 91 Family of supports, 36 I Fiber metric, 41 Ideal sheaf of A, 26 Fine, 24 Image sheaf, 22 Finitely sheeted, 7 Initial coefficient, 145 Finite Riemann surface, 111 Initial monomial, 145 First Chern class, 41 Initial term, 145 Flabby sheaves, 20 Integral closure, 140 Flat, 229 Invertible, 22 Flat coordinate, 229 Irreducible, 27 Flat metric, 229 Irreducible component, 27 Foliation, 205 Isomorphic, 11, 21 Foliation by rational curves, 208 Free resolution, 26 Free R-module of rank m, 22 J F -valued p-th cohomology classes, 23 Jacobian ideal, 141 Jacobi’s identity, 56

G Garuert potential, 72 K Germs, 20 Kähler manifold, 58 Grauert metric, 72 Kähler space, 71 Kernel, 22 Kodaira–Nakano vanishing theorem, 60 H Hartogs domain, 16 Hartogs’s continuation theorem, 3 L Hermitian complex space, 71 L2 ∂¯-cohomology groups, 53 Hermitian holomorphic vector bundle, 42 L2-dualizing sheaf, 68 Hermitian metric, 41, 71 Leaves, 205 Hironaka’s division theorem, 145 Lelong number, 152 Index 257

Levi flat hypersurface, 212 Oka’s normalization theorem, 30 Levi form, 16 Openness conjecture, 144 Levi problem, 16 L2 extension theorem, 120 L2 extension theorem on manifolds, 123 P Linear foliations, 209 Plurigenus, 127 Link, 86 Pluriharmonic, 12, 223 Local coordinates, 10 Pluripolar, 93 Local defining function, 26 Plurisubharmonic, 31 Local (holomorphic) frame, 32 Plurisubharmonic function, 9, 12 Locally closed, 11 Pompeiu’s formula, 35 Locally finitely generated, 25 Positive, 59 Locally free, 22 Possibly discontinuous sections, 19 Locally pseudoconvex domain over M, 17 (P, q)-convex-concave, 45 Local trivialization, 33 (P, q)-current, 38 Log canonical threshold, 144 Presheaf, 19 Long exact sequence, 25 Primary, 231 Prime element, 6 Projective algebraic manifolds, 13 M Projective algebraic sets, 27 Maitani–Yamaguchi theorem, 185 Pseudoconvex, 18, 31, 66 Meromorphic connection, 210 Pseudoconvex function, 9 Meromorphic functions, 21 Pseudoconvex manifolds, 17 Minimal local defining function, 6 P -th cohomology group, 23 Modification, 11 P -th direct image sheaf, 30 Morphism, 129 Multiplier dualizing sheaf, 68 Multiplier ideal sheaf, 64 Q Q-complete, 31 Q-concave, 87 Q-convex, 16, 31 N Q-convex map, 101 Nadel’s coherence theorem, 142 Quasi Stein manifold, 121 Nadel’s vanising theorem, 68 Nadel’s vanishing theorem on complex spaces, 68 R Nakano positive, 59 Ramadanov conjecture, 197 Nakano semipositive, 59 Reduced, 28 Nakano’s identity, 57 Reduction, 28 Nef, 69 Regular point, 27 Nemirovski’s hypersurface, 211 Reinhardt domain, 8 Neumann operator, 100 Relatively prime, 6 Nonsingular, 27 Remmert reduction, 45 Normal, 30, 42, 183 Remmert’s proper mapping theorem, 30 Reproducing kernel, 167 Reproducing kernel Hilbert space, 166 O Resolution, 23 Of diagonal type, 231 Restriction, 20 Of dimension n, 10 Restriction formula, 143 Of pure dimension n, 10 Riemann domain, 7 Of self-bounded gradient, 66 Riemann surfaces, 10 Oka–Grauert’s theorem, 17 Ringed space, 21 Oka’s coherence theorem, 25 R-modules, 21 Oka’s lemma, 9 Rückert’s Nullstellensatz, 26 258 Index

S T SBG, 66 Tautological line bundle, 33 Second fundamental form, 130 Toroidal groups, 91 Section(s), 11, 20 Toroidal reduction, 91 Semicharacter, 90 Torsion free, 22 Semipositive, 59 Totally real, 44 Semipositive curvature current, 64 Tower, 183 Serre’s duality theorem, 38 Transition function, 33 Set of interpolation, 122 Turbulent foliation, 210 Sheaf, 19 Sheafification, 20 U Sheaf projection, 20 Underlying space, 27 Singular fiber metric, 63 Uniformly discrete, 124 Singular holomorphic foliation, 207 Unique factorization domain, 6 Singular set, 207 Unit, 6 Stable set, 208 Upper uniform density, 124 Stalk, 20 Standard basis, 146 Stein factorization, 45 V Stein manifold, 13 Varieties, 27 Stein space, 28 Very ample, 60 Strictly plurisubharmonic, 16 Strictly positive curvature current, 64 W Strongly pseudoconvex, 18 Weakly q-convex, 16 Strongly pseudoconvex CR manifold, Weakly 1-complete, 66 106 Weakly 1-complete manifolds, 17 Strongly pseudoconvex domain, 18 Weakly pseudoconvex, 18 Strongly pseudoconvex manifolds, 17 Weierstrass division theorem, 6 Structure sheaf, 21, 27 Weierstrass polynomials, 5 Subadditivity theorem, 143 Weierstrass preparation theorem, 5 Subdomain, 7 Weighted log canonical threshold, 144 Subharmonic function, 3 Subsheaf, 20 Suita’s conjecture, 178 Z Support, 20, 145 Zero section, 19