On a Class of Algebraic Surfaces with Numerically Effective Cotangent Bundles

Home , Algebraic surface

ON A CLASS OF ALGEBRAIC SURFACES WITH NUMERICALLY EFFECTIVE COTANGENT BUNDLES

A Thesis

Presented in Partial Fulﬁllment of the Requirements for

the Degree Doctor of Philosophy in the

Graduate School of the Ohio State University

Hongyuan Wang, B.S., M.A., M.A.S.

*****

The Ohio State University 2006

Doctoral Examination Committee:

Fangyang Zheng, Advisor Approved by Zbigniew Fiedorowicz

Andrzej J. Derdzinski Advisor Department of Mathematics ABSTRACT

In this dissertation, we will present two new results in complex geometry. The ﬁrst one is for a borderline class of surfaces with nef cotangent bundle. The statement of the main theorem on this topic is as follows, suppose M is a general type surface,

2 0 such that the two Chern numbers satisfy c1 = c2, and P ic(M)/Pic (M) is generated by KM , then every nontrivial extension of the holomorphic tangent bundle TM by

KM will be Hermitian ﬂat. As a corollary, we obtain that every surface with the

1 above described properties and with h (T K∗ ) > 0 will be a generalized theta M ⊗ M divisor. The second result is an extended version of Hard Lefschetz theorem for the cohomology with values in Nakano semipositive vector bundles on a compact K¨ahler manifolds.

ii ACKNOWLEDGMENTS

I am indebted to my advisor, Prof. Fangyang Zheng, for his patience, expert guidance and constant encouragement which made this work possible. I am grateful for the numerous enlightening discussions and suggestions as well as the intellectual support he has provided.

I would like to thank my committee members Prof. Andrzej J. Derdzinski and

Prof. Zbigniew Fiedorowicz for their invaluable time and professional insight. Special thanks go to their inspiring classes, Ph.D candidate general test and enlightening discussions.

I would like to thank Prof. Herbert Clemens and his students for the group study and invaluable discussions.

I am thankful for the Department of Mathematics for their support expressed through special graduate assignments and fellowships that I enjoyed.

Finally, I wish to express my deep gratitude to my family, especially my wife, for all their love, understanding and support. Special gratitude goes to my older uncle for his education to me through the years, without him this degree will be impossible.

iii VITA

October, 1978 ...... Born - Wuxi, Jiangsu, P.R.China.

1993-1998 ...... B.S.,Department of Special Class of Gifted Young, University of Science and Tech- nology of China. Hefei Anhui, China.

2002 - present ...... Graduate Teaching and Research Asso- ciate, Department of Mathematics, The Ohio State University.

FIELDS OF STUDY

Major ﬁeld: Mathematics

Specialization: Complex Geometry, Compact Complex Surfaces

iv TABLE OF CONTENTS

Abstract...... ii

Acknowledgments...... iii

Vita ...... iv

ListofFigures ...... vi

CHAPTER PAGE

1 Introduction...... 1

2 Stabilityofvectorbundles ...... 11

2.1 Coherentsheaftheory...... 11 2.2 Semistable and stable sheaves ...... 18

3 Positivity of Hermitian vector bundles ...... 22

3.1 Curvatureofvectorbundles...... 22 3.2 Nakano positivity, Griﬃths positivity and Numerically effectiveness ...... 27 3.3 Amplevectorbundles ...... 32

4 Inequalities for Chern classes ...... 37 5 MaintheoremspartI...... 40

6 MaintheoremspartII ...... 60

Bibliography ...... 69

v LIST OF FIGURES

FIGURE PAGE

1.1 TheGeographyofChernNumbers ...... 5

vi CHAPTER 1

INTRODUCTION

A major goal in complex geometry is to classify compact complex manifolds in a given dimension up to birational equivalence classes. In early 19th century, Castel- nuovo, Enriques and many others had succeeded in creating an impressive geomet- ric theory of birational classiﬁcation for smooth algebraic surfaces. It was discov- ered that main birational invariants for an algebraic surface M are the ﬁrst Betti

1 number b1(M) = h (M, C) and the plurigenera Pn(M) of M, which is deﬁned by

0 n Pn(M)= h (M,KM⊗ ).

After sheaf theory had been developed, and applied by Hodge, Serre, Hirzebruch,

Grothendieck and others to analytic and algebraic geometry, the classiﬁcation questions got a decisive progress since 1950’s. In 1960’s and 1970’s Kodaira and others carried out the classiﬁcation project for projective surfaces, and also extended it to

1 cover all compact complex surfaces. It is now referred to in the literature as Enrique-

Kodaira classiﬁcation theory. Given a compact complex surface M, there are four possibilities for its plurigenera:

1), all Pn(M) vanish;

2), not all Pn(M) vanish, but all are either 0 or 1;

3), P (M) n; n ∼

4), P (M) n2. n ∼

Nowadays this fact is expressed by saying that the Kodaira dimension kod(M) of M is respectively , 0, 1, or 2. The algebraic surfaces in class 1) are formed by all −∞ the birationally ruled surfaces, namely, those M that is birational to P1 C, for any × compact complex curve C. The non-algebraic surfaces in class 1) are so called V II0- class, which are defined as compact complex surfaces with first Betti number b1 =1 and all Pn = 0. For instance, all Hopf surfaces (i.e. compact complex surface with universal cover C2 0 ) belong to this set. There are other examples that belongs to \{ } this set as well. However, an exhaustive list is yet to be established. For case 2), there are four types of Kählerian surfaces distinguished by their first Betti numbers and plurigenera, namely the birational classes of tori, hyperelliptic surfaces, K-3 surfaces and Enriques surfaces. The non-Kähler ones in case 2) consists of Kodaira surfaces

2 which are all affine quotients of C2. The case 3) are all elliptic surfaces, i.e. there is a smooth curve B and a surjective morphism p : M B whose generic fibre is an → ellpitic curve. Kodaira had studied all possible types of singular fibres, there are 7 of them, and gave a complete classification of this type of surface. Finally, surfaces in class 4), are simply called surfaces of general type. This is of course means 99.99% of the surfaces, and not much is known about this class except some very basic results.

It is known that, for any surfaces with non-negative Kodaira dimension, there is a unique surface in each birational equivalent class that is minimal, i.e. can not be blown down any further. This surface is called the minimal model of the birational class.

By Gieseker’s thoerem [Gies], p236, we know that there exists a quasi-projective coarse module scheme for minimal surfaces of general type with ﬁxed Chern numbers

2 c1 and c2. However, little is known about the structure of the Gieseker scheme in

2 general. Even the very basic questions such as, for which pair (c2,c1) is the Gieseker

2 scheme non-empty, are not completely answered. The known restrictions on (c2,c1) are expressed by

1.1. Theorem If M is any minimal surface of general type, then

1), c2(M)+ c (M) 0(12), 1 2 ≡

3 2 2), c1(M) > 0 and c2(M) > 0,

3), c2(M) 3c (M), 1 ≤ 2

4), 5c2(M) c (M)+36 0 (if c2(M) is even), 5c2(M) c (M)+30 0 (if c2(M) 1 − 2 ≥ 1 1 − 2 ≥ 1 is odd).

2 Let c1 = n and c2 = m. This theorem tells us that no integer pair (m, n) with n> 3m or 5n m + 36 < 0, can be the two Chern numbers of any minimal general − type surface. In light of this, the question now is following:

If D Z2 := (m, n) m Z, n Z is given by the inequalities 0 < n 3m, ⊂ { | ∈ ∈ } ≤ m 5n + 36 and 12 (n + m), then for which pair (m, n) D, does there exist a ≤ | ∈ 2 minimal surface of general type M with c1 = n and c2 = m?

One can divide D into two parts D := (m, n) D n 2m and D := 1 { ∈ | ≤ } 2 (m, n) D n> 2m . D is the region where τ = 1 p = 1 (c2 2c ) 0. Here τ is { ∈ | } 1 3 1 3 1 − 2 ≤ the signature of M and p1 is the ﬁrst Pontryagin number. Similarly, D2 is the region for those M with positive signature. The simple examples such as, the complete intersections or the theta divisors, which are the smooth ample divisors of the three dimensional abelian varieties, usually lie in the region D1. In particular, the theta

2 divisors have c1 = c2, so they are on the line n = m. Indeed for a long time only

4 very few examples with Chern pairs in D2 were known. The geography of the Chern numbers is plot as Figure 1.1.

2 n =3m n =2m n = m n = c1

D2 D

5n m + 36 0 − ≥

m = c2

Figure 1.1: The Geography of Chern Numbers

Although we can easily find lots of pairs in D, but it is difficult to show that all pairs in D can be represented. The combined efforts of Persson ([Pers]) for region D1 and Chen ([Che1] [Che2]) for the regions D1 and D2 gave rise to the following results

5 1.2. Theorem Given any pair (m, n) D, there exists a minimal general type surface ∈ 2 M with c1 = n and c2 = m, except maybe for points on the ﬁnitely many lines n 3m +4k = 0 with 0 k 347. In fact, for at most ﬁnitely many exceptions − ≤ ≤ on these lines all pairs in D1 occurs as the Chern numbers of minimal general type surfaces.

This thoerem does not give surfaces on the borderline n = 3m. By Yau’s deep theorem [Yau1] [Yau2] on K¨ahler-Einstein metrics

1.3. Theorem (Yau) Let M be a compact K¨ahler manifold of dimension n,

(a) if c > 0 (resp. c < 0), i.e. the (1, 1) cohomology class c (resp. c ) is 1 1 1 − 1 represented by some positive (1, 1)-form. Then in any given K¨ahler class, there

exists a K¨ahler metric with positive (resp. negative) Ricci curvature.

2 (b) If c1 = 0 in H (M, R), then in any given K¨ahler class, there exists a K¨ahler

metric with vanishing Ricci curvature, i.e. Ricci ﬂat.

r = 1. −

2 In particular, we know that any general type surface with c1 =3c2 are quotients of the unit ball in C2. A natural question would be, which points (m, n) on the line n =3m

6 are represented by the Chern numbers of surfaces, and how can we characterize the other lines.

2 What we are interested in is a class of general type surfaces on the line c1 = c2 with numerically effective cotangent (nef in short) bundle, which will be defined in chapter 3. The numerical effectiveness is a weaker type of semipositivity than the

Griffiths semipositivity, It is widely used in transcendental algebraic geometry because of its flexibility. It is clear that every Griffiths semipositive vector bundle is nef. But the converse is not true, a standard counter example is given in the Chap- ter 3, due originally to Demailly. The positivity of holomorphic vector bundles has played a very important role in algebraic geometry. It has been used as an essential tool to study and classify compact Kähler manifolds. For example, all the compact

Riemann surfaces which are not of general type are exactly those with semipositive curvature. In 1961, T. Frankel [Fran] proved that a compact K¨ahler surface with positive bisectional curvature is biholomorphic to P2. He conjectured that the same is true in higher dimensions, which was later is known as the Frankel’s Conjecture.

In 1973, Hartshorne raised a more general conjecture, which states that any compact

K¨ahler manifold M of dimension n with ample tangent vector bundle is Pn. Since tangent bundle is ample when the bisectional curvature is positive, Hartshorne’s conjecture is stronger than the Frenkel’s one. But it is still an open question whether

7 the ample vector bundle is Griﬃths positive. In his famous paper [Mori] Mori solved

Hartshorne’s conjecture. Around the same time Siu and Yau [SiYa] gave an analytic proof of the Frankel conjecture. Combining the algebraic and analytic tools,

Mok [Mok] classiﬁed all compact K¨ahler manifolds with semipositive holomorphic bisectional curvature.

Since nefness is a generalization of the Griﬃths semipositivity, it is natural to consider the class of projective manifolds M whose tangent bundles are nef. This has been done by Campana and Peternell [CaPe] for surface case, and Zheng [Zhen1] completed for the case of complex dimension 3. Later Demailly-Peternell-Schneider

[DPS1] investigated compact Kähler manifolds with nef tangent bundle, and reduced the problem to the simply connected cases. The conjecture is that such a manifold should be a product of compact simply connected homogenous Kähler manifolds, i.e. the Kähler C-spaces. On the other hand, very little is known for the compact Kähler manifolds with ample or nef cotangent bundles.

In this paper we are going to consider compact complex surfaces M with nef cotangent bundles. When M is not of general type, the nefness and the Enrique-

Kodaira classiﬁcation theory yield that M is either a complex torus, or a elliptic curve bundle over a curve of positive genus. So we should focus on general type surfaces with nef cotangent bundle. By Theorem 4.2 we can know that c2 c . In 1 ≥ 2

8 2 this dissertation, we will focus on the borderline case c1 = c2. For this borderline case we have the following conjecture

1.4. Conjecture Let M be a general type surface with nef cotangent bundle and

2 with c1 = c2. Then it must be a generalized theta divisor, namely, there exists an equivariant proper holomorphic immersion from the universal cover of M into C3.

We can show that the conjecture is true under some stronger conditions. The main theorem, which is presented in chapter 5, is the following

2 0 1.5. Theorem Let M be a general type surface with c1 = c2 such that P ic(M)/Pic (M) is generated by the canonical line bundle KM . Then every nontrivial holomorphic bundle extension

0 T E K 0 → M → → M → of the holomorphic tangent bundle TM by KM is Hermitian ﬂat.

2 0 1.6. Theorem Let M is a general type surface with c1 = c2 such that P ic(M)/Pic (M)

1 is generated by the canonical line bundle K . If h (T K∗ ) > 0, then M must be M M ⊗ M an immersed theta divisor.

Theorem 1.6 is a consequence of Theorem 1.5. As a direct corollary of Theorem

1.5, we know that any M satisfying the conditions of Theorem 1.6 always has nef cotangent bundle. A key lemma used in the proof of the Theorem 1.5 is the following

9 2 0 1.7. Lemma Let M be a general type surface with c1 = c2 and P ic(M)/Pic (M) is

0 1 generated by KM . Then the irregularity q(M)= h (M, ΩM ) can not be equal to 1 or

In the last chapter we prove an extended version of the Hard Lefschetz Theorem for cohomology with values in Nakano semipositive vector bundles over compact K¨ahler manifolds. The case of the line bundles was obtained by Mourougane [Mour] in 1999.

Demailly-Peternell-Schneider [DPS2] generalized it to pseudo eﬀective line bundles.

Our result is the following

1.8. Theorem Let (E, h) be a Nakano semipositive vector bundle of rank r over a compact K¨ahler manifold (M,ω) of dimension n. Then the wedge multiplication operator ωq induces a surjective morphism ∧·

q q 0 n q q n L = ω : H (M, Ω − E) H (M, Ω E) h ∧· M ⊗ → M ⊗ for any 0 q n. ≤ ≤

10 CHAPTER 2

STABILITY OF VECTOR BUNDLES

2.1 Coherent sheaf theory

Let M be a compact complex manifold of dimension n and the structure sheaf OM of M, i.e. the sheaf of germs of holomorphic functions on M. We write

p p M = M M O Oz ⊕···⊕O}| {

An analytic sheaf over M is a sheaf of -modules over M. OM

2.1. Deﬁnition locally ﬁnitely generated sheaf

We say that an analytic sheaf over M is locally ﬁnitely generated if, given any J point x of M, there exists a neighborhood U of x and ﬁnitely many sections of 0 0 JU that generate each stalk , x U, as an -module. This means that we have an Jx ∈ Ox exact sequence

p 0 (2.1.1) OU →JU →

In particular, each stalk is an -module of ﬁnite type. Jx Ox 11 2.2. Deﬁnition coherent sheaf

We say that an analytic sheaf over M is coherent if, given any point x of M, there J 0 exists a neighborhood U of x0 and an exact sequence

q p 0 (2.1.2) OU → OU →JU →

This means that the kernel of squence (2.1.1) is also ﬁnitely generated.

We have the following usful lemma, whose proof can be found in [GunR].

2.3. Oka’s Lemma The kernel of any homomorphism

j : q p O → O is locally ﬁnitely generated.

By this lemma we can get the local Syzygy theorem for coherent sheaves.

2.4. Syzygy Theorem Let be a coherent sheaf over M. Given any x M, there J 0 ∈ exists a small neighberhood U of x0, and a ﬁnite step free resolution of sheaves:

0 pd p1 p0 0 (2.1.3) → OU →···→OU → OU →JU → with d n. ≤

12 2.5. Deﬁnition homological dimension

The homological dimension of the stalk of coherent sheaf over M, denoted by Jx J dh( ), is deﬁned to be the length d of a minimal free resolution: Jx

0 pd p1 p0 0 → Ox →···→Ox → Ox →Jx → with d n, which is the stalk version of the Syzygy theorem. ≤

2.6. Deﬁnition homological codimension

The homological codimension of the stalk of coherent sheaf over M, denoted Jx J by codh( ), is deﬁned to be the maximal length p of a -sequence. The p is Jx Jx independent of the choice of the sequence. A sequence a , , a of the element of { 1 ··· p} maximal ideal m of is called an -sequence if, for each i, 0 i p 1, a is Jx Jx ≤ ≤ − i+1 not a zero divisor on /(a , , a ) . Then the sequnce of the ideals Jx 1 ··· i Jx

(a ) (a , a ) (a , a , , a ) 1 ⊂ 1 2 ⊂···⊂ 1 2 ··· i ⊂··· is strictly increasing and there is a maximal -sequence. Jx

2.7. Theorem For the stalk of coherent sheaf over M Jx J

dh( )+ codh( )= n Jx Jx

From the deﬁnition of the homological dimension we can know that is a free Jx -module if and only if dh( ) = 0, or equivalently, codh( )= n. Ox Jx Jx 13 2.8. Deﬁnition m-th sigularity set

For each integer m, 0 m n, the m-th singularity set of is deﬁned to be ≤ ≤ J

S ( )= x M; codh( ) m = x M; dh( ) n m (2.1.4) m J { ∈ Jx ≤ } { ∈ Jx ≥ − } we call Sn 1( )= x M; x is not free the singularity set of the . − J { ∈ J } J

2.9. Theorem the m-th singularity set S ( ) of the coherent sheaf is a closed m J J analytic set of M of dimension m. ≤

2.10. Corollary Let over M is a coherent sheaf, for any x M, if there exists J ∈ U, which is a small neighberhood of x, together with an exact sequence of sheaves:

0 p1 pk →JU → OU →···→OU then dimS ( ) m k. m J ≤ −

2.11. Deﬁnition rank of coherent sheaf

Let be a coherent sheaf over M. It is locally free outside of the singularity set J

Sn 1( ). We deﬁne the rank of coherent sheaf − J

rank = rank x x M Sn 1( ) J J ∈ \ − J

2.12. Deﬁnition torsion free coherent sheaf

We say that over M is a torsion free coherent sheaf, if is a torsion free -module J Jx Ox for each x M. ∈ 14 By the above deﬁnition we know that every locally free sheaf is obviously torsion free. any coherent subsheaf of a torsion free sheaf is again torsion free. Conversely, we have the following lemma

2.13. Lemma If over M is a torsion free coherent sheaf of rank r, then it is locally J a subsheaf of a free sheaf of rank r. i.e. for any x M, there exists a neighborhood ∈ U and an injective homomorphism j : r . JU → OU

From this lemma and Corollary (2.10) we get the following

2.14. Corollary Let be a torsion free coherent sheaf over M, then dimS ( ) J m J ≤ m 1, for all m. This means that a torsion free coherent sheaf is locally free outside − the singularity set of codimension at least 2.

Let us deﬁne the dual of a coherent sheaf to be ∗ = Hom( , ). There is a J J J O natural homomorphism σ from to its double dual ∗∗: J J

σ : ∗∗ J →J

2.15. Lemma the kernel of σ Ker(σ) consists exactly of torsion elements of and J it is a torsion subsheaf of . So σ is injective if and only if is torsion free. J J

2.16. Deﬁnition reﬂexive coherent sheaf

If σ : ∗∗ is bijective, we say that is reﬂexive coherent sheaf. J →J J 15 Every reﬂexive coherent sheaf is torsion free, and we have the following

2.17. Lemma The dual sheaf of any coherent sheaf is reﬂexive.

2.18. Lemma If is a reﬂexive coherent sheaf, then for any x M, there exists a J ∈ small neighberhood U of x0 and an exact sequence of sheaves:

0 p1 p2 →JU → OU → OU

By Corollary (2.10) we know that every reﬂexive coherent sheaf is locally free outside the singularity set of codimension at least 3.

2.19. Deﬁnition normal coherent sheaf

We say that coherent sheaf over M is normal, if for every open set U and every J analytic subset Z U of codimension at least 2, the restriction map ⊂

r : Γ (U, ) Γ (U Z, ) J → \ J is an isomorphism.

From Lemma (2.13) we know that the restriction map is injective if is torsion J free. Obviously the torsion free sheaf is not necessarily normal. Conversely, a normal sheaf is not necessarily torsion free, either. However, we do have the following relation between reﬂexive coherent sheaves and torsion free normal coherent sheaves.

16 2.20. Lemma Coherent sheaf is reﬂexive if and only if is torsion free and J J normal.

2.21. Lemma Let

0 0 →J →E→Q→ be a short exact sequence of sheaves. If is reﬂexive and is torsion free, then E Q J is normal.

As a corollary of the Lemma (2.18), we have the following

2.22. Lemma A reﬂexive coherent sheaf of rank 1 on a smooth compact complex surface M is a line bundle. A torsion free coherent sheaf of rank 1 on a smooth compact complex curve M is a line bundle.

2.23. Deﬁnition determinant line bundle

For a holomorphic vector bundle E of rank r over a compact complex manifold M, its determinent line bundle detE is defined by detE = rE. ∧ For a coherent sheaf of rank r over the M, its determinent line bundle det is J J defined by following: for any x M and let U is a small neighberhood of x such that ∈ there exist a finite step locally free resolution of sheaves:

0 0 (2.1.5) →En →···→E1 →E0 →JU →

17 let the E denotes the vector bundle correspoonding to the sheaf , we deﬁne i Ei

n ( 1)i det = (detE ) − JU i Oi=0

It can be showed that det is independent of the choice of the resolution (2.1.5) JU and that det is well deﬁned. We have the following properties of the det . J J

2.24. Proposition If is a torsion free coherent sheaf of rank r, then there are J canonical isomorphisms

r det =( )∗∗ J ∧ J

(det )∗ =(det ∗) J J

Having defined the determinant bundle det , we can define the first Chern class J of by J c ( )= c (det ) 1 J 1 J

2.2 Semistable and stable sheaves

Let be a torsion free coherent sheaf over a n dimensional compact Kähler manifold J (M,g) with ω the Kähler form, which is a real positive closed (1, 1)-form on M. Let c ( ) be the first Chern class of , it is represented by a real closed (1, 1)-form on 1 J J M.

18 2.25. Deﬁnition The degree (or ω-degree) of is deﬁned to be J

n 1 deg( )= c1( ) ω − J ZM J ∧

2.26. Deﬁnition The slope µ( ) is deﬁned to be J

µ( )= deg( )/rank( ) J J J

2.27. Deﬁnition we say that is semistable (or ω-semistable) if for every coherent J subsheaf ′ of with 0 < rank( ′) < rank( ), we have J J J J

µ( ′) µ( ) J ≤ J

If moreover for every coherent subsheaf ′ of with 0 < rank( ′) < rank( ), the J J J J strict inequality

µ( ′) <µ( ) J J holds, then we say that is stable (or ω-stable). J

A holomorphic vector bundle E over M is said to be semistable (resp. stable) if the sheaf = (E) of the E is semistable (resp. stable). From the deﬁnition above, E O we have to check all subsheaves of to show that is semistable (resp. stable). J J But the following theorem tell us that we don’t need to consider all subsheaves.

2.28. Theorem Let be a torsion free coherent sheaf over a compact K¨ahler man- J ifold (M,g), then

19 (a) is a semistable (resp. stable) if and only if µ( ′) (resp. <) µ( ) for every J J ≤ J

coherent subsheaf ′ with 0 < rank( ′) < rank( ) such that the quotient J J J

/ ′ is torsion free. J J

(b) is a semistable (resp. stable) if and only if µ( ) (resp. <) µ( ′′) for every J J ≤ J

torsion free quotient sheaf ′′ with 0 < rank( ′′) < rank( ). J J J

We also have following propositions about semistable or stable sheaves.

2.29. Proposition Let be a torsion free coherent sheaf over a compact K¨ahler J manifold (M,g), then

(a) If rank( )=1, then is stable. J J

(b) Let be a line bundle over M, then is stable (resp. semistable) if and L J ⊗L only if is stable (resp. semistable). J

2.30. Proposition Let and be two torsion free coherent sheaves over a compact J1 J2 K¨ahler manifold (M,g), then is semistable if and only if and are both J1 ⊕J2 J1 J2 semistable and µ( )= µ( ). J1 J2

20 2.31. Theorem (Jordan-Hölder) Given a semistable sheaf over a compact Kähler E manifold (M, g), there is a filtration of by subsheaves E

0= = Ek+1 ⊂Ek ⊂···⊂E1 ⊂E0 E such that / are stable, and µ( / )= µ( ) for i=0, 1, , k. Ei Ei+1 Ei Ei+1 E ···

21 CHAPTER 3

POSITIVITY OF HERMITIAN VECTOR BUNDLES

In this chapter we started with basic definition of curvature for complex vector bundles over smooth differential manifolds. Then we recall the definition and basic properties of Hermitian curvature of Hermitian vector bundles over compact complex manifold, which is the curvature of the unique Hermitian connection. After that we discuss Her- mitian flat bundles, positivities for Hermitian vector bundles, and Einstein-Hermitian vector bundles. In the last section we recall the definition for ample vector bundle, which is another type of positivity from the algebraic point of view. We also discuss the relation between ampleness and Griffiths positivity of vector bundle.

3.1 Curvature of vector bundles

Let E be a complex vector bundle of rank r over a smooth diﬀerentiable manifold

p p M with dimRM = n. Denote by A (E) = C∞(M, (T ∗ ) E), the space of C∞ ∧ M ⊗ p-forms on M with values in E. A connection D on E is a linear diﬀerential operator

22 of order 1

D : Ap(E) Ap+1(E) → such that

D(f u)= df u +( 1)degf f Du ∧ ∧ − ∧

q p for all forms f C∞(M, (T ∗ )), u A (E). On an open set U M where E ∈ ∧ M ∈ ∈ admits a trivialization θ : E ∼= U Cr, the connection D can be written as U → ×

Du = du + Γ u θ ∧

1 r r where Γ C∞(U, (T ∗ ) Hom(C , C )) is a matrix of 1-forms and d acts compo- ∈ ∧ M ⊗ nentwisely. Then we have

D2u = D(du + Γ u) ∧ = d(du + Γ u)+ Γ (du + Γ u) ∧ ∧ ∧ = dΓ u Γ du + Γ du + Γ Γ u ∧ − ∧ ∧ ∧ ∧ =(dΓ + Γ Γ ) u ∧ ∧ It is not hard to see that D2 is actually linear over A0(E). Since D2 is globally deﬁned operator, there is a global 2-form, called the curvature form of D,

2 2 Θ(D) A (Hom(E, E)) = A (E∗ E) ∈ ⊗

23 2 such that D u = Θ(D) u for every form u A∗(E). ∧ ∈

Next, let us assume that E is endowed with a C∞ Hermitian metric along the ﬁbres and e , , e is a local C∞ frame. we have a canonical sesquilinear pairing { 1 ··· r}

Ap(E) Aq(E) Ap+q(C) ⊗ → (u, v) u, v = u v¯ (e , e ) 7→ { } α ∧ β α β Xα,β here u = u e , v = u e , (e , e ) is the Hermitian inner product of E. α α ⊗ α β β ⊗ β α β P P We say that a connection D is compatible with the metric if it satisﬁes the additional property

d u, v = Du,v +( 1)degu u,Dv { } { } − { } assuming that e , , e is orthonormal, we will get Θ(D)∗ = Θ(D) and { 1 ··· r} −

√ 1Θ(D) A2(Herm(E, E)) − ∈

From now on, let us assume that M is the compact complex manifold with dimCM = n, and the bundle E is a holomorphic vector bundle of rank r. When E is equipped with a Hermitian metric we will call it a Hermitian vector bundle. The complexiﬁ-

′ ′′ cation of the real cotangent bundle splits as TC∗ (M) = T ∗ (M) T ∗ (M), into the ,z z ⊕ z p,q p ′ q ′′ holomorphic and the antiholomorphic part. A (E) = C∞(M, T ∗ T ∗ E), ∧ M ⊗ ∧ M ⊗ the space of C∞ (p, q)-forms on M with values in E. We can decompose connetion

24 p,q p+1,q p,q p,q+1 D = D′ + D′′, with D′ : A (E) A (E) and D′′ : A (E) A (E). There → → is a canonical choice of connection for Hermitian vector bundle,

3.1. Theorem There is a unique connecion D on a Hermitian vector bundle E, called the Hermitian connection, satisfying the following two conditions

(1) D is compatible with the complex structure, i.e. D′′ = ∂¯.

(2) D is compatible with the metric

The curvature of the Hermitian connection is called the Hermitian curvature. Its curvature form

√ 1Θ(E) := √ 1Θ(D) A1,1(Herm(E, E)) − − ∈

Let z , , z be holomorphic coordinates on open subset U M, e , , e an { 1 ··· n} ⊂ { 1 ··· r} orthonormal frame of EU . On U one can write

√ 1 √ 1 − Θ(E)= − R ¯ ¯dz dz¯ e∗ e (3.1.1) 2 2 ijαβ i ∧ j ⊗ α ⊗ β 1 Xi,j n 1 ≤α,β≤ r ≤ ≤ and identify this curvature form to the following curvature tensor

¯ Θ(E)(ξ, ξ)= Ri¯jαβ¯fi,αfj,β (3.1.2) 1 Xi,j n e 1 ≤α,β≤ r ≤ ≤ where ξ = f ∂/∂z e T M E. i,α i,α i ⊗ α ∈ ⊗ P If e , , e is a holomorphic frame of E , then in U we can write { 1 ··· r} U

1 √ 1Θ(E)= √ 1∂¯(h¯− D′h¯) (3.1.3) − − 25 here h is Hermtian matrix of the Hermitian metric of E. Under the frame, that is h =(hαβ)=(eα, eβ).

In general, it is impossible to ﬁnd a local frame e , , e of E that is simulta- { 1 ··· r} neously holomorphic and orthonormal. Because under such a frame, we would have the Hermitian curvature Θ(E)=0on U. On the other hand, if Hermitian curvature

Θ(E) = 0, we can always ﬁnd local frames that are holomorphic and orthonormal simultaneously.

3.2. Deﬁnition Hermitian ﬂat bundle

We say that a holomorphic vector bundle E of rank r over a compact complex manifold M of dimension n is Hermitian ﬂat if E has a ﬂat Hermitian structure, which is given by an open covering U and a system of local holomorphic frames { } s =(e , , e ) such that the transition functions g are all constant unitary { U 1 ··· r } { UV } matrices in U(r) := r r unitary matrices . { × }

3.3. Theorem Let E be holomorphic vector bundle of rank r over a compact complex manifold M of dimension n, the following conditions are equivalent,

(1) E is Hermitian ﬂat

(2) E admits a Hermitian structure h with ﬂat Hermitian connection, i.e. its cur-

vature vanishes.

26 (3) E is deﬁned by a representation ρ : π (M) U(r) of π (M). 1 → 1

We say that a holomorphic local frame e , , e is normal at point x U if { 1 ··· r} 0 ∈ h =(δαβ) at x0, and Γ =0 at x0.

The following proposition tell us that Hermitian vector bundles always admit normal local frames and Hermitian curvature tensor (3.1.2) is the obstruction to the existence of orthonormal holomorphic frames.

3.4. Proposition For every point x M and every holomorphic coordinate system 0 ∈ z , , z at x , there exists a holomorphic frame e , , e in a neighborhood { 1 ··· n} 0 { 1 ··· r} of x0 such that

n 3 h (z)=(e (z), e (z)) = δ R ¯ ¯z z¯ + O( z ) αβ α β αβ − ijαβ i j | | i,jX=1 where R ¯ ¯ are the coeﬃcients of the Hermitian curvature tensor at x M ijαβ 0 ∈

3.2 Nakano positivity, Griﬃths positivity and Numerically

eﬀectiveness

Let M be a compact complex manifold of complex dimension n and E a holomorphic vector bundle of rank r over M. We use the same notations as in the last section.

3.5. Deﬁnition

27 a. The Hermitian vector bundle E is said to be Nakano positive (resp. semiposi-

tive) if Θ(E)(ξ, ξ) > 0 (resp. 0) for all non zero tensors ξ = f ∂/∂z ≥ i,α i,α i ⊗ P e e T M E. α ∈ ⊗

b. The Hermitian vector bundle E is said to be Griﬃths positive (resp. semi-

positive) if Θ(E)(ξ, ξ) > 0 (resp. 0) for all non zero tensors ξ = ϕ v = ≥ ⊗ e f ∂/∂z v e T M E. i,α i i ⊗ α α ∈ ⊗ P

Nakano positivity (resp. semipositivity) implies Griffiths positivity (resp. semipositivity). We can view that Nakano positivity (resp. semipositivity) as being positive (resp. semipositive) definite on any nonzero n r matrix vector, while Griffiths × positivity (resp. semipositivity) is just being positive (resp. semipositive) definite on non-zero n r matrix vectors of rank 1. × When the rank of the Hermitian vector bundle E is 1, i.e. E is Hermitian line bundle, the Nakano positivity (resp. semipositivity) and Griffiths positivity (resp. semipositivity) are cioncide. So we will simply call the Griffiths (or Nakano) positive line bundles positive line bundles.

For positive line bundles we have the following vanishing thoerem is a major tool.

It is a direct consequence of the Bochner-Kodaira-Nakano inequality which we will discuss in the next section.

28 3.6. Theorem (Kodaira-Nakano Vanishing)

(a) If L is a positive line bundle over a compact K¨ahler manifold M of dimension

n, then

Hq(M, Ωp(L))=0 for p + q > n

(b) If L is a negative line bundle over a compact K¨ahler manifold M of dimension

n, then

Hq(M, Ωp(L))=0 for p + q < n

There are several other versions of the vanishing theorem, one of them is the following

3.7. Theorem Let L be a positive line bundle over a compact K¨ahler manifold M of dimension n. Then for any holomorphic vector bundle E, there exists a positive integer k0 such that

Hq(M, (Lk E))=0 for any q > 0 and any k k O ⊗ ≥ 0

A generalized version of the vanishing thoerem for numerically eﬀective line bundles is Theorem 6.12 in [Dema]

The numerically effectiveness is a weaker notion of positivity than Griffiths positive. It is widely used in transcendental algebraic geometry because it is more flexible.

The analytic deﬁnition of the numerically eﬀectiveness is the following

29 3.8. Definition Let (M,g) be compact complex manifold of dimension n with Her- mitian form ω, L is a holomorphic line bundle on M. We say that the line bundle L is numerically effective if for every ε> 0, there exists a smooth Hermitian metric hε on L such that the Hermitian curvature of L satisfies

√ 1Θ(L) εω − hε ≥ −

This means that the curvature of the L has an arbitrarily samll negative part.

3.9. Definition Let (M,g) be compact complex manifold of dimension n with Her- mitian form ω, E a Hermitian vector bundle on M. We denote by P(E) the projec- tivized bundle of hyperplanes of E, i.e. P(E) = x M P(Ex∗) = x M (Ex∗ 0)/C∗, ∈ ∈ − S S and denote by (1) the associated line bundle on P(E), which is the dual of the OE univeral (or tautological) line sub-bundle ( 1). We say that the vector bundle E OE − is numerically effective if the line bundle (1) is numerically effective. OE

From this definition one can see that Griffiths semipositive vector bundles are always numerically effective, but the converse is not true. A counter example is the following

3.10. Example Let C be a smooth elliptic curve, E a rank 2 vector bundle on C obtained as a non-trivial extension

0 E 0 →O→ →O→

30 by the trivial line bundle. Then E is numerically eﬀective, but is not Griﬃths semipositive.

Proof. We can see that E is numerically effective by the next proposition. On the other hand, if E is Griffiths semipositive, there will be a Hermitian metric h on E whose curvature tensor Θ(E) is Griffiths semipositive. Thus TrΘ(E)= Θ(detE) 0. ≥ e e e Since detE = , the TrΘ(E) = 0. We know that the curvature tensor Θ(E) = 0. O e e This implies that the above exact sequence of E will split holomorphically, contradicts our assumption. 2

The following properties and results about numerically eﬀective vector bundles will be used later.

3.11. Proposition Let X, Y be compact complex manifolds, f : Y X be a holo- → morphic map, E be a holomorphic vector bundle over X. If E is numerically eﬀective, then f ∗E is also numerical eﬀective.

3.12. Proposition Let 0 F E Q 0 be an exact sequence of holomorphic → → → → vector bundles over a compact complex manifold (M,g) with Hermitian form ω. Then

(a) If E is numerically eﬀective, then Q is numerically eﬀective.

(b) If F , Q are both numerically eﬀective, then E is numerically eﬀective.

31 3.3 Ample vector bundles

Another important notion on positivities is the ampleness, which captures the algebraic point of view.

3.13. Deﬁnition Let M be a compact complex manifold of dimension n. A holomorphic line bundle L over M is called an ample line bundle if there is a holomorphic embedding f : M ֒ PN →

k such that L = f ∗H for some integer k > 0, where H := (1) is the hyperplane line O bundle of PN .

3.14. Deﬁnition Let M be a compact complex manifold of dimension n. We say that a holomorphic vector bundle E over M is ample vector bundle, if the associated line bundle (1) on P(E) is ample. OE

By the following Kodaira embedding theorem one can see that for line bundles, the ampleness is equivalent to the positivity notion we discussed before. So we sometimes will call ample line bundles positive line bundles for this reason. But we can not tell from this thoerem if ampleness of vector bundles of rank > 1 is the same as the

Griﬃths positivity of vector bundles. It is not hard to show that a Griﬃths positive vector bundle is always ample by a straight forward computation of the curvature

32 Θ( (1))of the (1), which, at a given point (p, [v]) of P(E), is the sum of the OE OE curvature Θ(E)vv¯ with the Fubini-Study metric form of the ﬁbre. The converse was conjectured by Griﬃths and has not been solved yet.

3.15. Theorem (Kodaira embedding theorem) If L is a positive line bundle over a compact complex manifold M of dimension n, then there exists k0, such that for any k k , ≥ 0 N i k : M P L → is an embedding of the M.

k The map iLk is deﬁned by the linear system of the line bundle L . Blowing up techniques and vanishing theorems were used to prove this theorem. The detail of this theorem can be found from [GrHa].

Since we are only concerned with smooth projective varieties, it is important for us to give a different definition of numerically effectiveness for line bundles, as well as an equivalent definition for the ampleness of line bundle. These definitions only work for projective varieties.

3.16. Deﬁnition-Proposition Suppose M is a projective manifold of dimension n.

Then A holomorphic line bundle L over M is numerically eﬀective if and only if

L C = c1(L) 0 · ZC ≥

33 for any irreducible curve C M. ⊂

p p The ′′only if ′′ part is clear, and in fact, it can be shown that L Y = Y c1(L) 0 · R ≥ for any p-dimensional subvariety Y M. We refer the readers to [Hart] for details. ⊂ The converse is proved in Prop. 3.18 below. To relate ampleness with this, we have the following

3.17. Nakai-Moishezon Ampleness Criterion A holomorphic line bundle L over a projective manifold M of dimension n is ample if and only if

p p L Y = c1(L) > 0 · ZY for any p dimensional subvariety Y M. ⊂

The following proposition will tell us that this definition of the numerical effective of line bundle on projective manifold is equivalent to the analytic definition 3.8.

3.18. Proposition Let L be a holomorphic line bundle over a projective manifold

(M,g) with hermitian form ω of dimension n, H an ample line bundle on M. Then the following are equivalent,

p p (a) L Y = Y c1(L) 0 for any p-dimensional subvariety Y M. · R ≥ ⊂

(b) For any positive interger k, the line bundle kL H is ample. ⊗

34 (c) For any ε > 0, there is a metric h on L whose curvature satisﬁes Θ(L) ε hε ≥ εω. −

Proof. (a) (b). If L satisﬁes the condition (a) and H is ample, then clearly kL + H ⇒ will satisfy the Nakai Moishezon Ampleness Criterion. Hence kL + H is ample for any k > 0.

(b) (c). We may select a metric h on H with positive curvature and set ω = ⇒ H √ 1Θ(H). If kL + H is ample, then this bundle admits a Hermitian metric h − kL+H 1 1/k with positive curvature. Consider the metric h =(h h− ) on L, its curvature L kL+H ⊗ H is 1 1 √ 1Θ(L)= (√ 1Θ(kL + H) √ 1Θ(H)) √ 1Θ(H) − k − − − ≥ −k −

1 So for any given ε> 0, by letting k large enough, we can be sure that k <ε.

(c) (a). Under the condition (c), for any irreducible curve C M, we have ⇒ ⊆ √ 1 ε L C = − Θ(L) ω · ZC 2π ≥ −2π ZC for any ε> 0, so we get L C 0 by taking ε 0. Thus (a) is true. 2 · ≥ →

3.19. Deﬁnition Einstein-Hermitian vector bundle

Let (E, h) be a Hermitian vector bundle of rank r over a compact Hermitian manifold

(M,g) of dimension n. Deﬁne the mean curvature K of E by

α αγ¯ i¯j K =(Kβ )= h g Ri¯jβγ¯ X 35 αγ¯ 1 i¯j 1 here h =(hαγ¯)− and g =(gi¯j)− , Ri¯jβγ¯ are the coeﬃcients of curvature discussed

α α in (3.1.2). If K = cIE, i.e. Kβ = cδβ , c is a constant, then we say that E satisﬁes the Einstein condition and call E an Einstein-Hermitian vector bundle.

We have the following important theorems about Einstein-Hermitian vector bundles which will be used later.

3.20. Theorem (Donaldson, Uhlenbeck-Yau) Let M be a compact complex manifold with ample line bundle H and K¨ahler form ω representing the Chern class c1(H).

If a holomorphic vector bundle E over M is ω-stable, then E admits an Einstein-

Hermitian structure.

3.21. Theorem Let (E, h) be a Hermitian vector bundle over a compact Kähler manifold (M, g) of dimension n with Kähler form ω. If it satisfies the Einstein

2 condition and c1(E)=0 in H (M, R) then

n 2 c2(E) ω − 0 ZM ∧ ≥ and the equality holds if and only if (E, h) is ﬂat.

36 CHAPTER 4

INEQUALITIES FOR CHERN CLASSES

In this chapter, we will take a quick tour on the inequalities for Chern classes for ample vector bundles and more gererally numerically eﬀective vector bundles.

Let P Q[c , ..., c ] be a weighted homogeneous polynomial of degree n, where each ∈ 1 r variable ci is assigned with degree i. We say P is called a numerically positive polynomial for ample vector bundles if for any projective manifold M of dimension n, and any ample vector bundle E over M with rank r, the following inequality is satisﬁed

P (c1(E), ..., cr(E)) > 0 ZX

A thoerem of Bloch and Gieseker [BlGi] asserts that the Chern class cn is numerically positive for ample bundles provided that n r. ≤

A basis for the Q vector space Q[c1, ..., cr] is give by the so called Schur Polynomials, which is deﬁned as follows.

Denote by Λ(n, r) the set of all partitions of n by nonnegative integers r. Thus an ≤

37 element λ Λ(n, r) is speciﬁed by a sequence ∈

r λ1 λ2 λn 0, λi = n ≥ ≥ ≥···≥ ≥ X

Each λ Λ(n, r) gives rise to a Schur polynomial P Q[c , ..., c ] of degree n, ∈ λ ∈ 1 r deﬁned as the n n determinant ×

cλ1 cλ1+1 ... cλ1+n 1 −

cλ2 1 cλ2 ... cλ2+n 2 P = − − λ ......

cλn n+1 cλn n+2 ... cλn − − where by convention c = 1 and c =0 if i / [0,r]. Any polynomial P Q[c , ..., c ] 0 i ∈ ∈ 1 r can be written uniquely as

P = a (P )P , a (P ) Q λ λ λ ∈ λ XΛ(n,r) ∈

In particular for λ = (1, , 1) we get the n’th dual Segre class s (E∗). Fulton and ··· n Lazrasfeld [FuLa] showed the following

4.1. Theorem (Fulton and Lazrasfeld) The polynomial P is a numerically positive polynomial for ample vector bundles if and only if the coeﬃcients

a (P ) 0 for all λ Λ(n, r) λ ≥ ∈ and at least one of them is strictly positive.

38 Demailly-Peternell-Schneider [DePS] generalized the above result to the numerically eﬀective vector bundles, the main theorem is following

4.2. Theorem (Demailly-Peternell-Schneider) Let E be a numerically effective vector bundle over a compact Kähler manifold M equipped with a Kähler form ω.

Then for any Schur polynomial P of degree k and any analytic subset Y M of λ ⊂ dimension d k we have ≥ d k Pλ(c(E)) ω − 0 ZY ∧ ≥

For example, if M is a projective surface with numerically eﬀective cotangent bundle, we will have c2 c . 1 ≥ 2

39 CHAPTER 5

MAIN THEOREMS PART I

Notation In this chapter a surface means a compact complex manifold of complex dimension 2. All the vector bundles are holomorphic vector bundles. We will use divisor and the line bundle determined by the divisor interchangeably. We will not distinguish between a holomorphic vector bundle and the corresponding sheaf of the holomorphic vector bundle.

Let M be a surface, E a holomorphic vector bundle over M. We will ﬁx the following notations:

(D): the invertible sheaf (or holomorphic line bundle) corresponding to the divisor OM D.

Hq(M, E): the q-th cohomology group of the sheaf (E). OM hq(M, E): the complex dimension of the group Hq(M, E).

K : the canonical divisor, a divisor such that (K )= 2Ω1 . M OM M ∧ M P ic(M): the group of all holomorphic line bundles on M.

40 P ic0(M): the group of holomorphic line bundles with zero ﬁrst Chern class.

χ (M)= ( 1)ihi(M, R), the Euler number of M. top i − P

2 0 5.1. Main Lemma Let M be a general type surface with c1 = c2. If P ic(M)/Pic (M) is generated by the canonical line bundle K , then q(M)= h0(M, Ω1 ) =1 or 2. M M 6

Proof. If q(M) = 1, by Theorem V.13 [Beau] the Albanese map is p : M B, → where B is a smooth elliptic curve. By Theorem V.15 [Beau], p must be a fibration with connected fibres. So the generic fibre C will have self intersection C2 = 0

2 by Proposition I.8 [Beau], here C = kKM for some k > 0. So KM = 0, which will contradict the original condition c2 = c 6 by Noether’s formula χ( ) = 1 2 ≥ OM 1 2 12 (c1 + c2) and Theorem (1.1).

Now Let us assume that q(M) = 2. By Theorem V.13 the Albanese map is π : M → A, where A is a abelian surface. There are two possibilities for the map π.

1. the map π will contract some curve C, possible reducible, to a point.

2. the map π is a ﬁnite map onto A.

For Case 1, By Grauert’s criterion Theorem III.2.1 [BaPV] we know that the intersection matrix (C C ) is negative deﬁnite, here C = C with each C being i · j i i i P irreducible. Thus there must be some i such that C2 0. Write C = kK , k must i ≤ i M

41 be positive, so we get K2 0, contradicting to the fact that c2 = c 6 by Noether’s M ≤ 1 2 ≥ formula.

In Case 2, since the map π : M A is ﬁnite, we know that A must be a simple → abelian surface, which means that A does not contain any smooth elliptic curve C.

2 1 If it has such a curve C, then C = 0 by genus formula. So by looking at π− (C) we get a contradiction.

Let B be the branch locus of π, and R the ramiﬁcation locus. The Hurwitz formula gives

K = π∗K + (m 1)R M A i − i Xi where R = R is the decomposition into irreducible components and m 2 is i i i ≥ P the multiplicity of π at a generic point of Ri. Since KM is the generator of the

0 P ic(M)/Pic (M), we know that KM is irreducible. If it is not, let’s say KM =

D + D , here D = l K L , L P ic0(M), l > 0, i=1, 2. Then K K = 1 2 i i M ⊗ i i ∈ i M · M l K2 + l K2 , contradiction. We also know that K = , so we know that R = R 1 M 2 M A OA 1 must be irreducible, and m = 2. Suppose d is the degree of the map π : R B, 1 |R → then π : M A has degree 2d and K = R. → M Note that we could also assume that A has a principle polarization, i.e. an ample divisor L on A of type of (1, 1), see Deﬁnition 4.1 in [BiLa]. This is because if A does not have type (1, 1) polarization, then by Proposition 4.1.2 [BiLa] there will

42 be an isogeny p : A A′, onto a principle polarized abelian surface, that is, the p → is a surjective homomorphism with ﬁnite kernel from the complex torus A onto the complex torus A′, such that L = p∗L′, with L′ a type (1, 1) line bundle on A′. If that is the case, we could simply replace π : M A by the composition of π and p. → From now on, let us assume that L is a type (1, 1) line bundle on A. Since we have π∗B =2KM and KM is a generator, we get B = L or B =2L. If B =2L, then by using Proposition 4.1.2 [BiLa] again we can replace A by another abelian surface via an isogency, so we could always reduce to the B = L case. Now suppose B = L, by Proposition 4.5.2 of [BiLa] and the vanishing theorem Theorem 3.4.5 of [BiLa] for complex tori we know that the line bundle L is ample if and only if hi(L) = 0 for i = 1, 2 and L2 > 0. Since L is ample line bundle of type (1, 1), then by Riemann-

Roch Theorem 3.6.3 [BiLa] on complex tori A, we get χ(L) := χ( (L)) = 1 L2 (the OM 2 more general version of the Riemann-Roch theorem will be given in the proof of the

0 1 2 2 next lemma). This implies that h (A, L) = 2 L =1. So L = 2, and the arithmetic genus of B is L2 + K .L p (B)= p (L)=1+ A =2 a a 2 and we also have

2 2 2 2 4KM = π∗B.π∗B =2dB =2dL =4d, so KM = d

43 Since pa(B) = 2, by the Lemma (p505, [GrHa]) we know that the topological genus of B is k (k 1) g(B) p (B) i i − ≤ a − 2 Xi where the ki are the multiplicities of the singular points. Since pa(B) = 2, we know that B must be one of the following

i. B is a smooth genus 2 curve.

ii. B is a singular rational curve with two multiplicity 2 singular points or one

multiplicity 3 singular point.

iii. B is a singular elliptic curve with a multiplicity 2 singular point.

For case i, since B is a smooth genus 2 curve, we will have χ (B)=2 4= 2 top − − We also have

χ (M R)=2dχ (A B) top \ top \ χ (M)= 2dχ (B)+ χ (R) since χ (A)=0 top − top top top

2 KM = χtop(M)=4d + χtop(R)

4d +2 2(1 + K2 )+ (k 1)2 ≥ − M i − Xi

=2K2 + (k 1)2 M i − Xi

44 here we used the formula

χ (R) χ (R )+ (k 1)2 top ≥ top 0 i − Xi where ki is the multiplicity of the i-th singular point of R, and R0 is a smooth curve homologous to R, i.e. a smooth curve with the topological genus of pa(R). The

2 inequality on KM certainly provides a contradiction, so case i can not occur.

For case ii, if there is a singular rational curve in A, there will be a holomorphic map i : P1 A. We can lift this map to the universal cover of A, ˜i : P1 C2, which must → → be constant by maximum principle. So case ii can not occur either.

For case iii, the singular point of B is either a node or a cusp because it is of multiplicity 2. We divided our discussion into two subcases. a), If B has a cusp p, let f : A A be the blowing up of A at p, and B the strict → e e transformation of B. We have

1 χtop(B)= χtop(B) (# f − (p) 1) − X { } − e i

so χtop(B) = 0 since χtop(B) = 0. Therefore e χ (M R)=2dχ (A B) top \ top \

χtop(M)= χtop(R)

χ (R) 0 ≤ top ≤ e 45 since R has positive genus, This contradicts the inequality χ (M)= K2 6. top M ≥ e b), If B has a node p, similar to the above argument, we will have χ (B)= χ (B) top top − e 1= 1. Therefore − χ (M)= 2dχ (B)+ χ (R) top − top top

d =2d + χtop(R)

χ (R)= d top − But we also have the following

χtop(R)= dχtop(B) (ν(qi) 1) − −1 − qi Xπ (p) ∈

d = d + (ν(qi) 1) − − −1 − qi Xπ (p) ∈

ν(qi)=1 So we know that π : R B is an unbranched cover over B. Take a small open |R → set Z of p in A with local holomorphic coordinates (z , z ) such that Z = (z , z ) 1 2 { 1 2 ∈ 2 1 C : z < 1, z < 1 . Let U = π− (Z), so that π : U Z is a double | 1| | 2| } |U → 1 covering branched along z z = 0. Write Z∗ = Z z .z = 0 and U ∗ = π− (Z∗). 1 · 2 \{ 1 2 }

Then π and U are determined uniquely by the nonramiﬁed covering U ∗ Z∗ (see |U →

[Stei], Satz1), i.e. by the subgroup Γ = π ∗ (π (U ∗)) π (Z∗) of index 2, here |U 1 ⊂ 1

π (Z∗) = Z Z with generator w = (1, 0) and w = (0, 1), and w is the class of 1 × 1 2 i a positively oriented little loop around the zi-axis. We can pick generators for Γ as follows:

46 Since Γ (Z, 0) is nontrivial, there is some (n, 0) Γ generating this intersection, ∩ ∈ here n> 0. Then the quotient Γ/Z (n, 0) is isomorphic to Z, and there exists some · (q, m) Γ , 0 q < n, m > 0, such that (n, 0) and (q, m) generate Γ . If the q = 0, ∈ ≤ we know that either n = 1 and m = 2, or n = 2 and m = 1 since Γ is an index 2 subgroup. This will imply that π is not ramify over one of the z-axis. So we can |U conclude that q > 0. By The proof of Theorem III.5.2 in [BaPV], we know there must exist singularity on U. But that contradicts to the smoothness of surface M.

So the map π : M A can not be a ﬁnite map. → From Case 1 and 2, we conclude that q(M) =2. So q(M) =1 or 2. 2 6 6

2 5.2. Theorem Let M be a general type surface with c1 = c2 and suppose that

0 P ic(M)/Pic (M) is generated by KM . Then any nontrivial holomorphic bundle extension

0 T E K 0 → M → → M → of the holomorphic tangent bundle TM by KM is Hermitian ﬂat.

Proof. First, we will show that E is semistable. It suﬃces to show that E K∗ is ⊗ M semistable by Proposition 2.29.

Let

0 T E K 0 (5.0.1) → M → → M →

47 be a nontrivial extension, and twisting the exact sequence (5.0.1) by the line bundle

KM∗ , we get

0 T K∗ E K∗ 0 (5.0.2) → M ⊗ M → ⊗ M → OM →

By Whitney sum formula and Chern formula for tensor product, we get

c(E K∗ )= c(T K∗ )c( ) ⊗ M M ⊗ M OM

c (E K∗ )= c (T )+2c (K∗ )=3c (K∗ )= 3c (K ) 1 ⊗ M 1 M 1 M 1 M − 1 M and from (5.0.1)

c(E)=(1+ c1(TM )+ c2(TM ))(1 + c1(KM ))

=1+ c1(TM )+ c1(KM )+ c2(TM )+ c1(TM )c1(KM )

The two chern classes of E are

c1(E)= c1(TM )+ c1(KM )=0

c (E)= c (T )+ c (T )c (K )= c c2 =0 2 2 M 1 M 1 M 2 − 1 so the slope with respect to KM is

M c1(E KM∗ ) c1(KM ) 3KM .KM 2 2 µ(E K∗ )= ⊗ ∪ = − = K = c ⊗ M R 3 3 − M − 1

0 Step1. We want to show that H (M, E K∗ )=0 ⊗ M First, by the Serre duality and the vanishing theorem, we have

0 2 1 H (M, T K∗ )= H (M, Ω 2K )=0 M ⊗ M M ⊗ M 48 From (5.0.2) we get

0 i 0 π 0 δ 1 0 H (M, T K∗ ) H (M, E K∗ ) H (M, ) H (M, T K∗ ) → M ⊗ M → ⊗ M → OM → M ⊗ M →···

0 So if there exits 0 = σ H (M, E K∗ ), then π(σ)= m1 with 0 = m C, where 6 ∈ ⊗ M 6 ∈ 1 is the generator of H0(M, ). We have OM

0= δπ(σ)= δ(m1)= mδ(1)

Thus δ(1) = 0, contradicting our assumption that E is a non-trivial extension. Thus we must have σ = 0.

Step2. Let ∗ be a coherent rank 1 subsheaf with torsion free quotient, F ⊂E⊗KM here ∗ is the sheaf notation of vector bundle E K∗ . We want to show that E⊗KM ⊗ M the slope of is no greater than the slope of ∗ . F E⊗KM By Lemma 2.20, 2.21, 2.22, we know that is a line bundle on M, and we have a F short exact sequence of sheaves

0 ∗ 0 (5.0.3) →F→E⊗KM →Q→

Since P ic(M)/Pic0(M) is generated by K , we know that = (kK L), for M F O M ⊗ some L P ic0(M) and some k Z. We claim that k 1. ∈ ∈ ≤ − Twisting the short exact sequence (5.0.3) by the dual of the line bundle (kK L), O M ⊗ we get

0 ∗ (k + 1) ∗ ∗ k ∗ 0 (5.0.4) →O→E⊗L ⊗ KM →Q⊗L ⊗ KM → 49 Also, twisting the short exact sequence (5.0.2) by the dual of the line bundle kK L, M ⊗ we get

0 T L∗ (k + 1)K∗ E L∗ (k + 1)K∗ L∗ kK∗ 0 (5.0.5) → M ⊗ ⊗ M → ⊗ ⊗ M → ⊗ M →

By Theorems 3.18, 3.6 and Serre duality, we have

0 2 1 H (M, T L∗ (k +1)K∗ ) = H (M, Ω L (k +2)K )=0, fork 1 M ⊗ ⊗ M ∼ M ⊗ ⊗ M ≥ −

0 H (M, L∗ kK∗ )=0, fork 1 ⊗ M ≥

0 and H (M, L∗)=0, if k = 0 and L is not the trivial line bundle.

From the short exact sequence (5.0.5) we get that for any k 1 or k = 0 with L ≥ nontrivial,

0 0 H (M, E L∗ (k + 1)K∗ )= H (M, ∗ (k + 1) ∗ )=0 (5.0.6) ⊗ ⊗ M E⊗L ⊗ KM

When k = 0 and L is trivial, the short exact sequence (5.0.5) becomes

0 T K∗ E K∗ 0 → M ⊗ M → ⊗ M → OM →

0 0 In this case we also know that H (M, E K∗ )= H (M, ∗ ) = 0 by step 1. ⊗ M E⊗KM Combining this with (5.0.6), we know that

0 0 H (M, E L∗ (k + 1)K∗ )= H (M, ∗ (k + 1) ∗ )=0 for all k 0 ⊗ ⊗ M E⊗L ⊗ KM ≥

50 Therefore, the short exact sequence (5.0.4) leads us to the fact that k 1. ≤ − Since K L = 0, we get M ·

2 2 µ( )=(kK + L) K = kK + L K K = µ(E K∗ ) F M · M M · M ≤ − M ⊗ M

Step 3. It remains to consider the case ∗ , where is a coherent rank 2 F⊂E⊗KM F subsheaf with torsion free quotient. We want to show that µ( ) µ( K∗ ). F ≤ E ⊗ M We have

0 ∗ 0 (5.0.7) →F→E⊗KM →Q→ where is a torsion free sheaf with rank 1. Taking the dual of this short exact Q sequence, we have

0 ∗ ∗ ∗ (5.0.8) → Q →E ⊗KM →F

By Lemma 2.17, we know that ∗ is local free. So it can be written as ∗ = Q Q 0 ( kK L) for some L P ic (M) and some k Z. Twisting by (2K∗ L∗) O − M ⊗ ∈ ∈ O M ⊗ on the short exact sequence (5.0.8), we get

0 ∗ (2K∗ L∗) ∗ (K∗ L∗) ∗ (2K∗ L∗) (5.0.9) → Q ⊗ O M ⊗ →E ⊗ O M ⊗ →F ⊗ O M ⊗

Taking the dual of the exact sequence (5.0.2), we have

1 0 E∗ K Ω K 0 → OM → ⊗ M → M ⊗ M →

By twisting 2K∗ L∗ on the above exact sequence we get M ⊗

1 0 2K∗ L∗ E∗ K∗ L∗ Ω K∗ L∗ 0 (5.0.10) → M ⊗ → ⊗ M ⊗ → M ⊗ M ⊗ → 51 By Theorem 3.6 we get

0 1 0 H (M, Ω K∗ L∗)=0 and H (M, 2K∗ L∗)=0 M ⊗ M ⊗ M ⊗

So the exact sequence (5.0.10) give us

0 0 H (M, E∗ K∗ L∗)= H (M, ∗ (K∗ L∗))=0 ⊗ M ⊗ E ⊗ O M ⊗

We then get, from the exact sequence (5.0.9)

0 0 H (M, ∗ (2K∗ L∗)) = H (M, ( kK L) (2K∗ L∗)) Q ⊗ O M ⊗ O − M ⊗ ⊗ O M ⊗ = H0(M, ( k 2)K )=0 O − − M From this we can conclude that k 2 1, i.e. k 1. This leads to − − ≤ − ≥ −

µ(E K∗ )= µ( ∗ ) µ( ) ⊗ M E⊗KM ≤ Q

By short exact sequence (5.0.7), we have

M [c1( )+ c1( )] c1(KM ) M c1(E KM∗ ) c1(KM ) F Q ∪ = ⊗ ∪ = µ(E K∗ ) R 3 R 3 ⊗ M

2µ( )+ µ( )=3µ(E K∗ ) F Q ⊗ M

So we get from the above discussion that

2µ( ) 2µ(E K∗ ) i.e. µ( ) µ(E K∗ ) F ≤ ⊗ M F ≤ ⊗ M

Combining steps 1, 2, and 3, we know that E is semistable by Theorem 2.28.

52 Our next goal is to show that E is Hermitian ﬂat.

If E is stable, then by Theorems 3.20 and 3.21, we know that E must be Hermitian

ﬂat since c1(E)=0, c2(E) = 0 by our construction of E. So we may assume that E is semistable but not stable. By our main Lemma (5.1), we may assume that q(M) can not be 1 or 2. We will divide our discussion into the following two cases: q(M)=0 or q(M) 3. ≥ Case 1. q(M)=0

By Theorem 2.31, there exists a ﬁltration

0= = Ek+1 ⊂Ek ⊂···⊂E1 ⊂E0 E where µ( / )= µ( ) and / is stable for each i =0, 1, ,k. Ei Ei+1 E Ei Ei+1 ··· Since rank(E) = 3, there are three possibilities for the ﬁltration

i 0 rank( )=2 ⊂E1 ⊂E E1

ii 0 rank( )=1 ⊂E2 ⊂E E2

iii 0 rank( )=1, rank( )=2 ⊂E2 ⊂E1 ⊂E E2 E1

For case i, we will have

0 0 →E1 →E→Q→

53 where rank( ) = 1, is torsion free, and µ( ) = 0. Taking the dual of the above Q Q Q exact sequence we will have

0 ∗ ∗ ∗ (5.0.11) → Q →E →E1

Taking the dual of the exact sequence (5.0.1) we have the sequence

1 0 K∗ E∗ Ω 0 → M → → M →

0 0 0 1 So H (M, E∗) = 0 becuase H (M,KM∗ ) = 0 and H (M, ΩM )=0.

0 This and the short exact sequence (5.0.11) imply H (M, ∗) = 0. On the other hand, Q 0 we assumed that q(M)=0,so P ic (M) = 0. Thus µ( ∗) = 0 implies that ∗ = , Q Q OM 0 which contradicts the fact that H (M, ∗)=0. Q For case ii, we will have

0 0 →E2 →E→Q→ where rank( ) = 2 and is torsion free. Taking the dual of the above exact sequence Q Q we will have

0 ∗ ∗ ∗ (5.0.12) → Q →E →E2

By Lemmas 2.20, 2.21 and 2.22, we know that = . By Lemma 2.17, ∗ is a E2 OM Q rank 2 locally free sheaf.

54 Also by Theorem 2.31 and Proposition 2.29, we know that ∗ is stable. By (5.0.12), Q we get c ( ∗) = 0 since c (E) = 0. We also have 1 Q 1

τ 0 ker(τ) ∗ det( ∗) 0 → → Q → Q →

Here det( ∗)= by our assumption that q(M) = 0, which contradicts to the fact Q OM that ∗ is stable by Theorem 2.28. Q For case iii, we will have

0 0 →E1 →E→Q→

By the same reason as in Case i, we know that this case can not occur. So we conclude that when q(M) = 0, the case that E is not stable leads to a contradiction. So E must be stable.

Case 2. q(M) 3 Assume that E is not stable, taking the dual of the exact ≥ sequence (5.0.1) we get

1 0 K∗ E∗ Ω 0 → M → → M → which leads to the long exact sequence

0 0 0 1 1 0 H (M,K∗ ) H (M, E∗) H (M, Ω ) H (M,K∗ ) → M → → M → M →···

0 0 1 0 1 From this, we get H (M, E∗) ∼= H (M, ΩM ), since H (M,KM∗ ) = 0 and H (M,KM∗ ) ∼=

1 0 H (M, 2K ) = 0 by Theorem 3.6. So h (M, E∗) 3. But by Proposition (2.10) M ≥

55 0 [Take] or Proposition 1.16 [DSP1], we know h (M, E∗) = 3 and E is Hermitian ﬂat.

This concludes the proof that the vector bundle E is Hermitain ﬂat. 2

5.3. Deﬁnition Let M be a compact complex surface with ample canonical line bundle KM , we say M is an immersed theta divisor if it can be holomorphically immersed into a three dimensional abelian torus.

2 0 5.4. Theorem Let M is a general type surface with c1(TM )= c2(TM ) and P ic(M)/Pic (M)

1 is generated by K . If h (T K∗ ) > 0, then M will be an immersed theta divisor. M M ⊗ M

1 Proof. From h (T K∗ ) > 0, we will have a nontrivial holomorphic bundle exten- M ⊗ M sion

0 T E K 0 → M → → M →

Taking the dual of the above exact sequence, we get

i τ 1 0 K∗ E∗ Ω 0 → M → → M →

1 As a consequence we know that cotangent bundle ΩM is numerically eﬀective by

Proposition 3.12. Let π : M M be the universal cover of M, then we have the → f corresponding short exact sequence

i τ 1 0 K∗f E∗f Ωf 0 → M → M → M →

Since the vector bundle E∗ is Hermitian ﬂat on M by Theorem 5.3, we know that E∗f M is the trivial vector bundle on M. Let e , e , e be a parallel holomorphic basis for { 1 2 3} f 56 E∗f, and φ = τ(e ), i=1, 2, 3. they are global holomorphic 1-forms on M. For each M i i f 1 i 3, φ is closed since it is parallel. (Or by the following reason: since the φ is ≤ ≤ i i holomorphic 1 forms, dφi = ∂φi is holomorphic 2 forms and exact, and we have

dπ φi dπ φ¯i = d(π φi dπ φ¯i)=0 ZM ∗ ∧ ∗ ZM ∗ ∧ ∗

So we will get dφi = 0)

Locally, for any point x M, there exists a small neighborhood x U M, such ∈ ∈ x ⊂ f f that φi = dzi, where zi is a holomorphic function on Ux. This is guaranteed by the

1 Poincar´elemma, see Chapter 0 [GrHa], for instance. Since the map τ : E∗f Ωf is M → M surjective, we know that at least one of the conditions φ φ = 0, φ φ = 0 or 1 ∧ 2 6 1 ∧ 3 6 φ φ = 0 should hold. Thus we always have a holomorphic immersion: 2 ∧ 3 6

F =(z , z , z ) : U C3 x 1 2 3 x →

3 It is unique up to F ′ = F +(a , a , a ), here point (a , a , a ) C . By Theorem x x 1 2 3 1 2 3 ∈ VII.7.2 of [KoNo], there is an allowable immersion

F = M C3 → f

If we ﬁx a point x0, an open set Ux0 and a map Fx0 as above, then the map F is uniquely determined. Any two such allowable immersions F and F ′ diﬀer by a translation, i.e. F = F ′ +(a1, a2, a3).

57 Let γ Γ, where Γ = π (M) is the group of deck transformations of M, then we ∈ ∼ 1 f have

F γ = A F ◦ γ ◦

3 3 for some transformation Aγ(z) = z + aγ in C . Here aγ is a point in C . This leads to a homomorphism τ :Γ A(C3) ∗ → γ A 7→ γ where A(C3) is the group of all translations. Since the group A(Γ) = Im(τ (Γ)) C3 ∗ ⊂ is free abelian, we have

onto π (M) A(Γ) = Zρ 1 −−−→ ∼ onto x  2q π1( M) H1(M, Z) =Z + torsion −−−→ ∼ So we can conclude that ρ 2q(M). ≤ Next, we are going to show that ρ = 6.

We already know that R A(Γ) R6, If R A(Γ) R5, then there exists at least ⊗ ⊆ ⊗ ⊆ one real dimension in C3 which is invariant under the deck transformation. Without loss of generosity, we may assume that we have non-constant map

x : M R 3 → f where x = z +z ¯ , such that x γ = x for each γ in Γ. Thus x descends to a 3 3 3 3 ◦ 3 3 pluriharmonic function on M, as ∂∂x¯ 3 = ∂∂¯(z3 +¯z3)=0. So x3 must be a constant

58 function, contradicting the fact that F is an immersion.

Thus R A(Γ) = R6 and ρ 6. ⊗ ∼ ≥ By the fact that 2q ρ, we know that we must be in the case when q 3. Since ≥ ≥

E is Hermitian ﬂat, both E and E∗ are nef, thus any holomorphic section of E∗ can

0 not vanish by Proposition 1.16 in [DSP1]. Therefore, q = h (E∗) 3. Combining ≤ this with our previous result, q 3, we know that q = 3 and ρ = 6. So we have a ≥ holomorphic immersion f from M to T 3, which is a three dimensional abelian torus

, as following M F C3 −−−→ fπ π    f  M/y Γ C3/Ay (Γ) −−−→ f

f M T 3 −−−→ 2

59 CHAPTER 6

MAIN THEOREMS PART II

Notation In this chapter we are going to derive an extended version of Hard Lefschetz

Theorem for cohomology with values in a Nakano semipositive vector bundle of rank r on a compact K¨ahler manifold. To establish this, we need to obtain a Bochner type inequality.

First, let us ﬁx some notations which we will use throughout this chapter.

(M,g): a n-dimensional compact K¨ahler manifold with K¨ahler metric g.

(E, h): a rank r Hermitin vector bundle on (M,g) with Hermitian metric h.

p,q A : the space of C∞ (p, q)-forms on M.

p,q A (E): the space of C∞ (p, q)-forms on M with values in E.

r A (E): the space of C∞ r-forms on M with values in E.

D = D′ + ∂¯ : the Hermitian connection of (E, h).

¯¯ ¯ ¯ p,q = ∂∂h∗ + ∂h∗∂ : the Laplacian acting on A (E).

L = ω : Ap,q(E) Ap+1,q+1(E), here ω is the K¨alher form of the K¨alher metric g ∧· →

60 Λ= L∗: the adjoint operator of the L.

Θ := Θ(D)= D2: the curvature operator of the (E, h).

Hard Lefschetz Theorem: On a n-dimensional compact K¨ahler manifold (M,g), the map

k n k n+k L : H − (M) H (M) → is an isomorphism for any 1 k n. Also, if we deﬁne the primitive cohomology by ≤ ≤

n k k+1 n k n+k+2 P − = KerL : H − (M) H → then we have so-called Lefschetz decomposition,

r k r 2k H (M)= L P − (M) Mk 0 ≥ It is a well-known fact that the following inequality holds,

Bochner-Kodaira-Nakano inequality see [Dema] (4.6) for instance. Let (E, h) be a Hermitian vector bundle of rank r on a compact K¨ahler manifold (M,g). For any u Ap,q(E), it holds that ∈

2 2 ∂u¯ + ∂¯∗u ([√ 1Θ, Λ]u,u) k k k h k ≥ −

This inequality can be used to prove the vanishing theorems. Here we construct a new Bochner type inequality.

61 6.1. Lemma (a new Bochner type inequality) Let (E, h) be a Hermitian vector

n q,0 bundle of rank r on a compact K¨ahler manifold (M,g). For any v A − (E), let ∈ u = Lqv, then

2 2 2 q 1 q 0 ∂u¯ + ∂¯∗u = ∂v¯ + q(√ 1ΘL − v, L v) q 1 ≤k k k h k k k − ≥

Before proving this lemma, let us recall the deﬁnition of the Hodge star operators ∗ and , and the L2 norm on the space Ap,q(E) induced from g and h. ∗E The Hodge star operator is deﬁned as a linear(over R) isomorphism ∗

p,q n p,n q : A A − − ∗ → by requiring that for any ϕ, ψ Ap,q, ∈ ωn ϕ ψ = g(ϕ, ψ) ∧∗ n! here g(ϕ, ψ) is the induced Hermitian metric on Ap,q.

In order to write down explicitly, let ϕ , ,ϕ be a unitary frame of (1, 0)-forms. ∗ 1 ··· n For multi-index I = (i , , i ) in 1, , n , write ϕ = ϕ ϕ and ω = 1 ··· p { ··· } I i1 ∧···∧ ip √ 1 − ϕ ϕ . Also denotes by Iˆ the complement of I in 1, , n , so that 2 i i ∧ i { ··· } P

τ(I) ϕ ϕˆ =( 1) ϕ ϕ I ∧ I − 1 ∧···∧ n

Then for any multi-indices I, J with I = p and J = q, is given by | | | | ∗

n p+q n (fϕ ϕ )=(√ 1) 2 − ε fϕ¯ ˆ ϕ ˆ ∗ I ∧ J − IJ I ∧ J 62 1 n(n 1)+(n p)q+τ(I)+τ(J) where ε =( 1) 2 − − . It is straight forward to check that ψ = IJ − ∗∗ − p+q ωk ωn k ( 1) ψ for any (p, q)-form ψ, and ( k! )= (n k)! . − ∗ − Since M is compact, we can have a Hermitian inner product (the L2 norm) on each

Ap,q deﬁned by

(ϕ, ψ)= ϕ ψ ϕ,ψ Ap,q ZM ∧∗ ∈

p,q p,q 1 The operator ∂¯∗ = ∂¯ : A A − is adjoint to ∂¯ under this norm −∗ ∗ → The star operator is deﬁned similarly ∗E

p,q n p,n q : A (E) A − − (E∗) ∗E →

by sending ψ e to ( ψ ) e∗ , where each ψ is a (p, q)-form on M, α α ⊗ α α ∗ α ⊗ α α P P and e , , e is a unitary frame of E with e∗, , e∗ the dual frame. { 1 ··· r} { 1 ··· r} We can also deﬁne L2-norm over Ap,q(E) similarly as

p,q (ϕ, ψ)= ϕ Eψ ϕ,ψ A (E) ZM ∧∗ ∈

p,q p,q 1 2 The operator ∂¯∗ = ∂¯ : A A − is the adjoint of ∂¯ under the L -norm. h −∗E ∗E →

Proof. (Proof of the Lemma 6.1)

Since M is a K¨ahler manifold and E is a Hermitian vector bundle, we have the following K¨ahler identity

[Λ,D′]= √ 1∂¯∗ − h

63 From this we get the following equality from the deﬁnition of the Laplacian

√ 1 = ∂¯[Λ,D′]+[Λ,D′]∂¯ − (6.0.1)

= ∂¯ΛD′ ∂D¯ ′Λ+ΛD′∂¯ D′Λ∂¯ − − We also have the following. For any η Ar(E), since ω is closed, ∈

d(ω η)= ω dη ∧ ∧ so [L, d] = 0. Similarly, we have [L, D] = 0. Also, we have

LD′ = D′L, L∂¯ = ∂L¯ (6.0.2)

From Equations (6.0.1) and (6.0.2) we have

√ 1[, L]= √ 1L √ 1L − − − −

=(∂¯ΛD′ ∂D¯ ′Λ+ΛD′∂¯ D′Λ∂¯)L L(∂¯ΛD′ ∂D¯ ′Λ+ΛD′∂¯ D′Λ∂¯) − − − − −

= ∂HD¯ ′ ∂D¯ ′H + HD′∂¯ D′H∂¯ − − (6.0.3)

2n r where H = [Λ, L]= (n r)P , P is the projection map: A∗(E) A (E). r=0 − r r → P

64 Thus, for any ϕ Ap,q(E), we have ∈ √ 1[, L]ϕ = √ 1Lϕ √ 1Lϕ − − − −

=(n p q 1)∂D¯ ′ (n p q)∂D¯ ′+ − − − − − −

(n p q 2)D′∂¯ (n p q 1)D′∂¯ − − − − − − −

= (∂D¯ ′ + D′∂¯)ϕ − = Θϕ − So we obtain that

Lϕ Lϕ = √ 1Θϕ (6.0.4) − −

Since [L, D]=0andΘ= D2, we get [L, Θ] = 0.

n q,0 Now let v A − (E), q 1. By the equation (6.0.4), we get ∈ ≥

q q 1 q 1 L v = LL − v + √ 1ΘL − v −

2 q 2 q 1 = L L − v + 2(√ 1Θ)L − v − (6.0.5) = ······

q q 1 = L v + q√ 1ΘL − v − n q,0 n q,0 Since v A − (E), we know that v A − (E). The inner product ∈ ∈ (Lqv, Lqv)= ωq (v) (ωq v) ZM ∧ ∧∗ ∧

= (v) (v) (6.0.6) ZM ∧∗

=(v, v)

65 By Equations (6.0.5) and (6.0.6), we get the following

q q q q q 1 q (L v, L v)=(L v, L v)+ q(√ 1ΘL − v, L v) − (6.0.7) q 1 q =(v, v)+ q(√ 1ΘL − v, L v) − Let u = Lqv An,q(E), by Equation (6.0.7) we get ∈

2 2 2 2 q 1 q ∂u¯ + ∂¯∗u = ∂v¯ + ∂¯∗v + q(√ 1ΘL − v, L v) k k k h k k k k h k −

n q,0 But since v A − (E) is pure type, so we know that ∂¯∗v = 0. This leads to ∈ h

2 2 2 q 1 q ∂u¯ + ∂¯∗u = ∂v¯ + q(√ 1ΘL − v, L v) k k k h k k k −

and the lemma is proved.

6.2. Theorem Let (E, h) be a Nakano semipositive vector bundle of rank r on a compact K¨ahler manifold (M,ω). Then, for any 0 q n, the wedge multiplication ≤ ≤ operator ωq induces a surjective morphism ∧·

q q 0 n q q n L = ω : H (M, Ω − E) H (M, Ω E) h ∧· M ⊗ → M ⊗

Proof. Let e , , e be a local orthonormal basis of E, let ϕ , ,ϕ be the { 1 ··· r} { 1 ··· n} unitary coframe of (1, 0) forms of M. Write

q n u = f ϕ ϕ ϕˆ e H (M, Ω E) I,α 1 ∧···∧ n ∧ I ⊗ α ∈ M ⊗ IX=n q |1| α −r ≤ ≤ 66 n q,0 here ϕˆ = ϕ ϕ , q 1. Let v be an element in A − (E) such that I i1 ∧···∧ iq ≥ u = Lq (v)= ωq v, then h ∧

q v =2 f ϕ e ϕ = ϕ ϕ − I,α I ⊗ α I i1 ∧···∧ in q IX=n q |1| α −r ≤ ≤ where I =(i , , i ) in 1, , n . By the equation (3.1.1) we have 1 ··· p { ··· }

√ 1Θ = √ 1 R ¯ ¯ϕ ϕ e∗ e − − ijαβ i ∧ j ⊗ α ⊗ β 1 Xi,j n 1 ≤α,β≤ r ≤ ≤

The coefficients Ri¯jαβ¯ here is differ from the coefficients in (3.1.2) by a positive definite matrix. We have

q 1 q √ 1ΘL − v (L v)= √ 1Θv (Lv) − ∧∗ − ∧∗

2q =2 √ 1 R ¯ ¯ϕ ϕ e∗ e f ϕ e − ijαβ i ∧ j ⊗ α ⊗ β ∧ I,α I ⊗ α 1 Xi,j n IX=n q 1 ≤α,β≤ r |1| α −r ≤ ≤ ≤ ≤

n q+1 √ √ − \ \ ∗ ( 1) 1 2 εI i ,ˆiϕI i ϕˆi eα ∧ − − − ∪{ } ∪{ } ∧ ⊗ IX=n q 1 α| | r, 1− i n ≤ ≤ ≤ ≤ r n n+q+1 ω =2 fK j,αfK i,βRi¯jαβ¯ \ \ n! α,βX=1 K =Xn q+1 i,jXK | | − ∈ (6.0.8)

67 here I i =(i , , i , i) in 1, , n and i / I. ∪{ } 1 ··· p { ··· } ∈ r q 1 q n+q+1 (√ 1ΘL − v, L v)=2 fK j,αfK i,βRi¯jαβ¯ − \ \ α,βX=1 K =Xn q+1 i,jXK | | − ∈ r n+q+1 =2 fP i ,αfP j ,βRi¯jαβ¯ (6.0.9) ∪{ } ∪{ } α,βX=1 P =Xn q 1 i,jX /P | | − − ∈ r n n+q+1 P P =2 ( ξi,αξj,βRi¯jαβ¯) P =Xn q 1 α,βX=1 i,jX=1 | | − − here we deﬁne that

fP i ,α if i / P P ∪{ } ∈ ξi,α =   0 if i P  ∈  Let ξP = ξP , since that E is Nakano semi-positive, then by the equation (6.0.9) { i,α}i,α we get

q 1 q n+q+1 P P (√ 1ΘL − v, L v)=2 Θ(ξ , ξ ) 0 − ≥ P =Xn q 1 | | − − e By Lemma 6.1 we have

2 2 2 ∂u¯ + ∂¯∗u ∂v¯ k k k h k ≥k k

For any given cohomology class in Hq(M, Ωn E), one can pick a harmonic rep- M ⊗ n,q ¯ ¯ resentative u which is an element of A (E) satisfying ∂u = 0 and ∂h∗u = 0. Let

n q,0 q q v A − (E) be the element determined by u = L (v) = ω v. The above ∈ h ∧ Bochner-Kodaira-Nakano type inequality implies that ∂v¯ 0, thus ∂v¯ = 0. So k k ≤ 0 n q q v H (M, Ω − E), thus the map L is surjective. This completes the proof of the ∈ M ⊗ h theorem. 2

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