This is page Printer: Opaque this A Beginner’s Guide to Holomorphic Manifolds Andrew D. Hwang This is page i Printer: Opaque this Preface This book constitutes notes from a one-semester graduate course called “Complex Manifolds and Hermitian Differential Geometry” given during the Spring Term, 1997, at the University of Toronto. Its aim is not to give a thorough treatment of the algebraic and differential geometry of holomorphic manifolds, but to introduce material of current interest as quickly and concretely as possible with a minimum of prerequisites. There are several excellent references available for the reader who wishes to see subjects in more depth. The coverage includes standard introductory analytic material on holo- morphic manifolds, sheaf cohomology and deformation theory, differential geometry of vector bundles (Hodge theory, and Chern classes via curva- ture), and some applications to the topology and projective embeddability of K¨ahlerian manifolds. The final chapter is a short survey of extremal K¨ahler metrics and related topics, emphasizing the geometric and “soft” analytic aspects. There is a large number of exercises, particularly for a book at this level. The exercises introduce several specific but colorful ex- amples scattered through “folklore” and “the literature.” Because there are recurrent themes and varying viewpoints in the subject, some of the exercises overlap considerably. The course attendees were mostly advanced graduate students in math- ematics, but it is hoped that these notes will reach a wider audience, in- cluding theoretical physicists. The “ideal” reader would be familiar with smooth manifolds (charts, forms, flows, Lie groups, vector bundles), differ- ential geometry (metrics, connections, and curvature), and basic algebraic topology (simplicial and singular cohomology, the long exact sequence, and ii the fundamental group), but in reality the prerequisites are less strenuous, though a good reference for each subject should be kept at hand. Apologies are perhaps in order for the departure from standard termi- nology, see Table 1. A manifold M is “complex” if TM is a complex vector bundle, and is “holomorphic” if TM is a holomorphic vector bundle. This is in accord with the informal usage of “complex/holomorphic category,” as well as the standard usage for vector bundles. Similar considerations ap- ply to other types of structure: “complex” describes algebraic data, while “holomorphic” connotes integrability. In this usage, the real six-sphere S6 is a complex manifold, but it is not known whether or not S6 admits a holomorphic structure. Standard Term Replacement Complex Holomorphic Almost-complex Complex TABLE 1. Non-standard terminology. Acknowledgements The influence of the book of Griffiths-Harris is great, and that of several other authors (especially Chern, Kobayashi, Kobayashi-Nomizu, Kodaira, and Wells) is substantial. I am particularly indebted to E. Hironaka for the lecture on Singularities of Algebraic Varieties. In addition, it is a plea- sure to thank I. Graham for the opportunity to offer the original course, G. Valent for interest in and support of this project, and S. Kobayashi for mathematical and pedagogical inspiration. This project was supported by an NSERC Canada Individual Research Grant; I am also grateful to the LPTHE, Universit´ede Paris VII, for generous financial support. This is page iii Printer: Opaque this Contents 1 Holomorphic Functions and Atlases 2 1.1 FunctionsofSeveralComplexVariables . 3 1.2 HolomorphicManifolds . 5 2 Complex Structures and Integrability 13 2.1 ComplexLinearAlgebra . 13 2.2 ComplexManifolds. .. .. .. .. .. .. .. .. .. 15 2.3 IntegrabilityConditions . 19 3 Sheaves and Vector Bundles 26 3.1 PresheavesandMorphisms . 26 3.2 VectorBundles ......................... 30 4 Cohomology 38 4.1 CechCohomologyˇ ....................... 38 4.2 DolbeaultCohomology. 43 4.3 ElementaryDeformationTheory . 47 5 Analytic and Algebraic Varieties 50 5.1 The Local Structure of Analytic Hypersurfaces . 52 5.2 Singularities of Algebraic Varieties . 55 6 Divisors, Meromorphic Functions, and Line Bundles 62 6.1 DivisorsandLineBundles . 62 6.2 MeromorphicFunctions . 64 1 6.3 SectionsofLineBundles . 65 6.4 Chow’sTheorem ........................ 67 7 Metrics, Connections, and Curvature 73 7.1 HermitianandK¨ahlerMetrics. 74 7.2 ConnectionsinVectorBundles . 77 8 Hodge Theory and Applications 85 8.1 TheHodgeTheorem ...................... 85 8.2 TheHodgeDecompositionTheorem . 91 9 Chern Classes 98 9.1 ChernFormsofaConnection . 99 9.2 AlternateDefinitions . 101 10 Vanishing Theorems and Applications 107 10.1 AmplenessandPositivity . 107 10.2 The Kodaira-NakanoVanishing Theorem . 109 10.3 CohomologyofProjectiveManifolds . 110 10.4 TheKodairaEmbeddingTheorem . 113 10.5 TheHodgeConjecture . 115 11 Curvature and Holomorphic Vector Fields 118 11.1 RicciCurvature. 119 11.2 HolomorphicVectorFields. 120 12 Extremal K¨ahler Metrics 124 12.1 TheCalabiConjectures . 127 12.2 Positive Einstein-K¨ahler Metrics . 130 This is page 2 Printer: Opaque this 1 Holomorphic Functions and Atlases A function f : D C of one complex variable is (complex) differentiable in a domain D if the→ ordinary Newton quotient f(w) f(z) f ′(z) := lim − w z w z → − exists for every point z D. For present purposes, there are two other useful characterizations of this∈ condition. The first is to identify the complex line C with the real plane R2. The function f is complex differentiable if and only if the associated function f : D R2 has complex-linear derivative at every point, in which case f is said→ to be holomorphic. Concretely, there is a ring homorphism a b 2 2 (a + bi C ֒ R × , (1.1 ∈ → b a ∈ − so f ′ = Df is complex-linear if and only if u = Re f and v = Im f satisfy the Cauchy-Riemann equations. On the other hand, if f is holomorphic in a disk of radius > r centered at z , then for all z with z z < r, the Cauchy integral formula gives 0 | − 0| 1 f(w) dw f(z)= . 2πi w z0 =r w z Z| − | − Writing 1/(w z) as a geometric series in z z0 and integrating term- by-term shows− that a holomorphic function may− be expressed locally as 1.1 Functions of Several Complex Variables 3 a convergent power series. In words, a holomorphic function is complex- analytic. Intuitively, the averaging process effected by the contour integral makes the integrand smoother; if f is of class k, then the expression on the right is of class k+1. Since f (times a smoothC function) is the integrand, f itself must beC smooth. This is the prototypical bootstrap argument, and perhaps the most elementary example of “elliptic regularity.” 1.1 Functions of Several Complex Variables For functions of more than one variable, much of this philosophy carries over by the same reasoning. Let D Cn be an open set. A function f : D C is holomorphic if the Cauchy-Riemann⊂ equations hold on D. More precisely,→ write zα = xα + iyα and f = u + iv with u and v real-valued. Then u and v may be regarded as functions on a subset of R2n, and f is holomorphic if f is of class 1 and C ∂u ∂v ∂u ∂v = , = (1.2) ∂xα ∂yα ∂yα −∂xα at each point of D. Holomorphicity is related to “separate” holomorphic- ity (Osgood’s Lemma, Proposition 1.1 below), that is, holomorphicity of the functions obtained by fixing n 1 of the variables and varying the remaining one; the hypotheses allow− differentiation under the integral sign in the Cauchy integral formula. The continuity hypothesis may be dropped (Hartogs’ Theorem), though the proof becomes more difficult. Proposition 1.1 Let D Cn be a non-empty open set. If f : D C is continuous and separately⊂ holomorphic, then f is holomorphic. → Let r = (r1,...,rn) be a radius, that is, an n-tuple of positive real 1 n n numbers, and let z = (z ,...,z ) C . If r and r′ are radii, then r′ < r 0 0 0 ∈ is taken to mean rα′ < rα for α =1,...,n. The polydisk of radius r centered at z0 is, by definition, ∆ (z ) = z Cn : zα zα < r for α =1,...,n r 0 { ∈ | − 0 | α } = z Cn : z z < r . { ∈ | − 0| } Thus a polydisk is exactly a Cartesian product of ordinary disks. While polydisks are not generally domains of convergence for power series of sev- eral variables, they are often the most convenient sets to use for local purposes. Let D be a non-empty open set in Cn. A function f : D C is complex analytic if, for every z D, there is a complex power series→ centered at 0 ∈ z0 that converges and is equal to f on some polydisk ∆r(z0). Multi-indices 1.1 Functions of Several Complex Variables 4 simplify notation substantially; if I = (i1,...,in) is a multi-index, then set I = i + + i , zI = (z1)i1 (zn)in , and | | 1 · · · n · · · ∂f ∂kf fI = = . ∂zI (∂z1)i1 (∂zn)in · · · Analyticity means there is a polydisk ∆r(z0) such that ∞ 1 f(z)= f (z )(z z )I (1.3) k! I 0 − 0 k=0 I =k X |X| for all z ∆r(z0). As in the case of one variable, holomorphicity and analyticity∈ are equivalent, as is seen by using (an obvious generalization of) the Cauchy integral formula. The concepts of holomorphicity and analyticity extend in the obvious way to functions with values in Cm, which are usually called “holomorphic maps.” A holomorphic map between open subsets of Cn that possesses a holomorphic inverse is a biholomorphism. Remarkably, a one-to-one holo- morphic map between open subsets of Cn is a biholomorphism, see Theo- rem 5.3; this result has no analogue in the smooth category, even for real polynomial maps, as is shown by x x3. The set of biholomorphisms between open subsets of Cn is a pseudogroup:7→ The composite of two bi- holomorphisms is a biholomorphism wherever it is defined. It is sometimes useful to consider anti-holomorphic maps.