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MAT1360: Complex Manifolds and

Hermitian Differential Geometry

University of Toronto, Spring Term, 1997

Lecturer: Andrew D. Hwang

Contents

1 Holomorphic Functions and Atlases 1

1.1 Functions of Several Complex Variables . . . . . . . . . . . . . . . . . . . . . 11.2 Complex Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Almost-Complex Structures and Integrability 9

2.1 Complex Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Almost-Complex Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Integrability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Sheaves and Vector Bundles 20

3.1 Presheaves and Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Cohomology 30

4.1 Cech Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Dolbeault Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Elementary Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Analytic and Algebraic Varieties 40

5.1 The Local Structure of Analytic Hypersurfaces . . . . . . . . . . . . . . . . . 435.2 Singularities of Algebraic Varieties . . . . . . . . . . . . . . . . . . . . . . . . 45

6 Divisors, Meromorphic Functions, and Line Bundles 52

6.1 Divisors and Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.2 Meromorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.3 Sections of Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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6.4 Chows Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7 Metrics, Connections, and Curvature 60

7.1 Hermitian and Kahler Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 617.2 Connections in Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8 Hodge Theory and Applications 71

8.1 The Hodge Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718.2 The Hodge Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . . 77

9 Chern Classes 82

9.1 Chern Forms of a Connection . . . . . . . . . . . . . . . . . . . . . . . . . . 849.2 Alternate Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

10 Vanishing Theorems and Applications 90

10.1 Ampleness and Positivity of Line Bundles and Divisors . . . . . . . . . . . . 9010.2 The Kodaira-Nakano Vanishing Theorem . . . . . . . . . . . . . . . . . . . . 9110.3 Cohomology of Projective Manifolds . . . . . . . . . . . . . . . . . . . . . . 9310.4 The Kodaira Embedding Theorem . . . . . . . . . . . . . . . . . . . . . . . . 9610.5 The Hodge Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

11 Curvature and Holomorphic Vector Fields 100

11.1 Ricci Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10111.2 Holomorphic Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

I Metrics With Special Curvature 105

12 Einstein-Kahler Metrics 105

12.1 The Calabi Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10612.2 Positive Einstein-Kahler Metrics . . . . . . . . . . . . . . . . . . . . . . . . . 109

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Preface

These notes grew out of a course called Complex Manifolds and Hermitian DifferentialGeometry given during the Spring Term, 1997, at the University of Toronto. The intentis not to give a thorough treatment of the algebraic and differential geometry of complexmanifolds, but to introduce the reader to material of current interest as quickly as possible.

As a glance at the table of contents indicates, Part I treats standard introductory ana-lytic material on complex manifolds, sheaf cohomology and deformation theory, differentialgeometry of vector bundles (Hodge theory, and Chern classes via curvature), and some ap-plications to the topology and projective embeddability of Kahlerian manifolds. The intentis to provide a number of interesting and non-trivial examples, both in the text and in theexercises. Some details have been skipped, such as the a priori estimates in the proof of theHodge Theorem. When details are omitted, I have tried to provide ideas of proofs, partic-ularly when there is geometric intuition available, and to indicate what needs to be provenbut has not been.

Part II is a fairly detailed survey of results on Einstein and extremal Kahler metrics fromthe early 1980s to the present. It is hoped that this exposition will be of use to youngresearchers and other interested mathematicians and physicists by collecting results andreferences in one place, and by pointing out open questions. The results described in Part IIare due to T. Aubin, S. Bando, E. Calabi, S. Donaldson, A. Futaki, Z. D. Guan, N. Hitchin,S. Kobayashi, N. Koiso, C. Lebrun, T. Mabuchi, A. M. Nadel, H. Pedersen, Y.-S. Poon,Y. Sakane, S. R. Simanca, M. F. Singer, Y.-T. Siu, G. Tian, K. Uhlenbeck, S.-T. Yau. Ioffer my sincere apologies to authors whose work I have overlooked.

In Part I, my debt to the book of Griffiths-Harris is great, and to books of several otherauthors is substantial. The bibliography lists, among other works, the books from which thecourse packet was drawn. I hope readers find the exercises useful; while there are texts atthis level which contain exercises, it seems there are few which deal with the specific butcolourful examples scattered though folklore and the literature.

I have taken some care to ensure that the notationincluding signs and other constantfactorsis internally consistent, and maximally consistent with other works. Occasionallya concept is introduced informally, in which case the term being defined in enclosed inquotation marks. The subsequent formal definition contains the term in italics. Thefollowing lists the end-of symbols: occurs at the end of proofs, 2 denotes the end of anexample or remark, and signifies the end of an exercise.

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1 Holomorphic Functions and Atlases

A function f : D C of one complex variable is (complex) differentiable in a domain D ifthe ordinary Newton quotient

f (z) := limwz

f(w) f(z)w z

exists for every point z D. For present purposes, there are two other useful characteriza-tions 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 R2has complex-linear derivative at every point, in which case f is said to be holomorphic.Concretely, there is a ring homorphism

a+ bi C [

a bb a

] R22,

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 z0, then for allz with |z z0| < r, the Cauchy integral formula gives

f(z) =1

2i

|wz0|=r

f(w) dw

w z .

Writing 1/(w z) as a geometric series in z z0 and integrating term-by-term shows thata holomorphic function may be expressed locally as a convergent power series. In words, aholomorphic function is complex-analytic. Intuitively, the averaging process effected by thecontour integral makes the integrand smoother; if f is of class Ck, then the expression onthe right is of class Ck+1. Since f (times a smooth function) is the integrand, f itself mustbe smooth. This is the prototypical bootstrap argument, and perhaps the most elementaryexample 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 samereasoning. Let D Cn be an open set. A function f : D C is holomorphic if theCauchy-Riemann equations hold on D. More precisely, write z = x + iy and f = u + ivwith 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 C1 and

(1.1)u

x=

v

y,

u

y= v

x

1

at each point of D. Holomorphicity is related to separate holomorphicity (OsgoodsLemma, Proposition 1.1 below), that is, holomorphicity of the functions obtained by fix-ing n 1 of the variables and varying the remaining one. The continuity hypothesis may bedropped (Hartogs Theorem), though the proof becomes substantially more difficult.

Proposition 1.1 Let D Cn be a non-empty open set. If f : D C is continuous andseparately holomorphic, then f is holomorphic.

Let r = (r1, . . . , rn) be a radius, that is, an n-tuple of positive real numbers, and letz0 = (z

10 , . . . , z

n0 ) Cn. If r and r are radii, then r < r is taken to mean r < r for

= 1, . . . , n. The polydisk of radius r centered at z0 is, by definition,

r(z0) = {z Cn : |z z0 | < r for = 1, . . . , n} = {z Cn : |z z0| < r}.

Thus a polydisk is exactly a Cartesian product of ordinary disks. While polydisks are notgenerally domains of convergence for power series of several variables, they are nonethelessthe 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 z0 D, there is a complex power series centered at z0 which converges and isequal to f on some polydisk r(z0). In order to avoid purely notational complications, it isconvenient to use multi-indices. If I = (i1, . . . , in) is a multi-index, then set

|I| = i1 + + in, zI = (z1)i1 (zn)in , fI =f

zI=

kf

(z1)i1 (zn)in .

Analyticity means there is a polydisk r(z0

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