Lectures on Kähler Geometry

by Andrei Moroianu

• ISBN13: 9780521868914
• ISBN10: 0521868912
• Format: Hardcover
• Copyright: 2007-05-07
• Publisher: Cambridge University Press

Summary

Kahler geometry is a beautiful and intriguing area of mathematics, of substantial research interest to both mathematicians and physicists. This self-contained graduate text provides a concise and accessible introduction to the topic. The book begins with a review of basic differential geometry, before moving on to a description of complex manifolds and holomorphic vector bundles. Kahler manifolds are discussed from the point of view of Riemannian geometry, and Hodge and Dolbeault theories are outlined, together with a simple proof of the famous Kahler identities. The final part of the text studies several aspects of compact Kahler manifolds: the Calabi conjecture, Weitzenböck techniques, Calabi-Yau manifolds, and divisors. All sections of the book end with a series of exercises and students and researchers working in the fields of algebraic and differential geometry and theoretical physics will find that the book provides them with a sound understanding of this theory.

Table of Contents

Introduction	p. ix
Basics of differential geometry	p. 1
Smooth manifolds	p. 3
Introduction	p. 3
The tangent space	p. 4
Vector fields	p. 6
Exercises	p. 9
Tensor fields on smooth manifolds	p. 13
Exterior and tensor algebras	p. 13
Tensor fields	p. 15
Lie derivative of tensors	p. 17
Exercises	p. 19
The exterior derivative	p. 21
Exterior forms	p. 21
The exterior derivative	p. 21
The Cartan formula	p. 23
Integration	p. 24
Exercises	p. 26
Principal and vector bundles	p. 29
Lie groups	p. 29
Principal bundles	p. 31
Vector bundles	p. 33
Correspondence between principal and vector bundles	p. 33
Exercises	p. 35
Connections	p. 37
Covariant derivatives on vector bundles	p. 37
Connections on principal bundles	p. 39
Linear connections	p. 41
Pull-back of bundles	p. 41
Parallel transport	p. 42
Holonomy	p. 43
Reduction of connections	p. 44
Exercises	p. 45
Riemannian manifolds	p. 47
Riemannian metrics	p. 47
The Levi-Civita connection	p. 48
The curvature tensor	p. 49
Killing vector fields	p. 51
Exercises	p. 52
Complex and Hermitian geometry	p. 55
Complex structures and holomorphic maps	p. 57
Preliminaries	p. 57
Holomorphic functions	p. 59
Complex manifolds	p. 59
The complexified tangent bundle	p. 61
Exercises	p. 62
Holomorphic forms and vector fields	p. 65
Decomposition of the (complexified) exterior bundle	p. 65
Holomorphic objects on complex manifolds	p. 67
Exercises	p. 68
Complex and holomorphic vector bundles	p. 71
Holomorphic vector bundles	p. 71
Holomorphic structures	p. 72
The canonical bundle of CP [superscript m]	p. 74
Exercises	p. 75
Hermitian bundles	p. 77
The curvature operator of a connection	p. 77
Hermitian structures and connections	p. 78
Exercises	p. 80
Hermitian and Kahler metrics	p. 81
Hermitian metrics	p. 81
Kahler metrics	p. 82
Characterization of Kahler metrics	p. 83
Comparison of the Levi-Civita and Chern connections	p. 85
Exercises	p. 86
The curvature tensor of Kahler manifolds	p. 87
The Kahlerian curvature tensor	p. 87
The curvature tensor in local coordinates	p. 88
Exercises	p. 91
Examples of Kahler metrics	p. 93
The flat metric on C[superscript m]	p. 93
The Fubini-Study metric on the complex projective space	p. 93
Geometrical properties of the Fubini-Study metric	p. 95
Exercises	p. 97
Natural operators on Riemannian and Kahler manifolds	p. 99
The formal adjoint of a linear differential operator	p. 99
The Laplace operator on Riemannian manifolds	p. 100
The Laplace operator on Kahler manifolds	p. 101
Exercises	p. 104
Hodge and Dolbeault theories	p. 105
Hodge theory	p. 105
Dolbeault theory	p. 107
Exercises	p. 109
Topics on compact Kahler manifolds	p. 111
Chern classes	p. 113
Chern-Weil theory	p. 113
Properties of the first Chern class	p. 116
Exercises	p. 118
The Ricci form of Kahler manifolds	p. 119
Kahler metrics as geometric U[subscript m]-structures	p. 119
The Ricci form as curvature form on the canonical bundle	p. 119
Ricci-flat Kahler manifolds	p. 121
Exercises	p. 122
The Calabi-Yau theorem	p. 125
An overview	p. 125
Exercises	p. 127
Kahler-Einstein metrics	p. 129
The Aubin-Yau theorem	p. 129
Holomorphic vector fields on Kahler-Einstein manifolds	p. 131
Exercises	p. 133
Weitzenbock techniques	p. 135
The Weitzenbock formula	p. 135
Vanishing results on Kahler manifolds	p. 137
Exercises	p. 139
The Hirzebruch-Riemann-Roch formula	p. 141
Positive line bundles	p. 141
The Hirzebruch-Riemann-Roch formula	p. 142
Exercises	p. 145
Further vanishing results	p. 147
The Lichnerowicz formula for Kahler manifolds	p. 147
The Kodaira vanishing theorem	p. 149
Exercises	p. 151
Ricci-flat Kahler metrics	p. 153
Hyperkahler manifolds	p. 153
Projective manifolds	p. 155
Exercises	p. 156
Explicit examples of Calabi-Yau manifolds	p. 159
Divisors	p. 159
Line bundles and divisors	p. 161
Adjunction formulas	p. 162
Exercises	p. 165
Bibliography	p. 167
Index	p. 169
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