Lectures on Kaehler geometry

Andrei Moroianu9780521868914, 0521868912

Kähler 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. Kähler 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 Kähler identities. The final part of the text studies several aspects of compact Kähler 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.

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Table of contents :
Contents……Page 6
Introduction……Page 10
1.1. Introduction……Page 14
1.2. The tangent space……Page 15
1.3. Vector fields……Page 17
1.4. Exercises……Page 20
2.1. Exterior and tensor algebras……Page 24
2.2. Tensor fields……Page 26
2.3. Lie derivative of tensors……Page 28
2.4. Exercises……Page 30
3.2. The exterior derivative……Page 32
3.3. The Cartan formula……Page 34
3.4. Integration……Page 35
3.5. Exercises……Page 37
4.1. Lie groups……Page 40
4.2. Principal bundles……Page 42
4.4. Correspondence between principal and vector bundles……Page 44
4.5. Exercises……Page 46
5.1. Covariant derivatives on vector bundles……Page 48
5.2. Connections on principal bundles……Page 50
5.4. Pull-back of bundles……Page 52
5.5. Parallel transport……Page 53
5.6. Holonomy……Page 54
5.7. Reduction of connections……Page 55
5.8. Exercises……Page 56
6.1. Riemannian metrics……Page 58
6.2. The Levi–Civita connection……Page 59
6.3. The curvature tensor……Page 60
6.4. Killing vector fields……Page 62
6.5. Exercises……Page 63
7.1. Preliminaries……Page 68
7.3. Complex manifolds……Page 70
7.4. The complexified tangent bundle……Page 72
7.5. Exercises……Page 73
8.1. Decomposition of the (complexified) exterior bundle……Page 76
8.2. Holomorphic objects on complex manifolds……Page 78
8.3. Exercises……Page 79
9.1. Holomorphic vector bundles……Page 82
9.2. Holomorphic structures……Page 83
9.3. The canonical bundle of CPm……Page 85
9.4. Exercises……Page 86
10.1. The curvature operator of a connection……Page 88
10.2. Hermitian structures and connections……Page 89
10.3. Exercises……Page 91
11.1. Hermitian metrics……Page 92
11.2. Kähler metrics……Page 93
11.3. Characterization of Kähler metrics……Page 94
11.4. Comparison of the Levi–Civita and Chern connections……Page 96
11.5. Exercises……Page 97
12.1. The Kählerian curvature tensor……Page 98
12.2. The curvature tensor in local coordinates……Page 99
12.3. Exercises……Page 102
13.2. The Fubini–Study metric on the complex projective space……Page 104
13.3. Geometrical properties of the Fubini–Study metric……Page 106
13.4. Exercises……Page 108
14.1. The formal adjoint of a linear di erential operator……Page 110
14.2. The Laplace operator on Riemannian manifolds……Page 111
14.3. The Laplace operator on Kähler manifolds……Page 112
14.4. Exercises……Page 115
15.1. Hodge theory……Page 116
15.2. Dolbeault theory……Page 118
15.3. Exercises……Page 120
16.1. Chern–Weil theory……Page 124
16.2. Properties of the first Chern class……Page 127
16.3. Exercises……Page 129
17.2. The Ricci form as curvature form on the canonical bundle……Page 130
17.3. Ricci-flat Kähler manifolds……Page 132
17.4. Exercises……Page 133
18.1. An overview……Page 136
18.2. Exercises……Page 138
19.1. The Aubin–Yau theorem……Page 140
19.2. Holomorphic vector fields on Kähler–Einstein manifolds……Page 142
19.3. Exercises……Page 144
20.1. The Weitzenböck formula……Page 146
20.2. Vanishing results on Kähler manifolds……Page 148
20.3. Exercises……Page 150
21.1. Positive line bundles……Page 152
21.2. The Hirzebruch–Riemann–Roch formula……Page 153
21.3. Exercises……Page 156
22.1. The Lichnerowicz formula for Kähler manifolds……Page 158
22.2. The Kodaira vanishing theorem……Page 160
22.3. Exercises……Page 162
23.1. Hyperkähler manifolds……Page 164
23.2. Projective manifolds……Page 166
23.3. Exercises……Page 167
24.1. Divisors……Page 170
24.2. Line bundles and divisors……Page 172
24.3. Adjunction formulas……Page 173
24.4. Exercises……Page 176
Bibliography……Page 178
Index……Page 180