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.
Author(s): Andrei Moroianu
Series: London Mathematical Society Student Texts
Publisher: Cambridge University Press
Year: 2007
Language: English
Pages: 183
Series-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Introduction......Page 11
Part 1 Basics of differential geometry......Page 13
1.1. Introduction......Page 15
1.2. The tangent space......Page 16
1.3. Vector fields......Page 18
1.4. Exercises......Page 21
2.1. Exterior and tensor algebras......Page 25
2.2. Tensor fields......Page 27
2.3. Lie derivative of tensors......Page 29
2.4. Exercises......Page 31
3.2. The exterior derivative......Page 33
3.3. The Cartan formula......Page 35
3.4. Integration......Page 36
3.5. Exercises......Page 38
4.1. Lie groups......Page 41
4.2. Principal bundles......Page 43
4.4. Correspondence between principal and vector bundles......Page 45
4.5. Exercises......Page 47
5.1. Covariant derivatives on vector bundles......Page 49
5.2. Connections on principal bundles......Page 51
5.4. Pull-back of bundles......Page 53
5.5. Parallel transport......Page 54
5.6. Holonomy......Page 55
5.7. Reduction of connections......Page 56
5.8. Exercises......Page 57
6.1. Riemannian metrics......Page 59
6.2. The Levi–Civita connection......Page 60
6.3. The curvature tensor......Page 61
6.4. Killing vector fields......Page 63
6.5. Exercises......Page 64
Part 2 Complex and Hermitian geometry......Page 67
7.1. Preliminaries......Page 69
7.3. Complex manifolds......Page 71
7.4. The complexified tangent bundle......Page 73
7.5. Exercises......Page 74
8.1. Decomposition of the (complexified) exterior bundle......Page 77
8.2. Holomorphic objects on complex manifolds......Page 79
8.3. Exercises......Page 80
9.1. Holomorphic vector bundles......Page 83
9.2. Holomorphic structures......Page 84
9.3. The canonical bundle of CPm......Page 86
9.4. Exercises......Page 87
10.1. The curvature operator of a connection......Page 89
10.2. Hermitian structures and connections......Page 90
10.3. Exercises......Page 92
11.1. Hermitian metrics......Page 93
11.2. Kähler metrics......Page 94
11.3. Characterization of Kähler metrics......Page 95
11.4. Comparison of the Levi–Civita and Chern connections......Page 97
11.5. Exercises......Page 98
12.1. The Kählerian curvature tensor......Page 99
12.2. The curvature tensor in local coordinates......Page 100
12.3. Exercises......Page 103
13.2. The Fubini–Study metric on the complex projective space......Page 105
13.3. Geometrical properties of the Fubini–Study metric......Page 107
13.4. Exercises......Page 109
14.1. The formal adjoint of a linear di erential operator......Page 111
14.2. The Laplace operator on Riemannian manifolds......Page 112
14.3. The Laplace operator on Kähler manifolds......Page 113
14.4. Exercises......Page 116
15.1. Hodge theory......Page 117
15.2. Dolbeault theory......Page 119
15.3. Exercises......Page 121
Part 3 Topics on compact Kähler manifolds......Page 123
16.1. Chern–Weil theory......Page 125
16.2. Properties of the first Chern class......Page 128
16.3. Exercises......Page 130
17.2. The Ricci form as curvature form on the canonical bundle......Page 131
17.3. Ricci-flat Kähler manifolds......Page 133
17.4. Exercises......Page 134
18.1. An overview......Page 137
18.2. Exercises......Page 139
19.1. The Aubin–Yau theorem......Page 141
19.2. Holomorphic vector fields on Kähler–Einstein manifolds......Page 143
19.3. Exercises......Page 145
20.1. The Weitzenböck formula......Page 147
20.2. Vanishing results on Kähler manifolds......Page 149
20.3. Exercises......Page 151
21.1. Positive line bundles......Page 153
21.2. The Hirzebruch–Riemann–Roch formula......Page 154
21.3. Exercises......Page 157
22.1. The Lichnerowicz formula for Kähler manifolds......Page 159
22.2. The Kodaira vanishing theorem......Page 161
22.3. Exercises......Page 163
23.1. Hyperkähler manifolds......Page 165
23.2. Projective manifolds......Page 167
23.3. Exercises......Page 168
24.1. Divisors......Page 171
24.2. Line bundles and divisors......Page 173
24.3. Adjunction formulas......Page 174
24.4. Exercises......Page 177
Bibliography......Page 179
Index......Page 181
