Complex Manifolds and Geometric Algebraic Analysis

Preface ix

Chapter 1. Holomorphic Functions of Several Complex Variables 1
1.1. Notations, definitions and properties 1
1.2. Cauchy integral formula and applications 7
1.3. Power series in several variables 9
1.4. Various fundamental properties 20
1.5. Inverse mapping and implicit function theorems 28
1.6. Exercises 32

Chapter 2. Complex Manifolds and Analytic Varieties 37
2.1. Preliminaries 37
2.2. Examples of complex manifolds 42
2.2.1. Cn and open subsets 42
2.2.2. Complex algebraic curves or compact Riemann surfaces 43
2.2.3. Spheres 43
2.2.4. Projective spaces 47
2.2.5. Grassmannians 50
2.2.6. Tori 51
2.3. Tangent spaces and tangent bundles 52
2.4. Constant rank theorem 53
2.5. Submanifolds, subvarieties and examples 55
2.6. Exercises 58

Chapter 3. Sheaves and Cohomology 65
3.1. Sheaves 65
3.2. Cech cohomology 69
3.3. De Rham cohomology 71
3.4. Dolbeault cohomology 72
3.5. Connections and curvature 74
3.6. Curvature form, first Chern class of line bundles 76
3.7. The Poincare dual 77
3.8. Exercises 78

Chapter 4. Divisors and Line Bundles 79
4.1. Divisors 79
4.2. Line bundles 81
4.3. Sections of line bundles 83
4.4. Tori and Riemann form. 89
4.5. Line bundles on complex tori 91
4.6. Exercises 97

Chapter 5. Some Fundamental Theorems 101
5.1. Preliminaries and various notions 101
5.1.1. Projective variety 101
5.1.2. Tangent and cotangent bundles 103
5.1.3. Dolbeault cohomology group 103
5.1.4. First Chern class 106
5.1.5. Hodge forms, hermitian metrics, harmonic space, Hodge star operator and Hodget heorem 107
5.1.6. Kahler metric, Kahler form, Kahler manifold 109
5.2. The Kodaira–Nakano vanishing theorem 111
5.3. The Lefschetz theorem on hyperplane sections 113
5.4. The Lefschetz theorem on(1,1)-classes 114
5.5. The Kodaira embedding theorem 117
5.6. Exercises 126

Chapter 6. Abelian Varieties 133
6.1. Definitions and examples 133
6.2. Riemann conditions, polarization and Riemann form 138
6.3. Isogenies and reducible Abelian varieties 141
6.4. Dual Abelian varieties 142
6.5. Prym varieties 145
6.6. Projectivity normal embedding 154
6.7. Number of even and odd sections of a line bundle 155
6.8. Exercises 156

Chapter 7. Theta Functions and Complex Projective Tori 159
7.1. Meromorphic functions and theta functions 159
7.2. Lefschetz theorem 173
7.3. Exercises 179

Chapter 8. Algebraically Completely Integrable Systems 183
8.1. Preliminaries 183
8.2. The Henon–Heiles system. 190
8.3. Kowalewski's spinning top. 204
8.4. Kirchhoff's equations of motion of a solid in an ideal fluid 213
8.5. Exercises 216

Chapter 9. Appendix: Riemann Surfaces and Algebraic Curves 227

Chapter 10. Appendix: Elliptic Functions and Elliptic Integrals 233
10.1. Elliptic functions 233
10.2. Weierstrass functions 235
10.3. Elliptic integrals and Jacobi elliptic functions 240
10.4. Application: simple pendulum 244

References 247
Index 251