Complex Manifolds and Geometric Algebraic Analysis

Complex Manifolds and Geometric Algebraic Analysis is intended for graduate students in mathematics, physics, and beyond. The book is divided into ten chapters. Chapter 1 deals with the properties of holomorphic functions of several complex variables. Chapter 2 introduces tools for studying complex manifolds and analytic varieties, whilst Chapter 3 covers the foundational material from sheaves and cohomology. Chapter 4 concerns the study of divisors and line bundles on complex manifolds, and Chapter 5 is devoted to some fundamental theorems. Chapter 6 covers definitions and examples of abelian varieties, whilst Chapter 7 studies theta functions on complex projective tori. Lastly, the aim of Chapter 8 is to discuss an interesting interaction between complex algebraic geometry and dynamical systems. This book is supplemented with two appendices, one on Riemann surfaces and algebraic curves and the other covering elliptic functions and elliptic integrals. Additionally, various examples, exercises, and problems with solutions are provided throughout the book.

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 Inversemapping 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 complexmanifolds 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.ThePoincaré 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 andRiemann 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 Projectivevariety 101 5.1.2 Tangent and cotangent bundles 103 5.1.3 Dolbeault cohomology group 103 5.1.4 FirstChern class 106 5.1.5 Hodge forms, hermitian metrics, harmonic space, Hodge star operator and Hodge theorem 107 5.1.6.Kählermetric,Kähler form,Kählermanifold 109 5.2.TheKodaira–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.TheKodaira embedding theorem 117 5.6.Exercises 126 Chapter 6 Abelian Varieties 133 6.1.Definitions and examples 133 6.2.Riemann conditions, polarization andRiemann form 138 6.3 Isogenies and reducibleAbelian varieties 141 6.4.DualAbelian varieties 142 6.5 Prymvarieties 145 6.6 Projectively normal embedding 154 6.7 Number of even and odd sections of a line bundle 155 6.8.Exercises 156 Contents vii 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 Hénon–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