复几何导论(英文版)

复几何主要研究复流形 ，近年来弦论的发展使得 复几何成为数学家和理论 物理学家研究和关注的热 点。微分几何成为代数与 微分几何的交叉学科，在 数学领域扮演越来越重要 的角色。 《复几何导论(英文版) 》作者旨在写一部容易理 解的入门级教科书。书中 比较详细的综述了复几何 的基本概念，同时论及代 数以及矩阵方面的内容， 也有一些加深理解的练习 。各章附录可以对读者了 解科研动态的起到指引作 用。

1 Local Theory 1.1 Holomorphic Functions of Several Variables 1.2 Complex and Hermitian Structures 1.3 Differential Forms 2 Complex Manifolds 2.1 Complex Manifolds: Definition and Examples 2.2 Holomorphic Vector Bundles 2.3 Divisors and Line Bundles 2.4 The Projective Space 2.5 Blow-ups 2.6 Differential Calculus on Complex Manifolds 3 Kahler Manifolds 3.1 Kahler Identities 3.2 Hodge Theory on Kahler Manifolds 3.3 Lefschetz Theorems Appendix 3.A Formality of Compact Kahler Manifolds 3.B SUSY for Kahler Manifolds 3.C Hodge Structures 4 Vector Bundles 4.1 Hermitian Vector Bundles and Serre Duality 4.2 Connections 4.3 Curvature 4.4 Chern Classes Appendix 4.A Levi-Civita Connection and Holonomy on Complex Manifolds 4.B Hermite-Einstein and Kahler-Einstein Metrics 5 Applications of Cohomology 5.1 Hirzebruch-Riemann-Roch Theorem 5.2 Kodaira Vanishing Theorem and Applications 5.3 Kodaira Embedding Theorem Deformations of Complex Structures 6.1 The Maurer-Cartan Equation 6.2 General Results Appendix 6.A dGBV-Algebras A Hodge Theory on Differentiable Manifolds B Sheaf Cohomology References Index