代数几何原理(英文版)

代数几何是数学中最古老和发展比较快的学科之 一，它与投影几何、复分析、拓扑学、数论以及数学 领域的其它分支有着紧密的联系。然而近些年代数几 何不论是风格还是语言都发生了巨大的变化，本书展 示了相关理论的主要研究结果和计算工具的发展。

CHAPTER 0 FOUNDATIONAL MATERIAL 1. Rudiments of Several Complex Variables Cauchy's Formula and Applications Several Variables Weierstrass Theorems and Corollaries Analytic Varieties 2. Complex Manifolds Complex Manifolds Submanifolds and Subvarieties De Rham and Dolbeault Cohomology Calculus on Complex Manifolds 3. Sheaves and Cohomology Origins: The Mittag-Leffler Problem Sheaves Cohomology of Sheaves The de Rham Theorem The Dolbeault Theorem 4. Topology of Manifolds Intersection of Cycles Poincare Duality Intersection of Analytic Cycles 5. Vector Bundles, Connections, and Curvature Complex and Holomorphic Vector Bundles Metrics, Connections, and Curvature 6. Harmonic Theory on Compact Complex Manifolds The Hodge Theorem Proof of the Hodge Theorem I: Local Theory Proof of the Hodge Theorem II: Global Theory Applications of the Hodge Theorem 7. Kahler Manifolds The Kahler Condition The Hodge Identities and the Hodge Decomposition The Lefschetz Decomposition CHAPTER 1 COMPLEX ALGEBRAIC VARIETIES 1. Divisors and Line Bundles Divisors Line Bundles Chern Classes of Line Bundles 2. Some Vanishing Theorems and Corollaries The Kodaira Vanishing Theorem The Lefschetz Theorem on Hyperplane Sections Theorem B The Lefschetz Theorem on (1, l)-classes 3. Algebraic Varieties Analytic and Algebraic Varieties Degree of a Variety Tangent Spaces to Algebraic Varieties 4. The Kodaira Embedding Theorem Line Bundles and Maps to Projective Space Blowing Up