本书是一本关于复流形及其形变理论的导引教材。黎曼面复结构形变理论可以追溯到大数学家黎曼在1857年发表的关于阿贝尔函数的文章，计算出了形变所依赖的有效参数的个数。自黎曼发表的开创性文章后，关于黎曼面复结构形变理论的问题就一直是一个非常有趣的问题并被许多数学家所关注。

本书作者及其合作者D.C.Spencer构筑了紧致复流形的形变理论，其思想较为原始：因为紧致复流形是由有限个坐标邻域组成那么非常自然的想法就是，紧致复流形的无穷小形变应该能被其上同调群的元素表示出来。基于此想法，作者同Spencer一道倾起一生发展了紧致复流形形变理论。

全书分为七章：全纯函数、复流形、微分形式、向量从、层、无穷维共形存在定理、完备性定理与稳定性定理。复流形共形理论所用到的椭圆偏微分算子理论被放在附录中。

CHAPTER 1

Hoiomorphic Functions

 §1.1. Holomorphic Functions

 §1.2. Holomorphic Map

CHAPTER 2

Complex Manifolds

 §2.1. Complex Manifolds

 §2.2. Compact Complex Manifolds

 §2.3. Complex Analytic Family

CHAPTER 3

Differential Forms, Vector Bundles, Sheaves

 §3.1. Differential Forms

 §3.2. Vector Bundles

 §3.3. Sheaves and Cohomology

 §3.4. de Rham's Theorem and Dolbeault's Theorem

 §3.5. Harmonic Differential Forms

 §3.6. Complex Line Bundles

CHAPTER 4

Infinitesimal Deformation

 §4.1. Differentiable Family

 §4.2. Infinitesimal Deformation

CHAPTER 5

Theorem of Existence

 §5.1. Obstructions

 §5.2. Number of Moduli

 §5.3. Theorem of Existence

CHAPTER 6

Theorem of Completehess

 §6.1. Theorem of Completeness

 §6.2. Number of Moduli

 §6.3. Later Developments

CHAPTER 7

Theorem of Stability

 §7.1. Differentiable Family of Strongly Elliptic Differential Operators

 §7.2. Differentiable Family of Compact Complex Manifolds

APPENDIX

Elliptic Partial Differential Operators on a Manifold by Daisuke Fujiwara

 §1. Distributions on a Torus

 §2. Elliptic Partial Differential Operators on a Torus

 §3. Function Space of Sections of a Vector Bundle

 §4. Elliptic Linear Partial Differential Operators

 §5. The Existence of Weak Solutions of a Strongly Elliptic Partial Differential Equation

 §6. Regularity of Weak Solutions of Elliptic Linear Partial Differential Equations

 §7. Elliptic Operators in the Hilbert Space L2(X, B)

 §8. C∞ Difterentiability of φ(t)

Bibliography

Index