　　J-全纯曲线理论自其由Gromov于1985年引入以来，已经变得非常重要。在数学中，它的应用包括许多辛拓扑中的关键结果。它也是创立Floer同调的主要灵感之一。在数学物理中，它提供了一个自然的语境用以在其中定义镜像对称猜想的两个重要成分-Gromov-Witten不变量和量子上同调。《美国数学会经典影印系列：J-全纯曲线和辛拓扑（第2版影印版）》的主要目的是以充分和严格的细节来建立这个主题的基本定理。特别地，《美国数学会经典影印系列：J-全纯曲线和辛拓扑（第2版影印版）》包含关于球面的Gromov紧性定理、球面的黏合定理以及在半正情形下量子乘法的结合性的完整的证明。《美国数学会经典影印系列：J-全纯曲线和辛拓扑（第2版影印版）》也可以作为对辛拓扑当前工作的介绍：有两个关于应用的长的章节，一章专注于辛拓扑的经典结果，另一章涉及量子上同调。最后一章概述了Floer理论的一些新进展。《美国数学会经典影印系列：J-全纯曲线和辛拓扑（第2版影印版）》的五个附录提供了与线性椭圆算子的经典理论、Fredholm理论和Sobolev空间相关的必需的背景知识，以及关于零亏格稳定曲线模空间的讨论和四维流形中J·全纯曲线的交点的正性的证明。第二版澄清了各种争议，纠正了第1版中的几个错误，并包含了一些在第10章和附录C与D中的增加的结果，更新了对于新进展的参考文献。

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Preface to the second edition
Preface

Chapter 1． Introduction
1．1． Symplectic manifolds
1．2． Moduli spaces： regularity and compactness
1．3． Evaluation maps and pseudocycles
1．4． The Gromov-Witten invariants
1．5． Applications and further developments

Chapter 2． J-holomorpluc Curves
2．1． Almost complex structures
2．2． The nonlinear Cauchy-Riemann equations
2．3． Unique continuation
2．4． Criticalpoints
2．5． Somewhere injective curves
2．6． The adjunction inequality

Chapter 3． Moduli Spaces and Transversality
3．1． Moduli spaces of simple curves
3．2． Transversality
3．3． A regularity criterion
3．4． Curves with pointwise constraints
3．5． Implicit function theorem

Chapter 4． Compactness
4．1． Energy
4．2． The bubbling phenomenon
4．3． The mean value inequality
4．4． The isoperimetric inequality
4．5． Removal of singularities
4．6． Convergence modulo bubbling
4．7． Bubbles connect

Chapter 5． Stable Maps
5．1． Stable maps
5．2． Gromov convergence
5．3． Gromov compactness
5．4． Uniqueness of the limit
5．5． Gromov compactness for stable maps
5．6． The Gromov topology

Chapter 6． Moduli Spaces of Stable Maps
6．1． Simple stable maps
6．2． Transversality for simple stable maps
6．3． Transversality for evaluation maps
6．4． Semipositivity
6．5． Pseudocycles
6．6． Gromov-Witten pseudocycles
6．7． The pseudocycle of graphs

Chapter 7． Gromov-Witten Invariants
7．1． Counting pseudoholomorphic spheres
7．2． Variations on the definition
7．3． Counting pseudoholomorphic graphs
7．4． Rational curves in projective spaces
7．5． Axioms for Gromov-Witten invariants

Chapter 8． Hamiltonian Perturbations
8．1． Trivial bundles
8．2． Locally Hamiltonian fibrations
8．3． Pseudoholomorphic sections
8．4． Pseudoholomorphic spheres in the fiber
8．5． The pseudocycle of sections
8．6． Counting pseudoholomorphic sections

Chapter 9． Applications in Symplectic Topology
9．1． Periodic orbits of Hamiltonian systems
9．2． Obstructions to Lagrangian embeddings
9．3． The nonsqueezing theorem
9．4． Symplectic 4-manifolds
9．5． The group of symplectomorphisms
9．6． Hofer geometry
9．7． Distinguishing symplectic structures

Chapter 10， Gluing
10．1． The gluing theorem
10．2． Connected sums of J-holomorphic curves
10．3． Weighted norms
10．4． Cutoff functions
10．5． Construction of the gluing map
10．6． The derivative of the gluing map
10．7． Surjectivity of the gluing map
10．8． Proof of the splitting axiom
10．9． The gluing theorem revisited

Chapter 11， Quantum Cohomology
11．1． The small quantum cohomology ring
11．2． The Gromov-Witten potential
11．3． Four examples
……

Chapter 12． Floer Homology
Appendix A． Fredholm Theory
Appendix B． Elliptic Regularity
Appendix C． The Riemann-Roch Theorem
Appendix D． Stable Curves of Genus Zero
Appendix E． Singularities and Intersections （written with Laurent Lazzarini）
Bibliography
List of Symbols
Index
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