Gromov于1985年首次引进了J-全纯曲线，这对辛几何的研究是革命性的。通过量子上同调，数学物理中许多令人兴奋的新思想都与这些曲线有着某种关联。本书对J-全纯曲线理论进行了条理分明且全面充分的阐述，这个理论的各个细节目前分散在备类研究文章中。本书的前半部是关于该领域的一个说明性的陈述，解释了主要的技术方面。McDuff和Salamon给出了关于球面的Gromov紧性猜想和Gromov-Witten不变量存在性的完整证明。书的后半部集中于量子上同调的定义。作者证实了这里存在乘法，并给出了对于Ruan-Tian结果在适当流形上是结合的一个新证明；然后他们描述了对旗流形的量子上同调的Givental-Kim计算，这个上同调引向了量子陈类和对于Grassmann流形的Witten计算，而这些叉与Verlinde代数相关。讨论也涉及了Dubrovin联络，关于量子上同调的Gromov-Witten势以及曲线计数公式。本书最后概述了和Floer理论的联系。

1 Introduction
1.1 Symplectic manifolds
1.2 J-holomorphic curves
1.3 Moduli spaces
1.4 Compactness
1.5 Evaluation maps
1.6 The Gromov-Witten invariants
1.7 Quantum cohomology
1.8 Novikov rings and Floer homology
2 Local Behaviour
2.1 The generalised Cauchy-Riemann equation
2.2 Critical points
2.3 Somewhere injective curves
3 Moduli Spaces and Transversality
3.1 The main theorems
3.2 Elliptic regularity
3.3 Implicit function theorem
3.4 Transversality
3.5 A regularity criterion
4 Compactness
4.1 Energy
4.2 Removal of Singularities
4.3 Bubbling
4.4 Gromov compactness
4.5 Proof of Gromov compactness
5 Compactiflcation of Moduli Spaces
5.1 Semi-positivity
5.2 The image of the evaluation map
5.3 The image of the p-fold evaluation map
5.4 The evaluation map for marked curves
6 Evaluation Maps and Transversality
6.1 Evaluation maps are submersions
6.2 Moduli spaces of N-tuples of curves
6.3 Moduli spaces of cusp-curves
6.4 Evaluation maps for cusp-curves
7 Gromov-Witten Invariants
7.1 Pseudo-cycles
7.2 The invariant
7.3 Examples
7.4 The invariant
8 Quantum Cohomology
8.1 Witten's deformed cohomology ring
8.2 Associativity and composition rules
8.3 Flag manifolds
8.4 Grassmannians
8.5 The Gromov-Witten potential
9 Novikov PAngs and Calabi-Yau Manifolds
9.1 Multiply-covered curves
9.2 Novikov rings
9.3 Calabi-Yau manifolds
10 Floer Homology
10.1 Floer's cochain complex
10.2 Ring structure
10.3 A comparison theorem
10.4 Donaldson's quantum category
10.5 Closing remark
A Gluing
A.1 Cutoff functions
A.2 Connected sums of J-holomorphic curves
A.3 Weighted norms
A.4 An estimate for the inverse
A.5 Gluing
B Elliptic Regularity
B.1 Sobolev spaces
B.2 The Calderon-Zygmund inequality
B.3 Cauchy-Riemann operators
B.4 Elliptic bootstrapping
Bibliography
Indexes
Index of Notations