J-全純曲線理論自其由Gromov于1985年引入以來，已經(jīng)變得非常重要。在數(shù)學中，它的應用包括許多辛拓撲中的關(guān)鍵結(jié)果。它也是創(chuàng)立Floer同調(diào)的主要靈感之一。在數(shù)學物理中，它提供了一個自然的語境用以在其中定義鏡像對稱猜想的兩個重要成分——Gromov-Witten不變量和量子上同調(diào)。
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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