J-Holomorphic Curves and Symplectic Topology
J-Holomorphic Curves and Symplectic Topology
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发表时间：
2004
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通讯作者：
D. Mcduff;D. Salamon
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作者：
D. Mcduff;D. Salamon
自1985年Gromov提出J$-全纯曲线理论以来，J$-全纯曲线的研究一直具有重要意义.在数学中，它的应用包括辛拓扑学中的许多关键结果。这也是Floer同源性理论的主要灵感来源之一。在数学物理学中，它提供了一个自然的背景来定义格罗莫夫-威滕不变量和量子上同调，镜像对称猜想的两个重要组成部分。这本书的主要目标是建立在充分和严谨的细节主题的基本定理。特别是，这本书包含完整的证明格罗莫夫的紧致性定理领域的胶合定理领域，并associatively量子乘法在半正的情况下。这本书也可以作为一个介绍目前的工作辛拓扑：有两个很长的章节的应用，一个集中在经典成果辛拓扑和其他有关量子上同调。最后一章概述了弗洛尔理论的一些最新发展。这本书的五个附录提供了必要的背景有关的经典理论的线性椭圆算子，Fredholm理论，Sobolev空间，以及讨论的模空间的属零稳定曲线和证明的积极性交叉的$J$-全纯曲线在四维流形。第二版澄清了各种论点，纠正了第一版中的几个错误，包括第10章和附录C和D中的一些额外结果，并更新了对最近发展的参考。
The theory of $J$-holomorphic curves has been of great importance since its introduction by Gromov in 1985. In mathematics, its applications include many key results in symplectic topology. It was also one of the main inspirations for the creation of Floer homology. In mathematical physics, it provides a natural context in which to define Gromov-Witten invariants and quantum cohomology, two important ingredients of the mirror symmetry conjecture. The main goal of this book is to establish the fundamental theorems of the subject in full and rigourous detail. In particular, the book contains complete proofs of Gromov's compactness theorem for spheres, of the gluing theorem for spheres, and of the associatively of quantum multiplication in the semipositive case. The book can also serve as an introduction to current work in symplectic topology: there are two long chapters on applications, one concentrating on classical results in symplectic topology and the other concerned with quantum cohomology. The last chapter sketches some recent developments in Floer theory. The five appendices of the book provide necessary background related to the classical theory of linear elliptic operators, Fredholm theory, Sobolev spaces, as well as a discussion of the moduli space of genus zero stable curves and a proof of the positivity of intersections of $J$-holomorphic curves in four-dimensional manifolds. The second edition clarifies various arguments, corrects several mistakes in the first edition, includes some additional results in Chapter 10 and Appendices C and D, and updates the references to recent developments.