Kai Cieliebak and Yakov Eliashberg: “From Stein to Weinstein and Back. Symplectic Geometry of Affine Complex Manifolds”
Kai Cieliebak and Yakov Eliashberg: “From Stein to Weinstein and Back. Symplectic Geometry of Affine Complex Manifolds”
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Kai Cieliebak 和 Yakov Eliashberg：“从斯坦因到温斯坦再到仿射复流形的辛几何”。
DOI：
10.1365/s13291-013-0070-6
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
F. Schlenk
中科院分区：
文献类型：
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作者：
F. Schlenk
复几何是一个古老而丰富的领域，它起源于复函数的研究(在19世纪，复函数被认为比实函数更自然，例如，因为复多项式总是分解成线性因子)。Stein流形是仿射复流形。经典地，这种流形被研究到双全纯。这本书不是关于Stein流形的这种精细结构，而是关于它们的拓扑结构。这种方法是通过辛几何来实现的。辛几何是一个年轻得多的领域，它起源于经典力学和哈密顿系统，在这些系统中，相空间具有规范的辛结构。自从Gromov引入J-全纯方法和Floer创建了他的同调以来，辛几何与许多其他领域(复分析、代数几何、数学物理、低维拓扑等)联系在一起。在两个方向上进行交叉受精。这本书证明了辛几何也是支配Stein流形拓扑的结构。它是关于Stein流形的辛几何及其对Stein流形的复杂几何的启示。在续集中，我首先解释主要概念(斯坦流形和温斯坦流形)，然后描述在这本书中证明的主要结果，最后谈一谈论述的风格和这本书是如何产生的。
Complex geometry is an old and rich field with its origin in the study of complex functions (which in the nineteenth century were considered as much more natural than real functions, for instance because complex polynomials always decompose into linear factors). Stein manifolds are affine complex manifolds. Classically, such manifolds are studied up to biholomorphism. This book is not concerned with this fine structure of Stein manifolds, but with their topological structure. The approach is through symplectic geometry. Symplectic geometry is a much younger field, with its roots in classical mechanics and Hamiltonian systems, where the phase spaces carry a canonical symplectic structure. Since Gromov’s introduction of J -holomorphic methods and Floer’s creation of his homology, symplectic geometry is linked to many other fields (complex analysis, algebraic geometry, mathematical physics, low-dimensional topology, etc.) with cross-fertilization in both directions. This book demonstrates that symplectic geometry is also the structure that governs the topology of Stein manifolds. It is about the symplectic geometry of Stein manifolds and its implications for the complex geometry of Stein manifolds. In the sequel, I first explain the main notions (Stein and Weinstein manifolds), then describe the main results proven in this book, and finally say something about the style of exposition and how this book came to life.