在Eliashberg之后构造Stein流形
Constructing Stein manifolds after Eliashberg
摘要：
基于已发表的和待发表的文献，给出了所有维的Stein流形的存在性理论的统一总结。Eliashberg对承认Stein结构的流形的刻画需要一个在复维2中的额外的微妙假设，这可以通过传递到拓扑设置和调用弗里德曼理论来消除。情况是非常相似的，如果有人问，哪些开子集的一个固定的复流形可以斯坦由一个合痕。作为这些定理的一个应用，我们可以构造无数个奇异R^4的自同构类型，这些自同构类型被实现为C^2的Stein开子集（即全纯域）。更一般地说，C^2中的每个全纯全纯域都与其他这样的域拓扑同位素，从而实现了无数个全纯类型。复n-流形中任何驯服的n-复形都可以被合线化为Stein开子集的嵌套交，只要当n=2时合线是拓扑的。在后一种情况下，斯坦邻域是同胚的，但经常实现不可数的多个同胚类型。证明了在光滑的n-流形和Stein曲面上，每一个耗尽Morse函数都可以被细分，得到一个具有相同极大指数的局部有限凸体.
英文摘要：
A unified summary is given of the existence theory of Stein manifolds in all dimensions, based on published and pending literature. Eliashberg's characterization of manifolds admitting Stein structures requires an extra delicate hypothesis in complex dimension 2, which can be eliminated by passing to the topological setting and invoking Freedman theory. The situation is quite similar if one asks which open subsets of a fixed complex manifold can be made Stein by an isotopy. As an application of these theorems, one can construct uncountably many diffeomorphism types of exotic R^4's realized as Stein open subsets of C^2 (i.e. domains of holomorphy). More generally, every domain of holomorphy in C^2 is topologically isotopic to other such domains realizing uncountably many diffeomorphism types. Any tame n-complex in a complex n-manifold can be isotoped to become a nested intersection of Stein open subsets, provided the isotopy is topological when n=2. In the latter case, the Stein neighborhoods are homeomorphic, but frequently realize uncountably many diffeomorphism types. It is also proved that every exhausting Morse function can be subdivided to yield a locally finite handlebody of the same maximal index, both in the context of smooth n-manifolds and for Stein surfaces.
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