Stein structures and holomorphic mappings
Stein structures and holomorphic mappings
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斯坦因结构和全纯映射
DOI：
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发表时间：
2005
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通讯作者：
Marko Slapar
中科院分区：
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作者：
F. Forstnerič;Marko Slapar
证明了从Stein流形X到复流形Y的每个连续映射都可以通过X上的Stein结构和映射的同伦变形而成为全纯的，并且在没有拓扑障碍的情况下，全纯映射可以被选择为具有逐点最大秩.类似的结果对任何紧致Hausdorff映射族都成立，但对于非紧致映射族一般不成立。我们的主要结果实际上证明了光滑的几乎复源流形(X，J)具有正确的手柄结构。这篇论文包含了Eliashberg(Int J Math 1：29-46,1990)关于Stein流形的同伦刻画的另一个证明，以及对Gompf(Ann Math 148(2)：619-693,1998；J辛几何3：565-587,2005)关于奇异Stein曲面的构造的略微不同的解释。
We prove that every continuous map from a Stein manifold X to a complex manifold Y can be made holomorphic by a homotopic deformation of both the map and the Stein structure on X. In the absence of topological obstructions, the holomorphic map may be chosen to have pointwise maximal rank. The analogous result holds for any compact Hausdorff family of maps, but it fails in general for a noncompact family. Our main results are actually proved for smooth almost complex source manifolds (X,J) with the correct handlebody structure. The paper contains another proof of Eliashberg’s (Int J Math 1:29–46, 1990) homotopy characterization of Stein manifolds and a slightly different explanation of the construction of exotic Stein surfaces due to Gompf (Ann Math 148(2): 619–693, 1998; J Symplectic Geom 3:565–587, 2005).