Absolute neighbourhood retracts and spaces of holomorphic maps from Stein manifolds to Oka manifolds
Absolute neighbourhood retracts and spaces of holomorphic maps from Stein manifolds to Oka manifolds
复制标题
绝对邻域收缩和从 Stein 流形到 Oka 流形的全纯映射空间
DOI：
10.1090/s0002-9939-2014-12335-5
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
F. Lárusson
中科院分区：
文献类型：
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作者：
F. Lárusson
由于Gromov， Oka理论的基本结果表明，从Stein流形$S$到椭圆流形$X$的每一个连续映射$f$都可以变形为全纯映射。我们很自然地要问，是否可以一次对所有的$f$做到这一点，以一种连续依赖于$f$的方式，并且如果$f$一开始就是全纯的，那么它就会保持不变。换句话说，$\scrO(S，X)$是$\scrC(S，X)$的变形缩回吗？证明了$S$具有具有有限多个临界点的严格多次谐波莫尔斯耗竭；特别地，如果$S$是仿射代数。在证明中使用的X$的唯一性质是关于有限多面体的参数Oka性质的近似，因此我们的定理在X$是Oka流形的较弱假设下成立。除了Oka理论本身，我们的主要工具是绝对邻域收缩理论。我们还在拓扑空间的范畴上利用了混合模型结构。
The basic result of Oka theory, due to Gromov, states that every continuous map $f$ from a Stein manifold $S$ to an elliptic manifold $X$ can be deformed to a holomorphic map. It is natural to ask whether this can be done for all $f$ at once, in a way that depends continuously on $f$ and leaves $f$ fixed if it is holomorphic to begin with. In other words, is $\scrO(S,X)$ a deformation retract of $\scrC(S,X)$? We prove that it is if $S$ has a strictly plurisubharmonic Morse exhaustion with finitely many critical points; in particular, if $S$ is affine algebraic. The only property of $X$ used in the proof is the parametric Oka property with approximation with respect to finite polyhedra, so our theorem holds under the weaker assumption that $X$ is an Oka manifold. Our main tool, apart from Oka theory itself, is the theory of absolute neighbourhood retracts. We also make use of the mixed model structure on the category of topological spaces.
CW 同伦型的函数空间是希尔伯特流形
DOI：
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发表时间：
2009
期刊：
Proceedings of the American Mathematical Society 137
影响因子：
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作者：
Shirahada;K.;Niwa;K.;白肌 邦生・丹羽 清;Kunio Shirahada and Kiyoshi Niwa;Noriyuki Abe;Noriyuki Abe;Noriyuki Abe;Atsushi YAMASHITA
通讯作者：
Atsushi YAMASHITA