Complex analytic geometry and analytic-geometric categories
Complex analytic geometry and analytic-geometric categories
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复杂解析几何和解析几何类别
DOI：
10.1515/crelle.2009.002
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
S. Starchenko
中科院分区：
文献类型：
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作者：
Y. Peterzil;S. Starchenko
解析几何范畴的概念是由v. d. Dries和Miller在《几何范畴和o-极小结构》中提出的。J. 84(1996)，第2号。2, 497 - 540。]。它是实解析流形子集的一个范畴，扩展了子解析集的范畴。本文讨论了次解析范畴或更一般的解析几何范畴与复解析几何之间的联系。问题的性质如下：我们从复解析流形M的子集a开始，并假设a是解析几何范畴的对象（通过将M看作一个实数的二维解析流形）。然后，我们给出了A、它的闭包或它在全纯映射下的像是复解析集的条件。在论文的第二部分，我们考虑了复流形的概念，它推广了紧复流形的概念。在这种情况下，我们讨论了复流形族及其高阶全纯切束内参数的一致性。然后，我们证明了-流形的解析子集在射影空间中的一致嵌入的一个结果，该结果推广了Campana (F。Campana, g - 3 - 3 - 3, compacit<e:1> dans l' spaces des cycles d'un空间分析复合体，数学。Ann. 251 (1980), no。1,通气量。])和Fujiki ([Akira Fujiki，论范畴中的紧复空间的Douady空间？名古屋数学。J. 85(1982), 189-211。])在紧复流形上。
Abstract The notion of an analytic-geometric category was introduced by v. d. Dries and Miller in [Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540.]. It is a category of subsets of real analytic manifolds which extends the category of subanalytic sets. This paper discusses connections between the subanalytic category, or more generally analytic-geometric categories, and complex analytic geometry. The questions are of the following nature: We start with a subset A of a complex analytic manifold M and assume that A is an object of an analytic-geometric category (by viewing M as a real analytic manifold of double dimension). We then formulate conditions under which A, its closure or its image under a holomorphic map is a complex analytic set. In the second part of the paper we consider the notion of a complex -manifold, which generalizes that of a compact complex manifold. We discuss uniformity in parameters, in this context, within families of complex manifolds and their high-order holomorphic tangent bundles. We then prove a result on uniform embeddings of analytic subsets of -manifolds into a projective space, which extends theorems of Campana ([F. Campana, Algébricité et compacité dans l'espace des cycles d'un espace analytique complexe, Math. Ann. 251 (1980), no. 1, 7–18.]) and Fujiki ([Akira Fujiki, On the Douady space of a compact complex space in the category ?, Nagoya Math. J. 85 (1982), 189–211.]) on compact complex manifolds.