Kuranishi空间的叶状结构与紧复流形变形族的同构
Foliated Structure of The Kuranishi Space and Isomorphisms of Deformation Families of Compact Complex Manifolds
摘要：
考虑下面的均匀化问题。取紧致复流形的两个全纯（对于某些$p>0$，由定义在$\Bbb C^p$中的$0$的一个邻域上的某个解析集参数化）或可微（对于某些$p>0$，由$\Bbb R^p$中的$0$的一个开邻域参数化）变形族.假设它们是逐点同构的，即对于参数空间的每个点$t$，第一类$t$上的纤维与第二类$t$上的纤维是双全纯的。那么，在什么条件下这两个族在0点局部同构呢？本文在全纯族的情形下给出了一个充分条件。然后，我们表明，令人惊讶的是，这个条件是不充分的情况下，可微的家庭。我们还描述了不同类型的反例，并给出了反例的分类要素。这些结果依赖于紧复流形的Kuranishi空间的几何研究。
英文摘要：
Consider the following uniformization problem. Take two holomorphic (parametrized by some analytic set defined on a neighborhood of $0$ in $\Bbb C^p$, for some $p>0$) or differentiable (parametrized by an open neighborhood of $0$ in $\Bbb R^p$, for some $p>0$) deformation families of compact complex manifolds. Assume they are pointwise isomorphic, that is for each point $t$ of the parameter space, the fiber over $t$ of the first family is biholomorphic to the fiber over $t$ of the second family. Then, under which conditions are the two families locally isomorphic at 0? In this article, we give a sufficient condition in the case of holomorphic families. We show then that, surprisingly, this condition is not sufficient in the case of differentiable families. We also describe different types of counterexamples and give some elements of classification of the counterexamples. These results rely on a geometric study of the Kuranishi space of a compact complex manifold.
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