全纯Segre保持映射的参数化
Parametrization of holomorphic Segre preserving maps
摘要：
在本文中，我们探讨全纯Segre保持映射。首先，我们研究了将有限型的一般实解析子流形$M \subseteq \C^N$在某点$p$的复化$\mathcal{M}$转换为一般实解析子流形$M' \subseteq \C^{N'}$在某点$p'$非退化的复化$\mathcal{M}'$的全纯Segre保持映射。证明了对于固定的$M$和$M'$，将$(\M,(p,\bar{p}))$发送到$(\M',(p', \bar{p}'))$的Segre浸没全纯Segre保持映射的$(p,\bar{p})$上的芽可以用它们在$(p,\bar{p})$上的$r$ -喷流来参数化，对于某些固定的$r$只依赖于$M$和$M'$ . (If证明了$M$和$M'$都是实代数的，则任何这样的映射都是全纯代数的。从这个参数化可以得出，一个实解析子流形的复化$\mathcal{M}$的全纯Segre保持自同构$\mathcal{H}$的芽的集合是一个代数复李群，它在某个点$p$是非退化的和有限型的，并且使得$\mathcal{H}$固定$(p,\bar{p})$。然后，我们探讨这个自同构群和$p$的$M$的自同构群之间的关系。
英文摘要：
In this paper, we explore holomorphic Segre preserving maps. First, we investigate holomorphic Segre preserving maps sending the complexification $\mathcal{M}$ of a generic real analytic submanifold $M \subseteq \C^N$ of finite type at some point $p$ into the complexification $\mathcal{M}'$ of a generic real analytic submanifold $M' \subseteq \C^{N'}$, finitely nondegenerate at some point $p'$. We prove that for a fixed $M$ and $M'$, the germs at $(p,\bar{p})$ of Segre submersive holomorphic Segre preserving maps sending $(\M,(p,\bar{p}))$ into $(\M',(p', \bar{p}'))$ can be parametrized by their $r$-jets at $(p,\bar{p})$, for some fixed $r$ depending only on $M$ and $M'$. (If, in addition, $M$ and $M'$ are both real algebraic, then we prove that any such map must be holomorphic algebraic.) From this parametrization, it follows that the set of germs of holomorphic Segre preserving automorphisms $\mathcal{H}$ of the complexification $\mathcal{M}$ of a real analytic submanifold finitely nondegenerate and of finite type at some point $p$, and such that $\mathcal{H}$ fixes $(p,\bar{p})$, is an algebraic complex Lie group. We then explore the relationship between this automorphism group and the group of automorphisms of $M$ at $p$.
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