Parametrization of holomorphic Segre preserving maps

TL;DR

This paper studies holomorphic Segre preserving maps between complexifications of real analytic submanifolds, showing they can be parametrized by jets and that their automorphism groups form algebraic Lie groups, with implications for real algebraic cases.

Contribution

It establishes a jet parametrization for holomorphic Segre preserving maps and proves that automorphism groups are algebraic complex Lie groups, extending understanding of symmetries of real submanifolds.

Findings

Germs of Segre preserving maps are parametrized by finite jets.

Automorphism groups form algebraic complex Lie groups.

In real algebraic cases, maps are holomorphic algebraic.

Abstract

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 $p$.