Family of intersecting totally real manifolds of $(C^n ,0)$ and germs of holomorphic diffeomorphisms

TL;DR

This paper studies invariant complex analytic sets under abelian groups of holomorphic diffeomorphisms and explores conditions for holomorphic linearization and straightening of intersecting totally real submanifolds in complex space.

Contribution

It characterizes invariant analytic sets and provides conditions for their holomorphic linearization and for straightening families of totally real submanifolds.

Findings

Existence and characterization of invariant complex analytic sets under abelian groups.

Conditions for holomorphic linearization near fixed points.

Methods to straighten or construct analytic sets intersecting real submanifolds.

Abstract

We prove the existence (and give a characterization) of a germ of complex analytic set left invariant by an abelian group of germs of holomorphic diffeomorphisms at a common fixed point.We also give condition that ensure that such a group can be linearized holomorphically near the fixed point. It rests on a "small divisors condition" of the family of linear parts. The second part of this article is devoted to the study families of totally real intersecting n-submanifolds of (C n , 0). We give some conditions which allow to straighten holomorphically the family. If this is not possible to do it formally, we construct a germ of complex analytic set at the origin which interesection with the family can be holomorphically straightened. The second part is an application of the first.