Holomorphic linearization of commuting germs of holomorphic maps

TL;DR

This paper establishes conditions under which commuting germs of holomorphic maps can be simultaneously linearized, providing new criteria for formal and holomorphic linearization in several complex variables.

Contribution

It introduces a Brjuno-type condition for simultaneous linearization and proves holomorphic linearization results for commuting germs with diagonalizable linear parts.

Findings

Commuting germs with almost simultaneously Jordanizable linear parts are formally linearizable.

A Brjuno-type condition ensures holomorphic simultaneous linearization for germs with diagonalizable linear parts.

Answers a multi-dimensional version of Moser's problem on linearization of holomorphic germs.

Abstract

Let $f_1, ..., f_h$ be $h\ge 2$ germs of biholomorphisms of $\C^n$ fixing the origin. We investigate the shape a (formal) simultaneous linearization of the given germs can have, and we prove that if $f_1, ..., f_h$ commute and their linear parts are almost simultaneously Jordanizable then they are simultaneously formally linearizable. We next introduce a simultaneous Brjuno-type condition and prove that, in case the linear terms of the germs are diagonalizable, if the germs commutes and our Brjuno-type condition holds, then they are holomorphically simultaneously linerizable. This answers to a multi-dimensional version of a problem raised by Moser.