Simultaneous linearization of holomorphic germs in presence of resonances

TL;DR

This paper establishes conditions under which multiple commuting holomorphic germs in complex space can be simultaneously linearized, linking the problem to the existence of an invariant complex manifold.

Contribution

It provides new criteria for simultaneous linearization of commuting germs based on eigenvalue arithmetic conditions and resonance restrictions.

Findings

Simultaneous linearization is equivalent to the existence of an invariant complex manifold.

Arithmetic conditions on eigenvalues are crucial for linearization.

Resonance restrictions influence the linearizability of germs.

Abstract

Let $f_1, ..., f_m$ be $m\ge 2$ germs of biholomorphisms of $\C^n$, fixing the origin, with $(\d f_1)_O$ diagonalizable and such that $f_1$ commutes with $f_h$ for any $h=2,..., m$. We prove that, under certain arithmetic conditions on the eigenvalues of $(\d f_1)_O$ and some restrictions on their resonances, $f_1, ..., f_m$ are simultaneously holomorphically linearizable if and only if there exists a particular complex manifold invariant under $f_1,..., f_m$.