Torus actions in the normalization problem

TL;DR

This paper investigates how torus actions influence the normalization and linearization of biholomorphisms in complex space, providing a detailed, computable framework for understanding the structure and torsion phenomena involved.

Contribution

It offers a complete, computable characterization of the conditions under which torus actions enable holomorphic normalization and linearization of germs, linking eigenvalues to the action's weight matrix.

Findings

Characterization of torus actions that allow Poincaré-Dulac normalization

Link between eigenvalues and the weight matrix of the torus action

Analysis of torsion phenomena affecting normalization

Abstract

Let $f$ be a germ of biholomorphism of $\C^n$, fixing the origin. We show that if the germ commutes with a torus action, then we get information on the germs that can be conjugated to $f$, and furthermore on the existence of a holomorphic linearization or of a holomorphic normalization of $f$. We find out in a complete and computable manner what kind of structure a torus action must have in order to get a Poincar\'e-Dulac holomorphic normalization, studying the possible torsion phenomena. In particular, we link the eigenvalues of $df_O$ to the weight matrix of the action. The link and the structure we found are more complicated than what one would expect; a detailed study was needed to completely understand the relations between torus actions, holomorphic Poincar\'e-Dulac normalizations, and torsion phenomena. We end the article giving an example of techniques that can be used to construct torus actions.