Torus actions in the normalization problem

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.