On holomorphic $\mathbb{C}^*$-actions

TL;DR

This paper investigates holomorphic actions of the complex multiplicative group on complex manifolds near fixed points, establishing linearization results under certain conditions, advancing the understanding of such group actions in complex geometry.

Contribution

It provides new linearization theorems for holomorphic ^*-actions around fixed points, extending previous results in the field.

Findings

Linearization of the germ of the action achieved under specific conditions.

Extension of linearization results from germs to entire actions.

Builds upon and generalizes prior work by M. Suzuki and others.

Abstract

In this paper we study holomorphic actions of the complex multiplicative group on complex manifolds around a singular (fixed) point. We prove linearization results for the germ of action and also for the whole action under some conditions on the manifold. This can be seen as a follow-up to previous works of M. Suzuki and other authors.