\'Etale triviality of finite equivariant vector bundles

TL;DR

This paper proves that under certain conditions, H-equivariant holomorphic vector bundles over a complex space become trivial after a finite étale cover, extending the understanding of their structure and triviality conditions.

Contribution

It establishes a criterion for the triviality of H-equivariant vector bundles via finite étale covers in complex analytic spaces with specific invariance properties.

Findings

H-equivariant bundles are finite if and only if they become trivial after a finite étale cover.

The result applies to complex spaces with invariant functions constant on the reduced space.

Provides a characterization of equivariant bundle triviality in terms of étale covers.

Abstract

Let H be a complex Lie group acting holomorphically on a complex analytic space X such that the restriction to X_{\mathrm{red}} of every H-invariant regular function on X is constant. We prove that an H-equivariant holomorphic vector bundle E over X is $H$-finite, meaning f_1(E)= f_2(E) as H-equivariant bundles for two distinct polynomials f_1 and f_2 whose coefficients are nonnegative integers, if and only if the pullback of E along some H-equivariant finite \'etale covering of X is trivial as an H-equivariant bundle.