On equivariant Serre problem for principal bundles

TL;DR

This paper proves that under certain conditions, equivariant principal bundles over complex affine varieties are trivial or classify as products, especially in the context of toric varieties, extending the Serre problem.

Contribution

It establishes conditions under which equivariant principal bundles are trivial or can be classified, generalizing the Serre problem to equivariant settings over complex varieties.

Findings

Equivariant principal bundles over contractible varieties with dense orbits are trivial if the acting group is reductive.

Torus-equivariant principal G-bundles over affine toric varieties are always trivial.

Provides a classification of torus-equivariant principal G-bundles over complex toric varieties.

Abstract

Let $E_G$ be a $\Gamma$--equivariant algebraic principal $G$--bundle over a normal complex affine variety $X$ equipped with an action of $\Gamma$, where $G$ and $\Gamma$ are complex linear algebraic groups. Suppose $X$ is contractible as a topological $\Gamma$--space with a dense orbit, and $x_0 \in X$ is a $\Gamma$--fixed point. We show that if $\Gamma$ is reductive, then $E_G$ admits a $\Gamma$--equivariant isomorphism with the product principal $G$--bundle $X \times_{\rho} E_G(x_0)$, where $\rho\,:\, \Gamma \, \longrightarrow\, G$ is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal $G$-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal $G$-bundles over any complex toric variety.