Equivariant principal bundles and logarithmic connections on toric varieties

TL;DR

This paper establishes a precise correspondence between T-equivariant structures on principal G-bundles over toric varieties and the existence of logarithmic connections with singularities over the boundary divisor, providing canonical connections in the equivariant case.

Contribution

It proves that T-equivariance of principal G-bundles on toric varieties is equivalent to the existence of a logarithmic connection singular over the boundary divisor, and shows that equivariant bundles have canonical integrable logarithmic connections.

Findings

Equivariant principal G-bundles admit logarithmic connections singular over the boundary divisor.

Existence of a T-equivariant structure is equivalent to having a logarithmic connection.

Equivariant principal H-bundles have canonical integrable logarithmic connections.

Abstract

Let $M$ be a smooth complex projective toric variety equipped with an action of a torus $T$, such that the complement $D$ of the open $T$--orbit in $M$ is a simple normal crossing divisor. Let $G$ be a complex reductive affine algebraic group. We prove that an algebraic principal $G$--bundle $E_G\to M$ admits a $T$--equivariant structure if and only if $E_G$ admits a logarithmic connection singular over $D$. If $E_H\to M$ is a $T$-equivariant algebraic principal $H$--bundle, where $H$ is any complex affine algebraic group, then $E_H$ in fact has a canonical integrable logarithmic connection singular over $D$.