Logarithmic connections on principal bundles over normal varieties

TL;DR

This paper introduces and studies the concept of logarithmic connections on principal bundles over normal varieties, establishing conditions for their existence and relating them to covariant derivatives, especially over toric varieties.

Contribution

It defines logarithmic connections on principal bundles over normal varieties and characterizes their existence in relation to associated bundles and toric structures.

Findings

Logarithmic connections exist if and only if associated covariant derivatives exist when the logarithmic tangent sheaf is locally free.

A principal G-bundle admits a logarithmic connection iff its adjoint bundle admits one for semisimple G.

Existence of logarithmic connections on toric varieties is equivalent to the bundle being torus equivariant.

Abstract

Let $X$ be a normal projective variety over an algebraically closed field of characteristic zero. Let $D$ be a reduced Weil divisor on $X$. Let $G$ be a reductive linear algebraic group. We introduce the notion of a logarithmic connection on a principal $G$-bundle over $X$, which is singular along $D$. The existence of a logarithmic connection on the frame bundle associated with a vector bundle over $X$ is shown to be equivalent to the existence of a logarithmic covariant derivative on the vector bundle if the logarithmic tangent sheaf of $X$ is locally free. Additionally, when the algebraic group $G$ is semisimple, we show that a principal $G$-bundle admits a logarithmic connection if and only if the associated adjoint bundle admits one. We also prove that the existence of a logarithmic connection on a principal bundle over a toric variety, singular along the boundary divisor, is equivalent to the existence of a torus equivariant structure on the bundle.