Equivariant vector bundles and logarithmic connections on toric varieties

TL;DR

This paper establishes a correspondence between equivariant vector bundles and logarithmic connections on smooth complete complex toric varieties, providing tools for computing Chern classes and extending results to log parallelizable G-pairs.

Contribution

It proves that equivariant vector bundles on toric varieties admit tautological logarithmic connections and extends this to G-pairs, advancing understanding of their geometric structures.

Findings

Equivariant vector bundles admit tautological integrable logarithmic connections.

The result aids in computing Chern classes of equivariant bundles.

Extension of the theory to log parallelizable G-pairs.

Abstract

Let X be a smooth complete complex toric variety such that the boundary is a simple normal crossing divisor, and let E be a holomorphic vector bundle on X. We prove that E admits an equivariant structure if and only if E admits a logarithmic connection singular over D. More precisely, we show that an equivariant vector bundle on X has a tautological integrable logarithmic connection singular over D. This is used in computing the Chern classes of the equivariant vector bundles on X. We also prove a version of the result for holomorphic vector bundles on log parallelizable G-pairs (X,D), where G is a simply connected complex affine algebraic group.