Equivariant vector bundles and logarithmic connections on toric varieties

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 the following three statements are equivalent:

The holomorphic vector bundle E admits an equivariant structure.

The holomorphic vector bundle E admits an integrable logarithmic connection singular over D.

The holomorphic vector bundle E admits a logarithmic connection singular over D.

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 above result for holomorphic vector bundles on log parallelizable G-pairs (X, D), where G is a simply connected complex affine algebraic group