Goresky-Pardon lifts of Chern classes and associated Tate extensions

TL;DR

This paper develops geometric conditions for extending Chern classes of vector bundles on complex varieties, refines known theorems in the context of automorphic bundles, and reveals that these extensions can have non-zero imaginary parts, leading to Tate extensions.

Contribution

It introduces new geometric criteria for Chern class extensions, refines Goresky-Pardon's theorem for automorphic bundles, and demonstrates the existence of non-real Chern class lifts in certain compactifications.

Findings

Extended Chern classes with high Hodge level under specific conditions.

Refined Goresky-Pardon theorem for automorphic bundles on Baily-Borel compactifications.

Existence of non-zero imaginary parts in Chern class extensions, leading to Tate extensions.

Abstract

Let X be an irreducible complex variety, S a stratification of X and F a holomorphic vector bundle on the open statum. We give geometric conditions on S and F that produce a natural extension of the k-th Chern class F as a class in the complex cohomology of X of Hodge level at least k. When X is the Baily-Borel compactification of a locally symmetric variety with its stratification by boundary components, and F an automorphic bundle on its interior, then this recovers and refines a theorem of Goresky-Pardon. In passing we define a class of simplicial resolutions of the Baily-Borel compactification that can be used to define its mixed Hodge structure. Finally, we use the Beilinson regulator for the rationals to show that when X is the Satake (=Baily-Borel) compactification of A_g and F the Hodge bundle (with g large compared to k), the Goresky-Pardon k-th Chern class extension has nonzero imaginary part and gives rise to a basic Tate extension. This also shows that the lifts of Chern classes constructed by Goresky and Pardon need not be real and thus answers (negatively) a question asked by these authors.