Nonabelian Hodge Theory for klt spaces and descent theorems for vector bundles

TL;DR

This paper extends nonabelian Hodge theory to singular projective varieties with klt singularities, establishing descent theorems for vector bundles and new restriction results for Higgs sheaves.

Contribution

It generalizes Simpson's correspondence to klt spaces and proves a descent theorem for numerically flat bundles along resolutions.

Findings

Descent theorem for numerically flat vector bundles on klt varieties

Extension of nonabelian Hodge correspondence to singular spaces

New restriction theorem for semistable Higgs sheaves

Abstract

We generalise Simpson's nonabelian Hodge correspondence to the context of projective varieties with klt singularities. The proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety X and any resolution of singularities, then any vector bundle on the resolution that appears to come from X numerically, does indeed come from X. Furthermore and of independent interest, a new restriction theorem for semistable Higgs sheaves defined on the smooth locus of a normal, projective variety is established.