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

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.

Final version. To appear in Compositio Mathematica

References in corpus (1)

• Semistability vs. nefness for (Higgs) vector bundles

Cited by in corpus (5)

• Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities
• Projectively flat KLT varieties
• Harmonic metrics on Higgs sheaves and uniformization of varieties of general type
• Bogomolov's inequality and Higgs sheaves on normal varieties in positive characteristic
• Reductive quotients of klt singularities