Nonabelian Hodge theory for stacks and a stacky P=W conjecture

TL;DR

This paper extends the P=W conjecture to stacks, relating homology of representation stacks and Higgs bundle stacks, and proves the conjecture for genus zero and one.

Contribution

It introduces a stacky version of the P=W conjecture, constructs a canonical isomorphism between their homology groups, and proves the conjecture for low genus cases.

Findings

Proposes a stacky P=W conjecture relating different homology groups.

Constructs a canonical isomorphism between these homology groups.

Proves the conjecture for genus zero and one.

Abstract

We introduce a version of the P=W conjecture relating the Borel-Moore homology of the stack of representations of the fundamental group of a genus g Riemann surface with the Borel-Moore homology of the stack of degree zero semistable Higgs bundles on a smooth projective complex curve of genus $g$. In order to state the conjecture we propose a construction of a canonical isomorphism between these Borel-Moore homology groups. We relate the stacky P=W conjecture with the original P=W conjecture concerning the cohomology of smooth moduli spaces of twisted objects, and the PI=WI conjecture concerning the intersection cohomology groups of singular moduli spaces of untwisted objects. In genus zero and one, we prove the conjectures that we introduce in this paper.