Cohomological integrality for symmetric quotient stacks

TL;DR

This paper proves a conjecture relating to the purity of BPS sheaves and Borel-Moore homology in the context of symmetric quotient stacks, with applications to moduli stacks of Higgs bundles.

Contribution

It establishes the sheafified cohomological integrality conjecture for quotient stacks and proves the purity of BPS sheaves in specific geometric settings.

Findings

Proved the sheafified cohomological integrality conjecture for quotient stacks.

Established the purity of BPS sheaves in symplectic algebraic varieties.

Confirmed the purity of Borel-Moore homology for moduli stacks of Higgs bundles.

Abstract

In this paper, we establish the sheafified version of the cohomological integrality conjecture for stacks obtained as a quotient of a smooth affine symmetric algebraic variety by a reductive algebraic group equipped with an invariant function. A crucial step is the definition of the BPS sheaf as a complex of monodromic mixed Hodge modules. We prove the purity of the BPS sheaf when the situation arises from a smooth affine weakly symplectic algebraic variety with a weak moment map. This situation gives local models for 1-Artin derived stacks with self-dual cotangent complex. We then apply these results to prove a conjecture of Halpern-Leistner predicting the purity of the Borel-Moore homology of $0$-shifted symplectic stacks (or more generally, derived stacks with self-dual cotangent complex) having a proper good moduli space. One striking application is the purity of the Borel--Moore homology of the moduli stack of principal Higgs bundles over a smooth projective curve for a reductive group.