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arXiv · 2408.15786

Cohomological integrality for symmetric quotient stacks

Lucien Hennecart
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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.

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Lucien Hennecart. 2025-03-01. Cohomological integrality for symmetric quotient stacks. https://arxiv.org/abs/2408.15786

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