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Cohomological integrality for symmetric quotient stacks

Lucien Hennecart

TL;DR

This work extends cohomological integrality from type A settings to symmetric quotient stacks by developing sheafified cohomological integrality for smooth affine $G$-varieties and their critical loci. The authors construct a BPS sheaf as a complex of monodromic mixed Hodge modules and prove its purity in weak Hamiltonian reductions, enabling purity results for BM-homology of $0$-shifted symplectic stacks with proper good moduli spaces. The framework relies on étale slices, parabolic induction, and a perverse filtration to decompose pushforwards into Wely-group-equivariant pieces indexed by partitions, with explicit isotypic components controlled by signs $oldsymbol{ar W}_{( extλ, extα)}$ and characters $oldsymbol{ extvarepsilon}_{X,( extλ, extα)}$. A major consequence is the Halpern-Leistner purity conjecture for 1-Artin, $0$-shifted symplectic stacks (and its derived-generalization), with applications to moduli of Higgs bundles and related derived moduli problems. The results provide a unified, largely non-A-type approach to geometric representation-theoretic invariants across symmetric, critical, and weakly Hamiltonian settings, offering new tools for studying the topology of moduli spaces and their cohomological invariants.

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.

Cohomological integrality for symmetric quotient stacks

TL;DR

This work extends cohomological integrality from type A settings to symmetric quotient stacks by developing sheafified cohomological integrality for smooth affine -varieties and their critical loci. The authors construct a BPS sheaf as a complex of monodromic mixed Hodge modules and prove its purity in weak Hamiltonian reductions, enabling purity results for BM-homology of -shifted symplectic stacks with proper good moduli spaces. The framework relies on étale slices, parabolic induction, and a perverse filtration to decompose pushforwards into Wely-group-equivariant pieces indexed by partitions, with explicit isotypic components controlled by signs and characters . A major consequence is the Halpern-Leistner purity conjecture for 1-Artin, -shifted symplectic stacks (and its derived-generalization), with applications to moduli of Higgs bundles and related derived moduli problems. The results provide a unified, largely non-A-type approach to geometric representation-theoretic invariants across symmetric, critical, and weakly Hamiltonian settings, offering new tools for studying the topology of moduli spaces and their cohomological invariants.

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 -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.
Paper Structure (78 sections, 114 theorems, 127 equations)

This paper contains 78 sections, 114 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Semisimplicity
  4. Sheafified cohomological integrality for symmetric quotient stacks
  5. Sheafified cohomological integrality for critical symmetric quotient stacks
  6. Purity of the BPS sheaf for weak Hamiltonian reductions
  7. A purity conjecture of Halpern-Leistner
  8. Connection to other works
  9. Notations and conventions
  10. Acknowledgements
  11. Absolute cohomological integrality for symmetric representations of reductive groups
  12. Approximation by proper maps for affine varieties with reductive group action
  13. Semisimplicity
  14. The set of partitions of a $G$-variety
  15. The set of partitions
  16. ...and 63 more sections

Key Result

Theorem 1.1

The complex of mixed Hodge modules $\pi_*\underline{\mathbf{Q}}_{X/G}\in\mathcal{D}^+(\mathrm{MMHM}(X/\!\!/ G))$ is pure of weight zero, and therefore semisimple.

Theorems & Definitions (217)

  • Theorem 1.1: = Proposition \ref{['proposition:semisimplicitypistar']}
  • Corollary 1.2
  • Corollary 1.3
  • proof
  • Corollary 1.4
  • Theorem 1.5: = Theorem \ref{['theorem:sheafcohint']}
  • Proposition 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Lemma 1.9: Support lemma = Lemma \ref{['lemma:sectionsupport']}
  • ...and 207 more