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BPS Lie algebras and the less perverse filtration on the preprojective CoHA

Ben Davison

Abstract

The affinization morphism for the stack $\mathfrak{M}(Π_Q)$ of representations of a preprojective algebra $Π_Q$ is a local model for the morphism from the stack of objects in a general 2-Calabi-Yau category to the good moduli space. We show that the derived direct image of the dualizing complex along this morphism is pure, and admits a decomposition in the sense of the Beilinson-Bernstein-Deligne-Gabber decomposition theorem. We introduce a new perverse filtration on the Borel-Moore homology of $\mathfrak{M}(Π_Q)$, using this decomposition. We show that the zeroth piece of the resulting filtration on the cohomological Hall algebra built out of the Borel-Moore homology of $\mathfrak{M}(Π_Q)$ is isomorphic to the universal enveloping algebra of an associated BPS Lie algebra $\mathfrak{g}_{Π_Q}$. This Lie algebra is defined via the Kontsevich-Soibelman theory of critical cohomological Hall algebras for 3-Calabi-Yau categories. We then lift this Lie algebra to a Lie algebra object in the category of perverse sheaves on the coarse moduli space of $Π_Q$-modules, and use this algebra structure to prove results about the summands appearing in the above decomposition theorem. In particular, we prove that the intersection cohomology of singular spaces of semistable $Π_Q$-modules provide "cuspidal cohomology" - a conjecturally complete subspace of canonical generators for $\mathfrak{g}_{Π_Q}$.

BPS Lie algebras and the less perverse filtration on the preprojective CoHA

Abstract

The affinization morphism for the stack of representations of a preprojective algebra is a local model for the morphism from the stack of objects in a general 2-Calabi-Yau category to the good moduli space. We show that the derived direct image of the dualizing complex along this morphism is pure, and admits a decomposition in the sense of the Beilinson-Bernstein-Deligne-Gabber decomposition theorem. We introduce a new perverse filtration on the Borel-Moore homology of , using this decomposition. We show that the zeroth piece of the resulting filtration on the cohomological Hall algebra built out of the Borel-Moore homology of is isomorphic to the universal enveloping algebra of an associated BPS Lie algebra . This Lie algebra is defined via the Kontsevich-Soibelman theory of critical cohomological Hall algebras for 3-Calabi-Yau categories. We then lift this Lie algebra to a Lie algebra object in the category of perverse sheaves on the coarse moduli space of -modules, and use this algebra structure to prove results about the summands appearing in the above decomposition theorem. In particular, we prove that the intersection cohomology of singular spaces of semistable -modules provide "cuspidal cohomology" - a conjecturally complete subspace of canonical generators for .

Paper Structure

This paper contains 70 sections, 34 theorems, 299 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Cuspidal cohomology
  5. Halpern--Leistner's conjecture
  6. The algebras $\mathop{\mathrm{\mathbf{U}}}\nolimits(\mathfrak{g}_{C})$ and $\mathop{\mathrm{\mathbf{U}}}\nolimits(\mathfrak{g}_{\Sigma_g})$
  7. Notation and conventions
  8. Acknowledgements
  9. Background on CoHAs
  10. Monodromic mixed Hodge modules
  11. Mixed Hodge modules
  12. Monodromic mixed Hodge modules
  13. Six functors for MMHMs
  14. MMHM complexes on stacks
  15. Weight filtrations
  16. ...and 55 more sections

Key Result

Theorem A

There is an isomorphism of complexes of mixed Hodge modules and each $\mathop{\mathrm{\mathcal{H}}}\nolimits^n(\mathop{\mathrm{\mathcal{RA}}}\nolimits_{\Pi_Q})$ is pure of weight $n$, i.e. $\mathop{\mathrm{\mathcal{RA}}}\nolimits_{\Pi_Q}$ is pure. As a consequence, we may write where $Z_i\subset \mathcal{M}(\Pi_Q)$ are locally closed smooth connected subvarieties, and $\mathcal{IC}_{\overline{Z_

Theorems & Definitions (73)

  • Theorem A: Corollary \ref{['relPurity']}
  • Theorem B: Proposition \ref{['respProp']}, Theorem \ref{['UEAthm']}
  • Theorem C: Theorem \ref{['thmBdone']}
  • Theorem \oldthetheorem: Saito90
  • proof
  • Remark \oldthetheorem
  • Example \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • ...and 63 more