Purity and 2-Calabi-Yau categories

Abstract

For various 2-Calabi-Yau categories $\mathcal{C}$ for which the stack of objects $\mathfrak{M}$ has a good moduli space $p:\mathfrak{M}\to \mathcal{M}$, we establish purity of the mixed Hodge module complex $p_{!}{\underline{\mathbb{Q}}}_{\mathfrak{M}}$. We do this by using formality in 2CY categories, along with étale neighbourhood theorems for stacks, to prove that the morphism $p$ is modelled étale-locally by the semisimplification morphism from the stack of modules of a preprojective algebra. Then via the integrality theorem in cohomological Donaldson-Thomas theory we prove purity of $p_{!}{\underline{\mathbb{Q}}}_{\mathfrak{M}}$. It follows that the Beilinson-Bernstein-Deligne-Gabber decomposition theorem for the constant sheaf holds for the morphism $p$, despite the possibly very singular and stacky nature of $\mathfrak{M}$. We use this to define cuspidal cohomology for $\mathfrak{M}$, which is conjecturally a complete space of generators for the BPS algebra associated to $\mathcal{C}$. We prove purity of the Borel-Moore homology of the moduli stack $\mathfrak{M}$, provided its good moduli space $\mathcal{M}$ is projective, or admits a suitable contracting ${\mathbb{C}}^{\ast }$-action. In particular, when $\mathfrak{M}$ is the moduli stack of Gieseker-semistable sheaves on a K3 surface, this proves a conjecture of Halpern-Leistner. We use these results to moreover prove purity for several stacks of coherent sheaves that do not admit a good moduli space. Without the usual assumption that $r$ and $d$ are coprime, we prove that the Borel-Moore homology of the stack of semistable degree $d$ rank $r$ Higgs sheaves is pure and carries a perverse filtration with respect to the Hitchin base, generalising the usual perverse filtration for the Hitchin system to the case of singular stacks of Higgs sheaves.

v1: 80 pages, comments welcome v2: minor edits, including a reference v3: more minor edits v4: more minor edits/corrections. v5: many corrections, and improvements in results, with thanks to an anonymous referee

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• Calabi-Yau algebras
• Hochschild homology and semiorthogonal decompositions
• An overview of Morihiko Saito's theory of mixed Hodge modules
• A young person's guide to mixed Hodge modules
• Dimensional reduction in cohomological Donaldson-Thomas theory
• Homotopy L-infinity spaces
• On the P=W conjecture for ${\mathrm{SL}}_n$
• Calabi-Yau structures for multiplicative preprojective algebras
• Intersection cohomology of rank two character varieties of surface groups
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Cited by in corpus (2)

• Coherent sheaves on surfaces, COHAs and deformed $W_{1+\infty }$-algebras
• Crepant resolutions of stratified varieties via gluing