Purity and 2-Calabi–Yau categories

Abstract For various 2-Calabi–Yau categories C C for which the classical stack of objects M M has a good moduli space p M M p: M → M, we establish purity of the mixed Hodge module complex p!{Q} {₌} p ! Q _ M. We do this by using formality in 2CY categories, along with étale neighbourhood theorems for stacks, to prove that the morphism p p is modelled étale-locally by the semisimplification morphism from the stack of modules of a preprojective algebra. Via the integrality theorem in cohomological Donaldson–Thomas theory we then prove purity of p!{Q} {₌} p ! Q _ M. It follows that the Beilinson–Bernstein–Deligne–Gabber decomposition theorem for the constant sheaf holds for the morphism p p, despite the possibly singular and stacky nature of {M} M, and the fact that p p is not proper. We use this to define cuspidal cohomology for {M} M, which conjecturally provides a complete space of generators for the BPS algebra associated to C C. We prove purity of the Borel–Moore homology of the moduli stack M M, provided its good moduli space ℳ is projective, or admits a suitable contracting C^* C ∗ -action. In particular, when M 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 r and d d are coprime, we prove that the Borel–Moore homology of the stack of semistable degree d d rank r 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.