Purity and 2-Calabi-Yau categories

Abstract

For various 2-Calabi-Yau categories C for which the stack of objects M has a good moduli space p : M → M, we establish purity of the mixed Hodge module complex 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 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 prove purity of p ! Q M . It follows that the Beilinson-Bernstein-Deligne-Gabber decomposition theorem for the constant sheaf holds for the morphism p, despite the possibly singular and stacky nature of M. We use this to define cuspidal cohomology for M, which is conjecturally a complete space of generators for the BPS algebra associated to C .We prove purity of the Borel-Moore homology of the moduli stack M, provided its good moduli space M is projective, or admits a suitable contracting C * -action. In particular, when 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.

Cohomological $χ$-independence for Higgs bundles and Gopakumar-Vafa invariants

“…The above theorem implies that there exists an embedding BPS r,m ֒→ H(p S * DQ M ss S (r,m) ) ⊗ L r 2 (g−1)+1 . The purity of the right-hand side is proved in [Davc,Proposition 7.20], so we obtain the claim.…”

“…In [Dava,Conjecture 7.7], Davison formulated a conjecture relating the BPS cohomology and the intersection cohomology of the moduli space of representations of preprojective algebras. Using an étale neighborhood theorem [Davc,Theorem 5.11] which is also due to Davison, this conjecture is expected to imply similar statements for the Dolbeault and Betti moduli spaces. Once this conjecture is established, it would be possible to prove the equivalence of two versions of the P=W conjectures via the BPS cohomology and via the intersection cohomology, and that the χindependence of the intersection cohomology of the Dolbeault moduli space would follow from Theorem 1.2 as long as gcd(r, m) = gcd(r, m ′ ) holds.…”

“…Here H n denotes the perverse t-structure on D b c (X). This assumption is automatically satisfied when X is of the form [Y /G] for some scheme Y and a linear algebraic group G (see [Davc,§2.3.2]). Let p : X → X be a morphism to a scheme.…”

Derived categories of hearts on Kuznetsov components

“…In [BZ19] the authors proved the formality conjecture for semistable objects in the bounded derived category of a K3 surface, using Orlov's result on strongly uniqueness of the enhancement. In the case of cubic fourfolds and Gushel-Mukai varieties of even dimension, the formality conjecture follows from the general results in [Dav21]. Nevertheless, Theorem 1.2 could be useful to provide a direct and simpler proof of this conjecture in these cases.…”

Hodge-to-singular correspondence for reduced curves

“…The cohomology of M (n, d) has been extensively studied in the literature, especially under the assumption that n and d are coprime, namely when M (n, d) is smooth; see for instance [40,33,36,35,38,54,31,16,54,19]. Recently studies about the intersection cohomology IH * (M (n, d), Q) of singular Dolbeault moduli spaces have started to emerge; see for instance [26,27,48,50,51,49,14,13,43,64]. 1 From this viewpoint, the decomposition theorem for the Hitchin map is a key tool to investigate the intersection cohomology of M (n, d): it allows to decompose IH * (M (n, d), Q) into building blocks, which are cohomology of some perverse sheaves on A n .…”