Hitchin fibrations, abelian surfaces, and the P=W conjecture

Cataldo, Mark Andrea A. de; Maulik, Davesh; Shen, Junliang

doi:10.48550/arxiv.1909.11885

Cited by 11 publications

(28 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This perverse filtration has been the object of intense study for some time, in particular as a result of the P=W conjecture of de Cataldo, Hausel and Migliorini [dCHM12] (see e.g. [dCMS19,dCMS20,CHS20] for recent progress on this conjecture). This conjecture relates the perverse filtration on H(M Dol r,d (C), Q) with the weight filtration on H(M Betti g,r , Q) under the isomorphism in cohomology induced by the diffeomorphism…”

mentioning

confidence: 99%

Purity and 2-Calabi-Yau categories

Davison¹

2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

mentioning

confidence: 99%

Purity and 2-Calabi-Yau categories

Davison¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Another conjectural cohomological identity is the P=W conjecture [dCHM], predicting that the morphism in cohomology induced by the non-abelian Hodge correspondence exchanges the preverse filtration associated to the Hitchin fibration with the weight filtration on the associated character variety. This was proven by de Cataldo-Hausel-Migliorini [dCHM] in the rank 2 case, and by de Cataldo-Maulik-Shen [dCMS1,dCMS2] in the case of base curves of genus 2. The Donagi-Ein-Lazarsfeld degeneration in the case of abelian surfaces was a key element in the work of de Cataldo-Maulik-Shen.…”

mentioning

confidence: 81%

“…We summarize all of the above in the following theorem. This is a generalization to arbitrary smooth surfaces of the non-linear deformation that appeared first in [DEL] for K3 surfaces and in [dCMS1] for abelian surfaces.…”

Section: The Involutions ζ±mentioning

confidence: 99%

“…Sawon's conjecture was the motivation for our work as such degeneration of Prymian integrable system and that of the associated natural Lagrangian subverieties, could open the door for a cohomological study of pairs of (BBB) and (BAA)-branes in the Hitchin system, leading perhaps to strong evidence of their duality, by means of the specialization morphism in cohomology given in [dCMS1,dCMS2].…”

mentioning

confidence: 94%

See 1 more Smart Citation

Degeneration of natural Lagrangians and Prymian integrable systems

Franco¹

2021

Preprint

View full text Add to dashboard Cite

Starting from an anti-symplectic involution on a K3 surface, one can consider a natural Lagrangian subvariety inside the moduli space of sheaves over the K3. One can also construct a Prymian integrable system following a construction of Markushevich-Tikhomirov, extended by Arbarello-Saccà-Ferretti, Matteini and Sawon-Chen. In this article we address a question of Sawon, showing that these integrable systems and their associated natural Lagrangians degenerate, respectively, into fix loci of involutions considered by Heller-Schaposnik, Garcia-Prada-Wilkins and Basu-Garcia-Prada.Along the way we find interesting results such as the proof that the Donagi-Ein-Lazarsfeled degeneration is a degeneration of symplectic varieties, a generalization of this degeneration, originally described for K3 surfaces, to the case of an arbitrary smooth projective surface, and a description of the behaviour of certain involutions under this degeneration.

show abstract

“…From our perspective, this is a powerful approach to studying the Hitchin system. For instance, in a recent paper [7], de Cataldo, Maulik and Shen prove the P=W conjecture for g = 2 by means of the corresponding specialization map on cohomology.…”

Section: Introductionmentioning

confidence: 99%