Perverse filtrations, Hilbert schemes, and the $P=W$ Conjecture for parabolic Higgs bundles

Shen, Junliang; Zhang, Zili

doi:10.14231/ag-2021-014

Cited by 7 publications

(1 citation statement)

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“…The topology -specifically the homology and cohomology -of Hitchin moduli spaces and character varieties has been the subject of much work beginning with a study of SYZ-type [1028] mirror symmetry and its relation to Langlands duality [1029]. In another direction, a very general conjecture known as the "P = W " conjecture has been extensively studied in [1030][1031][1032][1033][1034][1035][1036][1037][1038][1039][1040][1041][1042]. The results can be interpreted very nicely using BPS states associated with string theory compactification on local Calabi-Yau manifolds [1043][1044][1045][1046].…”

Section: Hyperkähler and Quaternionic Kähler Geometrymentioning

confidence: 99%

A Panorama Of Physical Mathematics c. 2022

Bah¹,

Freed²,

Moore³

et al. 2022

Preprint

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What follows is a broad-brush overview of the recent synergistic interactions between mathematics and theoretical physics of quantum field theory and string theory. The discussion is forwardlooking, suggesting potentially useful and fruitful directions and problems, some old, some new, for further development of the subject. This paper is a much extended version of the Snowmass whitepaper on physical mathematics [1].

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Section: Hyperkähler and Quaternionic Kähler Geometrymentioning

confidence: 99%

A Panorama Of Physical Mathematics c. 2022

Bah¹,

Freed²,

Moore³

et al. 2022

Preprint

View full text Add to dashboard Cite

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

Cataldo

Maulik

Shen

2021

J. Amer. Math. Soc.

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We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus 2 2 curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration. Our main technique is to study the Hitchin fibration as a degeneration of the Hilbert–Chow morphism associated with the moduli space of certain torsion sheaves on an abelian surface, where the symmetries induced by Markman’s monodromy operators play a crucial role.

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