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Cohomological $\chi$–independence for moduli of one-dimensional sheaves and moduli of Higgs bundles

Davesh Maulik and Junliang Shen

Geometry & Topology 27 (2023) 1539–1586
Abstract

We prove that the intersection cohomology (together with the perverse and the Hodge filtrations) for the moduli space of one-dimensional semistable sheaves supported in an ample curve class on a toric del Pezzo surface is independent of the Euler characteristic of the sheaves. We also prove an analogous result for the moduli space of semistable Higgs bundles with respect to an effective divisor D of degree deg (D) > 2g 2. Our results confirm the cohomological χ–independence conjecture by Bousseau for 2, and verify Toda’s conjecture for Gopakumar–Vafa invariants for certain local curves and local surfaces.

For the proof, we combine a generalized version of Ngô’s support theorem, a dimension estimate for the stacky Hilbert–Chow morphism, and a splitting theorem for the morphism from the moduli stack to the good GIT quotient.

Keywords
moduli of sheaves, perverse sheaves, support theorem, Gopakumar–Vafa invariants
Mathematical Subject Classification
Primary: 14H60, 14J60, 14N35
Secondary: 32S60
References
Publication
Received: 10 January 2021
Revised: 1 October 2021
Accepted: 3 November 2021
Published: 15 June 2023
Proposed: Dan Abramovich
Seconded: Gang Tian, Richard P Thomas
Authors
Davesh Maulik
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Junliang Shen
Department of Mathematics
Yale University
New Haven, CT
United States

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