Cohomology of the moduli of Higgs bundles on a curve via positive characteristic

  • Mark Andrea de Cataldo

    Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA
  • Davesh Maulik

    Massachusetts Institute of Technology, Cambridge, MA 02139, USA
  • Junliang Shen

    Department of Mathematics, Yale University, New Haven, CT 06511, USA
  • Siqing Zhang

    Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA
Cohomology of the moduli of Higgs bundles on a curve via positive characteristic cover

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Abstract

For a curve of genus and any two degrees coprime to the rank, we construct a family of ring isomorphisms parameterized by the complex Lie group between the cohomology of the moduli spaces of stable Higgs bundles which preserve the perverse filtrations. As a consequence, we prove two structural results concerning the cohomology of Higgs moduli which are predicted by the P = W Conjecture in Non-Abelian Hodge Theory: (1) Galois conjugation for character varieties preserves the perverse filtrations for the corresponding Higgs moduli spaces. (2) The restriction of the Hodge–Tate decomposition for a character variety to each piece of the perverse filtration for the corresponding Higgs moduli space also gives a decomposition. Our proof uses reduction to positive characteristic and relies on the non-abelian Hodge correspondence in characteristic between Dolbeault and de Rham moduli spaces.

Cite this article

Mark Andrea de Cataldo, Davesh Maulik, Junliang Shen, Siqing Zhang, Cohomology of the moduli of Higgs bundles on a curve via positive characteristic. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1393