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

Abstract

We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus $2$ 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.

47 pages. Minor revision. Accepted at Journal of the American Mathematical Society

References in corpus (2)

• Cell decompositions of character varieties
• P=W for Lagrangian fibrations and degenerations of hyper-Kähler manifolds

Cited by in corpus (6)

• Purity and 2-Calabi-Yau categories
• Intersection cohomology of rank two character varieties of surface groups
• An Enumerative Approach to $P=W$
• The nilpotent cone in the Mukai system for rank 2 and genus 2
• Perverse Leray filtration and specialization with applications to the Hitchin morphism
• The Norm map on the compactified Jacobian, the Prym stack and Spectral data for G-Higgs pairs