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

TL;DR

This paper proves the P=W conjecture for genus 2 curves and arbitrary rank, using abelian surfaces and Hitchin fibrations, and reduces the conjecture to a question about the multiplicativity of the perverse filtration.

Contribution

It establishes the P=W conjecture for genus 2 and higher genus cases for specific cohomology subalgebras, linking Hitchin fibrations with abelian surface degenerations.

Findings

Proves P=W for genus 2 curves and arbitrary rank.

Shows tautological classes lie in predicted perverse filtration pieces.

Reduces P=W conjecture to multiplicativity of the perverse filtration.

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.