%Ade Cataldo, Mark%AMaulik, Davesh%AShen, Junliang%BJournal Name: Journal of the American Mathematical Society %D2021%I %JJournal Name: Journal of the American Mathematical Society %K %MOSTI ID: 10325819 %PMedium: X %THitchin fibrations, abelian surfaces, and the P=W conjecture %XWe 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. %0Journal Article