  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.
