  Let $X$ be a complex abelian variety. We prove an analogue of both the
(cohomological) $P=W$ conjecture and the geometric $P=W$ conjecture connecting
the finer topological structure of the Dolbeault moduli space of topologically
trivial semistable Higgs bundles on $X$ and the Betti moduli space of
characters of the fundamental group of $X$. The geometric heart of our approach
is the spectral data morphism for Dolbeault moduli spaces on abelian varieties
that naturally factors the Hitchin morphism and whose target is not an affine
space of pluricanonical sections, but a suitable symmetric product.
