$P=W$ phenomena on abelian varieties

TL;DR

This paper establishes an analogue of the $P=W$ conjecture for complex abelian varieties, linking the topology of Higgs bundle moduli spaces with character varieties through spectral data morphisms.

Contribution

It introduces a spectral data morphism for Dolbeault moduli spaces on abelian varieties, generalizing the Hitchin morphism and connecting topological and geometric structures.

Findings

Proves an analogue of the $P=W$ conjecture for abelian varieties.

Constructs a spectral data morphism factoring the Hitchin morphism.

Links the topology of Higgs moduli spaces with character varieties.

Abstract

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.