On the geometric P=W conjecture

TL;DR

This paper introduces the geometric P=W conjecture for singular character varieties, proves it for genus one surfaces, and explores its relation to the cohomological P=W conjecture using advanced geometric techniques.

Contribution

It formulates the geometric P=W conjecture for singular varieties, proves it for genus one, and clarifies its connection to the cohomological version.

Findings

Established the conjecture for genus one surfaces.

Obtained partial results for higher genus.

Clarified the relation between geometric and cohomological P=W conjectures.

Abstract

We formulate the geometric P=W conjecture for singular character varieties. We establish it for compact Riemann surfaces of genus one, and obtain partial results in arbitrary genus. To this end, we employ non-Archimedean, birational and degeneration techniques to study the topology of the dual boundary complex of certain character varieties. We also clarify the relation between the geometric and the cohomological P=W conjectures.