On the geometric P=W conjecture

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 character varieties. We also clarify the relation between the geometric and the cohomological P=W conjectures.