P=W conjectures for character varieties with a symplectic resolution

Description:
P=W conjectures for character varieties with a symplectic resolution

Character varieties parametrise representations of the fundamental group of a curve.
In general these moduli spaces are singular, therefore it is customary to slightly change the moduli problem and consider smooth analogues, called twisted character varieties.
In this setting, the P=W conjecture by de Cataldo, Hausel, and Migliorini suggests a surprising connection between the topology of Hitchin systems and Hodge theory of character varieties.
In a joint work with M. Mauri we establish (and in some cases formulate) analogous P=W phenomena in the singular case .
In particular we show that the P=W conjecture holds for character varieties which admit a symplectic resolution, namely in genus 1 and arbitrary rank and in genus 2 and rank 2.

(30 minutes general talk followed by 45 minutes research talk)