Perversity equals weight for Painlev\'e spaces

TL;DR

This paper verifies the $P=W$ conjecture for Painlevé systems by showing the agreement of perverse and weight filtrations on relevant moduli spaces, thus providing evidence for a broader geometric conjecture.

Contribution

It confirms the $P=W$ conjecture in Painlevé cases and proves the Geometric $P=W$ conjecture for these systems, establishing their equivalence.

Findings

Perverse and weight filtrations agree in Painlevé cases

Proved the Geometric $P=W$ conjecture for Painlevé systems

Showed the Geometric $P=W$ conjecture implies the $P=W$ conjecture in these cases

Abstract

We provide further evidence to the $P=W$ conjecture of de Cataldo, Hausel and Migliorini, by checking it in the Painlev\'e cases. Namely, we compare the perverse Leray filtration induced by the Hitchin map on the cohomology spaces of the Dolbeault moduli space and the weight filtration on the cohomology spaces of the irregular character variety corresponding to each of the Painlev\'e $I-VI$ systems. We find that the two filtrations agree. Along the way, we prove the Geometric $P=W$ conjecture of Katzarkov, Noll, Pandit and Simpson in the Painlev\'e cases, and show that in these cases the Geometric $P=W$ conjecture implies the $P=W$ conjecture.