Perversity equals weight for Painlevé spaces

Abstract

We provide further evidence to the $P=W$ conjecture of de Cataldo, Hausel and Migliorini, by checking it in the Painlevé 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 Painlevé $I-VI$ 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 Painlevé cases, and show that in these cases the Geometric $P=W$ conjecture implies the $P=W$ conjecture.

Final version, to appear in Adv. Math., containing the material of 1906.01856