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Perversity equals weight for Painlev\'e spaces
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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.
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Torus fibers and the weight filtration
A single real torus inside a log Calabi-Yau complement (or a Calabi-Yau degeneration) computes the odd weight filtration, yielding P=W type results for rational surfaces and K3 surfaces.
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