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.
Simpson's geometric P=W conjecture in the Painlev\'e VI case via abelianization
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abstract
We use abelianization of Higgs bundles near infinity to prove the homotopy commutativity assertion of Simpson's geometric P=W conjecture in the Painlev\'e VI case.
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2019 1verdicts
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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.