Limit mixed Hodge structures of hyperk\"ahler manifolds

Andrey Soldatenkov

Limit mixed Hodge structures of hyperk\"ahler manifolds

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arxiv 1807.04030 v3 pith:IDRZWQZR submitted 2018-07-11 math.AG

Andrey Soldatenkov This is my paper · ORCID

classification math.AG

keywords hodgestructureslimitmixedahlerhyperkmanifoldsaction

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abstract

This note is inspired by the work of Deligne on the local behavior of Hodge structures at infinity. We study limit mixed Hodge structures of degenerating families of compact hyperk\"ahler manifolds. We show that when the monodromy action on $H^2$ has maximal index of unipotency, the limit mixed Hodge structures on all cohomology groups are of Hodge-Tate type.

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Torus fibers and the weight filtration
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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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