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Hodge numbers from Picard-Fuchs equations. (English) Zbl 1366.14014

If \(C\) is a smooth quasiprojective curve which bears an \(\mathbb{R}\)-VHS (variation of Hodge structure) of weight \(k\), \(\mathbb{V}\) the underlying local system and \(j\) the embedding of \(C\) into its smooth completion, then the parabolic cohomology of \(\mathbb{V}\) bears a pure Hodge structure of weight \(k+1\) (shown by S. Zucker [Ann. Math. (2) 109, 415–476 (1979; Zbl 0446.14002)]). The paper aims at computing Hodge numbers of parabolic groups of VHS when the Picard-Fuchs equation is known. For a VHS over \(\mathbb{P}^1\) with Hodge numbers \((1,1,\ldots ,1)\) the authors show how to compute the degrees of the Deligne extension of its Hodge bundles, following a method of Eskin-Kontsevich-Möller-Zorich, by using the local exponents of the corresponding Picard-Fuchs equation. This allows to compute the Hodge numbers of Zucker’s Hodge structure on the corresponding parabolic cohomology groups. The result is applied to families of elliptic curves, \(K3\) surfaces and Calabi-Yau threefolds.

MSC:

14D07 Variation of Hodge structures (algebro-geometric aspects)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)

Citations:

Zbl 0446.14002
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Full Text: DOI arXiv

References:

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