### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 045, 23 pages      arXiv:1612.09439      https://doi.org/10.3842/SIGMA.2017.045
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

### Hodge Numbers from Picard-Fuchs Equations

Charles F. Doran a, Andrew Harder b and Alan Thompson cd
a) Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, AB, T6G 2G1, Canada
b) Department of Mathematics, University of Miami, 1365 Memorial Drive, Ungar 515, Coral Gables, FL, 33146, USA
c) Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, UK
d) DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

Received January 20, 2017, in final form June 12, 2017; Published online June 18, 2017

Abstract
Given a variation of Hodge structure over $\mathbb{P}^1$ with Hodge numbers $(1,1,\dots,1)$, we show how to compute the degrees of the Deligne extension of its Hodge bundles, following Eskin-Kontsevich-Möller-Zorich, by using the local exponents of the corresponding Picard-Fuchs equation. This allows us to compute the Hodge numbers of Zucker's Hodge structure on the corresponding parabolic cohomology groups. We also apply this to families of elliptic curves, K3 surfaces and Calabi-Yau threefolds.

Key words: variation of Hodge structures; Calabi-Yau manifolds.

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