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Picard-Fuchs equations of special one-parameter families of invertible polynomials. (English) Zbl 1302.14034

Laza, Radu (ed.) et al., Arithmetic and geometry of \(K3\) surfaces and Calabi-Yau threefolds. Proceedings of the workshop, Toronto, Canada, August 16–25, 2011. New York, NY: Springer (ISBN 978-1-4614-6402-0/hbk; 978-1-4614-6403-7/ebook). Fields Institute Communications 67, 285-310 (2013).
Summary: We calculate the Picard-Fuchs equation of hypersurfaces defined by certain one-parameter families associated to invertible polynomials. For this we deduce the Picard-Fuchs equation from the GKZ system. As consequences of our work and facts from the literature, we show a relation between the Picard-Fuchs equation, the Poincaré series and the monodromy in the space of period integrals.
For the entire collection see [Zbl 1267.14002].

MSC:

14J33 Mirror symmetry (algebro-geometric aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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References:

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