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September 2021 Gauss–Manin connection in disguise: Dwork family
H. Movasati, Y. Nikdelan
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J. Differential Geom. 119(1): 73-98 (September 2021). DOI: 10.4310/jdg/1631124264

Abstract

We study the moduli space $\mathsf{T}$ of the Calabi–Yau $n$‑folds arising from the Dwork family and enhanced with bases of the $n$‑th de Rham cohomology with constant cup product and compatible with Hodge filtration. We also describe a unique vector field $\mathsf{R}$ in $\mathsf{T}$ which contracted with the Gauss–Manin connection gives an upper triangular matrix with some non-constant entries which are natural generalizations of Yukawa couplings. For $n=1,2$ we compute explicit expressions of $\mathsf{R}$ and give a solution of $\mathsf{R}$ in terms of quasi-modular forms. The moduli space $\mathsf{T}$ is an affine variety and for $n = 4$ we give explicit coordinate system for $\mathsf{T}$ and compute the vector field $\mathsf{R}$ and the $q$-expansion of its solution.

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H. Movasati. Y. Nikdelan. "Gauss–Manin connection in disguise: Dwork family." J. Differential Geom. 119 (1) 73 - 98, September 2021. https://doi.org/10.4310/jdg/1631124264

Information

Received: 16 December 2017; Accepted: 27 November 2019; Published: September 2021
First available in Project Euclid: 10 September 2021

Digital Object Identifier: 10.4310/jdg/1631124264

Subjects:
Primary: 11Y55 , 14J15 , 14J32

Keywords: $q$-expansion , Dwork family , Gauss–Manin connection , Hodge filtration , Picard–Fuchs equation , quasi-modular form

Rights: Copyright © 2021 Lehigh University

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Vol.119 • No. 1 • September 2021
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