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Moscow Mathematical Journal

Volume 20, Issue 1, January–March 2020  pp. 127–151.

Modular Vector Fields Attached to Dwork Family: $\mathfrak{sl}_2(\mathbb{C})$ Lie algebra

Authors:  Younes Nikdelan (1)
Author institution:(1) Universidade do Estado do Rio de Janeiro (UERJ), Instituto de Matemática e Estatística (IME), Departamento de Análise Matemática: Rua São Francisco Xavier, 524, Rio de Janeiro, Brazil / CEP: 20550-900


This paper aims to show that a certain moduli space $\mathsf{T}$, which arises from the so-called Dwork family of Calabi–Yau $n$-folds, carries a special complex Lie {algebra} containing a copy of $\mathfrak{sl}_2(\mathbb{C})$. In order to achieve this goal, we introduce an algebraic group $\mathsf{G}$ acting from the right on $\mathsf{T}$ and describe its Lie algebra $\mathrm{Lie}(\mathsf{G})$. We observe that $\mathrm{Lie}(\mathsf{G})$ is isomorphic to a Lie subalgebra of the space of the vector fields on $\mathsf{T}$. In this way, it turns out that $\mathrm{Lie}(\mathsf{G})$ and the modular vector field $\mathsf{R}$ generate another Lie algebra $\mathfrak{G}$, called AMSY-Lie algebra, satisfying $\dim (\mathfrak{G})=\dim (\mathsf{T})$. We find a copy of $\mathfrak{sl}_2(\mathbb{C})$ containing $\mathsf{R}$ as a Lie subalgebra of $\mathfrak{G}$. The proofs are based on an algebraic method calling “Gauss–Manin connection in disguise”. Some explicit examples for $n=1,2,3,4$ are stated as well.

2010 Math. Subj. Class. 32M25, 37F99, 14J15, 14J32.

Keywords: Complex vector fields, Gauss–Manin connection, Dwork family, Hodge filtration, modular form.

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