Dwork families and $\mathcal{D}$-modules

TL;DR

This paper investigates the algebraic structure of Dwork families, focusing on their Gauss-Manin cohomology invariants under automorphisms, using algebraic $ ext{D}$-modules and hypergeometric theory.

Contribution

It provides an algebraic computation of the invariant part of Gauss-Manin cohomology for Dwork families using $ ext{D}$-modules and hypergeometric methods, a novel approach in this context.

Findings

Computed invariant Gauss-Manin cohomology for Dwork families.

Applied algebraic $ ext{D}$-modules and hypergeometric theory.

Enhanced understanding of automorphism actions on cohomology.

Abstract

A Dwork family is a one-parameter monomial deformation of a Fermat hypersurface. In this paper we compute algebraically the invariant part of its Gauss-Manin cohomology under the action of certain subgroup of automorphisms. To achieve that goal we use the algebraic theory of $\mathcal{D}$-modules, especially one-dimensional hypergeometric ones.