Exponents of some one-dimensional Gauss-Manin cohomologies

TL;DR

This paper offers an algebraic approach to determine the exponents of one-dimensional Gauss-Manin cohomologies, extending to broader systems and applying to hyperplane arrangements relevant in Dwork families and mirror symmetry.

Contribution

It introduces a purely algebraic characterization of exponents for certain Gauss-Manin systems, linking cohomology vanishing to Koszul complex properties.

Findings

Algebraic criteria for exponents of Gauss-Manin cohomologies.

Explicit calculation of exponents for hyperplane arrangements.

Relevance to Dwork families and mirror symmetry applications.

Abstract

In this paper we provide a purely algebraic characterization of the exponents of one-dimensional direct images of a structure sheaf by a rational function, related to the vanishing of the cohomologies of a certain Koszul complex associated with such a morphism. This can be extended to a more general family of Gauss-Manin systems. As an application, we calculate a set of possible exponents of the Gauss-Manin cohomology of some arrangements of hyperplanes with multiplicities, relevant to Dwork families and mirror symmetry.