Exponents of Some One-Dimensional Gauss–Manin Systems

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.