Period integrals associated to an affine Delsarte type hypersurface

TL;DR

This paper computes period integrals for affine Delsarte hypersurfaces, linking Mellin transforms, monodromy groups, and quantum cohomology, revealing new connections between algebraic geometry and hypergeometric functions.

Contribution

It provides explicit calculations of period integrals, monodromy groups, and establishes a novel link between oscillating integrals and quantum cohomology for Delsarte hypersurfaces.

Findings

Explicit formulas for period integrals using Mellin transforms

Determination of the monodromy group of hypergeometric solutions

Establishment of a relation between Stokes and Gram matrices

Abstract

We calculate the period integrals for a special class of affine hypersurfaces (deformed Delsarte hypersurfaces) in an algebraic torus by the aid of their Mellin transforms. A description of the relation between poles of Mellin transforms of period integrals and the mixed Hodge structure of the cohomology of the hypersurface is given. By interpreting the period integrals as solutions to Pochhammer hypergeometric differential equation, we calculate concretely the irreducible monodromy group of period integrals that correspond to the compactification of the affine hypersurface in a complete simplicial toric variety. As an application of the equivalence between oscillating integral for Delsarte polynomial and quantum cohomology of a weighted projective space $\mathbb{P}_{\bf B}$, we establish an equality between its Stokes matrix and the Gram matrix of the full exceptional collection on $\mathbb{P}_{\bf B}$.