# Moscow Mathematical Journal

Volume 22, Issue 1, January–March 2022 pp. 133–168.

Period Integrals Associated to an Affine Delsarte Type Hypersurface

**Authors**:
Susumu Tanabé (1)

**Author institution:**(1) Department of Mathematics,
Galatasaray University,
Çırağan cad. 36,
Beşiktaş, Istanbul, 34357, Turkey

**Summary: **

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 $\mathbb{P}_{\mathbf{B}}$, we establish an equality between its Stokes matrix and the Gram matrix of the full exceptional collection on $\mathbb{P}_{\mathbf{B}}$.

2020 Math. Subj. Class. Primary: 14M99; Secondary: 32S25, 32S40

**Keywords:**Affine hypersurface, Hodge structure, hypergeometric function.

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