Geometric realisation of hypergeometric local systems

TL;DR

This paper proves a realization theorem for hypergeometric local systems over the rationals using families of affine varieties in algebraic tori, with results depending on fiber dimension and monodromy assumptions.

Contribution

It provides a new geometric realization of hypergeometric local systems, extending previous work and connecting to mirror symmetry.

Findings

Unconditional realization for fibers of dimension one or even

Conditional realization for odd-dimensional fibers with monodromy assumption

Connects hypergeometric local systems to algebraic tori in mirror symmetry

Abstract

We prove a realisation theorem for irreducible hypergeometric local systems defined over the rational numbers in terms of families of affine varieties in algebraic tori. The families we consider have been studied extensively in the literature and appear in mirror symmetry. Our result holds unconditionally for families with one-dimensional or even-dimensional fibres. It holds under a monodromy assumption for families with fibres of odd dimension greater than one.