Periods of Hodge cycles and special values of the Gauss' hypergeometric function

TL;DR

This paper computes periods of Fermat variety perturbations, explores Hodge cycles with arithmetic conditions, and links these to algebraic expressions of Gauss hypergeometric functions.

Contribution

It introduces a method to analyze Hodge cycles via periods and provides explicit algebraic expressions for hypergeometric functions.

Findings

Explicit bounds for the dimension of Hodge cycle subspaces

Examples illustrating the arithmetic conditions on Hodge cycles

Explicit algebraic expressions for certain hypergeometric functions

Abstract

We compute periods of perturbations of a Fermat variety. This allows us to consider a subspace of the Hodge cycles defined by "simple" arithmetic conditions. We explore some examples and give an upper bound for the dimension of this subspace. As an application, we find explicit expressions involving some Gauss' hypergeometric functions which are algebraic over the field of rational functions in one variable.