Convolution of Picard-Fuchs equations

TL;DR

This paper develops an algorithmic approach to compute new fuchsian differential equations through convolution, generalizing classical methods and relating cohomology groups with solutions of these equations.

Contribution

It introduces explicit generators for a cohomology group derived from fuchsian equations and relates it to local system cohomology, extending classical convolution techniques.

Findings

Provides explicit generators for cohomology groups from fuchsian solutions

Describes an algorithm to generate new fuchsian equations from multi-parameter cases

Generalizes classical convolution of solutions for fuchsian differential equations

Abstract

We determine explicit generators for a cohomology group constructed from a solution of a fuchsian linear differential equation and describe its relation with cohomology groups with coefficients in a local system. In the parameterized case, this yields into an algorithm which computes new fuchsian differential equations from those depending on multi-parameters. This generalizes the classical convolution of solutions of fuchsian differential equations.