Unipotent extensions and differential equations (after Bloch-Vlasenko)

TL;DR

This paper explores the properties of motivic Gamma functions associated with variations of Hodge structure, linking them to monodromy, motivic cohomology, and solutions of inhomogeneous Picard-Fuchs equations.

Contribution

It extends the theory of motivic Gamma functions by analyzing their relation to unipotent extensions, monodromy, and motivic cohomology, providing new insights into their structure.

Findings

Relation between motivic Gamma functions and monodromy

Connection to motivic cohomology established

Solutions to inhomogeneous Picard-Fuchs equations analyzed

Abstract

S. Bloch and M. Vlasenko recently introduced a theory of \emph{motivic Gamma functions}, given by periods of the Mellin transform of a geometric variation of Hodge structure, which they tie to the monodromy and asymptotic behavior of certain unipotent extensions of the variation. Here we further examine these Gamma functions and the related \emph{Ap\'ery and Frobenius invariants} of a VHS, and establish a relationship to motivic cohomology and solutions to inhomogeneous Picard-Fuchs equations.