Communications in Number Theory and Physics

Volume 16 (2022)

Number 4

Unipotent extensions and differential equations (after Bloch–Vlasenko)

Pages: 801 – 849



Matt Kerr (Department of Mathematics, Washington University in St. Louis, Missouri, U.S.A.)


S. Bloch and M. Vlasenko recently introduced a theory of motivic Gamma functions, given by periods of the Mellin transform of a geometric variation of Hodge structure. They tie properties of these functions to the monodromy and asymptotic behavior of certain unipotent extensions of the variation. In this article, we further examine their Gamma functions and the related Apéry and Frobenius invariants of a VHS, and establish a relationship to motivic cohomology and solutions to inhomogeneous Picard–Fuchs equations.


variations of Hodge structure, periods, Picard–Fuchs equations, motivic gamma functions, Apéry constants, Frobenius constants, normal functions, motivic cohomology

2010 Mathematics Subject Classification

14C30, 14D07, 19E15, 32G20, 32S40

This work was partially supported by Simons Collaboration Grant 634268 and NSF Standard Grant DMS-2101482.

Received 14 January 2021

Accepted 21 September 2022

Published 21 October 2022