Arithmetic of degenerating principal variations of Hodge structure: examples arising from mirror symmetry and middle convolution

TL;DR

This paper investigates the arithmetic properties of degenerating Hodge structures in various geometric examples, providing evidence for a conjecture and exploring the role of Mumford-Tate groups and boundary components.

Contribution

It offers new evidence supporting a conjecture on the arithmetic of limiting mixed Hodge structures and analyzes examples from mirror symmetry and middle convolution.

Findings

Confirmed the conjecture for hypergeometric Calabi-Yau threefolds.

Extended analysis to non-hypergeometric examples in dimensions 1 to 6.

Identified the Mumford-Tate group of type G2 as crucial in the 6-fold case.

Abstract

We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi-Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions 1 through 6 arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (of type G2) of the family of 6-folds, and the theory of boundary components of Mumford-Tate domains.