Some Identities For Periods of Hulek-Verrill Threefolds

TL;DR

The paper studies the Hulek-Verrill Calabi–Yau threefolds by exploiting their birational equivalence to fibred products of elliptic surfaces. It develops a framework where threefold periods are computed as integrals of products of elliptic periods over contours in the base, and verifies this approach numerically for several instances. By analyzing the AESZ 34 Picard–Fuchs system, monodromy of elliptic periods, and vanishing cycles at special parameter values $\varphi=1/9$ and $1/25$, the authors construct integral-monodromy period vectors $\Pi(\varphi)$ and relate them to explicit elliptic-period integrals. They also identify a holomorphic $3$-form period at the large-complex-structure point $\varphi=0$ via a $T^3$-cycle, consistent with a geometric SYZ-like fibration by constant-phase contours on the base. These results yield concrete identities connecting high-dimensional period data to products of elliptic periods, enriching the understanding of the Hulek–Verrill family and its moduli.

Abstract

We study the Hulek--Verrill families of Calabi--Yau threefolds. They are birationally equivalent to fibred products of elliptic surfaces, so we expect to be able to compute periods on these threefolds by integrating products of elliptic periods over a contour on $\mathbb{P}^1$. We numerically verify this in several examples. This article was submitted to MATRIX Annals (2024) for inclusion in the proceedings of the conference "The Geometry of Moduli Spaces in String Theory", held 2--13 September 2024.