A note on periods of Calabi--Yau fractional complete intersections

TL;DR

This paper proves that the GKZ D-module associated with Calabi-Yau fractional complete intersections fully captures their period integrals, establishing an equivalence with the Picard--Fuchs system and providing explicit formulas for certain threefolds.

Contribution

It demonstrates the completeness of the GKZ D-module for Calabi-Yau fractional complete intersections and links it to the Picard--Fuchs system, with explicit period formulas for specific threefolds.

Findings

GKZ D-module is complete for these intersections

Solutions are exactly the period integrals

Explicit formulas for Calabi-Yau threefold periods

Abstract

We prove that the GKZ $\mathscr{D}$-module $\mathcal{M}_{A}^{\beta}$ arising from Calabi--Yau fractional complete intersections in toric varieties is complete, i.e., all the solutions to $\mathcal{M}_{A}^{\beta}$ are period integrals. This particularly implies that $\mathcal{M}_{A}^{\beta}$ is equivalent to the Picard--Fuchs system. As an application, we give explicit formulae of the period integrals of Calabi--Yau threefolds coming from double covers of $\mathbf{P}^{3}$ branch over eight hyperplanes in general position.