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

We prove that the GKZ $\mathcal{D}$-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.