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 \(\textbf{P}^{3}\) branched over eight hyperplanes in general position.
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Acknowledgements
The author thanks Bong H. Lian, Hui-Wen Lin, Chin-Lung Wang, and Shing-Tung Yau for their constant encouragement, their interests in this work, and providing him many useful comments. He thanks Dingxin Zhang for many valuable discussions. He thanks anonymous referees for their careful reading and valuable suggestions. He also would like to thank CMSA at Harvard for hospitality while working on this project.
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Lee, TJ. A note on periods of Calabi–Yau fractional complete intersections. Math. Z. 304, 60 (2023). https://doi.org/10.1007/s00209-023-03321-7
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DOI: https://doi.org/10.1007/s00209-023-03321-7