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On the monodromy of the moduli space of Calabi–Yau threefolds coming from eight planes in \({\mathbb{P}^3}\)

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Abstract

It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi–Yau threefolds coming from eight planes in \({\mathbb{P}^3}\) does not have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is naturally isomorphic to wedge product of a variation of Hodge structures of weight one. It turns out the hyperelliptic locus does not extend to a Shimura subvariety of type III (Siegel space) within the moduli space. Besides general Hodge theory, representation theory and computational commutative algebra, one of the proofs depends on a new result on the tensor product decomposition of complex polarized variations of Hodge structures.

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Authors and Affiliations

  1. Mathematisches Institut, Universität München, 80333, Munich, Germany

    Ralf Gerkmann

  2. Department of Mathematics, East China Normal University, Shanghai, 200062, People’s Republic of China

    Mao Sheng

  3. Institut für Mathematik, Universität Mainz, 55099, Mainz, Germany

    Duco van Straten & Kang Zuo

Authors
  1. Ralf Gerkmann
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  2. Mao Sheng
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  3. Duco van Straten
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  4. Kang Zuo
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Corresponding author

Correspondence to Ralf Gerkmann.

Additional information

This work was supported by the SFB/TR 45 Periods, Moduli Spaces and Arithmetic of Algebraic Varieties of the DFG (German Research Foundation).

M. Sheng is supported by a postdoctoral fellowship in the East China Normal University and is also partially supported by the Program for Changjiang Scholars and Innovative Research Team in University.

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Gerkmann, R., Sheng, M., van Straten, D. et al. On the monodromy of the moduli space of Calabi–Yau threefolds coming from eight planes in \({\mathbb{P}^3}\) . Math. Ann. 355, 187–214 (2013). https://doi.org/10.1007/s00208-012-0779-z

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  • DOI: https://doi.org/10.1007/s00208-012-0779-z

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