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On the monodromy of the moduli space of Calabi-Yau threefolds coming from eight planes in \({\mathbb{P}^3}\). (English) Zbl 1260.14048

A fundamental problem in geometry is to decide which moduli spaces of polarized algebraic varieties are embedded via period maps as Zariski open subsets of locally Hermitian symmetric domains. In this article, it is shown that the moduli space of Calabi-Yau threefolds arising from eight planes in \({\mathbb{P}}^3\) does not have this property. More concretely, the techniques of characteristic varieties that lead to the necessary conditions for this to happen are described. Then these conditions are applied to the moduli space of a particularly interesting Calabi-Yau threefolds, e.g., double octics ramified over an arrangement of eight planes in \({\mathbb{P}}^3\).
An arrangement \({\mathcal{A}}\) of eight planes in general position in \({\mathbb{P}}^3\) determines a double cover \(X\), which is a Calabi-Yau variety with singularities along \(28\) lines. A resolution \(\tilde{X}\) has \(\dim H^3(\tilde{X})=20\) and carries a weight \(3\) polarized Hodge structure with Hodge numbers \((1,9,9,1)\). Varying the arrangement \({\mathcal{A}}\) in a good family, one obtains an irreducible weight three \({\mathbb{Q}}\)-PVHS \({\mathbb{V}}\) of CY-type over a smooth nine dimensional base \(S\). The paper proves several theorems about \({\mathbb{V}}\).
Theorem 1. \({\mathbb{V}}\) does not factor canonically.
Theorem 2. Let \(s\in S\) be a base point and let \(\tau: \pi_1(S,s)\to \mathrm{Sp}(20,{\mathbb{Q}})\) be the monodromy representation associated to \({\mathbb{V}}\). Then the image of \(\tau\) is Zariski dense.
Corollary 3. The special Mumford-Tate group of a general member of the coarse moduli space \({\mathcal{M}}_{CY}\) of \(\tilde{X}\) is \(\mathrm{Sp}(20,{\mathbb{Q}})\).
Moreover, a natural sublocus, which is called the hyperelliptic locus, is studied, and it is shown that over such locus, the variation of Hodge structure is naturally isomorphic to wedge product of a variation of Hodge structures of weight one. It turns out that the hyperelliptic locus does not extend to a Shimura subvariety of type III (Siegel space) within the moduli space.
The methods used establishing these results are based mostly on classical Hodge theory, representation theory and computational commutative algebra. However, the proof of Theorem 2 depends on a new result about the tensor product decomposition of complex polarized variations of Hodge structures.

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)

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