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

TL;DR

This paper investigates the monodromy and geometric properties of the moduli space of Calabi-Yau threefolds derived from eight planes in projective three-space, revealing it does not embed as a Hermitian symmetric domain and analyzing its monodromy group and special loci.

Contribution

It proves that the moduli space does not embed as a Zariski open subset of a Hermitian symmetric domain and introduces new results on tensor product decompositions of Hodge structures.

Findings

Monodromy group is Zariski dense in the symplectic group

Hyperelliptic locus does not extend to a Shimura subvariety

Moduli space lacks Hermitian symmetric domain structure

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 {\em 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.