  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.
