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Let $\mathcal{M}_{n,2n+2}$ be the coarse moduli space of CY manifolds arisingfrom a crepant resolution of double covers of $\mathbb{P}^n$ branched along$2n+2$ hyperplanes in general position. We show that the monodromy group of agood family for $\mathcal{M}_{n,2n+2}$ is Zariski dense in the correspondingsymplectic or orthogonal group if $n\geq 3$. In particular, the period map doesnot give a uniformization of any partial compactification of the coarse modulispace as a Shimura variety whenever $n\geq 3$. This disproves a conjecture ofDolgachev. As a consequence, the fundamental group of the coarse moduli spaceof $m$ ordered points in $\mathbb{P}^n$ is shown to be large once it is not apoint. Similar Zariski-density result is obtained for moduli spaces of CYmanifolds arising from cyclic covers of $\mathbb{P}^n$ branched along $m$hyperplanes in general position. A classification towards the geometricrealization problem of B. Gross for type $A$ bounded symmetric domains isgiven.

The monodromy groups of Dolgachev's CY moduli spaces are Zariski dense