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

Let Mn,2n+2Mn,2n+2 be the coarse moduli space of CY manifolds arising from a crepant resolution of double covers of PnPn branched along 2n+22n+2 hyperplanes in general position. We show that the monodromy group of a good family for Mn,2n+2Mn,2n+2 is Zariski dense in the corresponding symplectic or orthogonal group if n≥3n≥3. In particular, the period map does not give a uniformization of any partial compactification of the coarse moduli space as a Shimura variety whenever n≥3n≥3. This disproves a conjecture of Dolgachev. As a consequence, the fundamental group of the coarse moduli space of m   ordered points in PnPn is shown to be large once it is not a point. Similar Zariski-density result is obtained for moduli spaces of CY manifolds arising from cyclic covers of PnPn branched along m hyperplanes in general position. A classification towards the geometric realization problem of B. Gross for type A bounded symmetric domains is given.