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

Abstract Let M n , 2 n + 2 be the coarse moduli space of CY manifolds arising from a crepant resolution of double covers of P n branched along 2 n + 2 hyperplanes in general position. We show that the monodromy group of a good family for M n , 2 n + 2 is Zariski dense in the corresponding symplectic or orthogonal group if n ≥ 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 ≥ 3 . This disproves a conjecture of Dolgachev. As a consequence, the fundamental group of the coarse moduli space of m ordered points in P n 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 P n 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.