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The monodromy groups of Dolgachev's CY moduli spaces are Zariski dense
Let be the coarse moduli space of CY manifolds arising
from a crepant resolution of double covers of branched along
hyperplanes in general position. We show that the monodromy group of a
good family for is Zariski dense in the corresponding
symplectic or orthogonal group if . In particular, the period map does
not give a uniformization of any partial compactification of the coarse moduli
space as a Shimura variety whenever . This disproves a conjecture of
Dolgachev. As a consequence, the fundamental group of the coarse moduli space
of ordered points in 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 branched along
hyperplanes in general position. A classification towards the geometric
realization problem of B. Gross for type bounded symmetric domains is
given.Comment: 48 page
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