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

TL;DR

This paper proves that the monodromy groups of certain Calabi-Yau moduli spaces are Zariski dense, disproving a conjecture and revealing their complex structure, with implications for the geometry of moduli spaces and period maps.

Contribution

It demonstrates Zariski density of monodromy groups for Dolgachev's CY moduli spaces when n≥3, challenging previous conjectures and extending results to related CY moduli spaces.

Findings

Monodromy groups are Zariski dense in symplectic or orthogonal groups for n≥3.

The period map does not uniformize any partial compactification as a Shimura variety for n≥3.

Fundamental groups of moduli spaces of points in projective space are large.

Abstract

Let $\mathcal{M}_{n,2n+2}$ be the coarse moduli space of CY manifolds arising from 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 a good family for $\mathcal{M}_{n,2n+2}$ is Zariski dense in the corresponding symplectic or orthogonal group if $n\geq 3$. In particular, the period map does not give a uniformization of any partial compactification of the coarse moduli space as a Shimura variety whenever $n\geq 3$. This disproves a conjecture of Dolgachev. As a consequence, the fundamental group of the coarse moduli space of $m$ ordered points in $\mathbb{P}^n$ 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 $\mathbb{P}^n$ 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.