Consider the set $M$ of all (rank $g$) lattices in $g$-dimensional complex affine space $C^g$.
Does M identify in some way with Siegel upper half space $H_g$?
Let's say a lattice has CM if it has endomorphism ring bigger than the integers. (This is not the correct definition.)
Is the set of CM lattices dense?
What if we mod out $M$ by some group?