Let $A\_g$ be the moduli space of principally polarised abelian varieties of dimension $g$ over the complex numbers. (EDIT: I mean the coarse moduli space.) Is this smooth?


Since $A\_g$ is the quotient of Siegel upper half space by $\mathrm{Sp}\_{2g}(\mathbb{Z})$ and this group has torsion elements, it seems likely that the answer is no.
On the other hand $\mathrm{SL}\_2(\mathbb{Z})$ already has torsion elements and $A\_1$ is smooth.


Any attempt I make to search for information on this question only leads to information about the smoothness of the boundaries in compactifications of $A\_g$.



Is the moduli space of ppAVs smooth?
The answer is no, it is not smooth for any $g \geq 2$. For $g \geq 3$ the singular locus is precisely the locus of PPAVs with automorphism group greater than $\pm \mathrm{id}$, as proven in Oort, Frans: "Singularities of coarse moduli schemes". For $g=2$ there is IIRC a unique singular point which is in $M\_2$ (the open subvariety of $A\_2$ of Jacobians), I think this is in Igusa's paper "Arithmetic variety of moduli for genus two".


