ag.algebraic geometry - projective subvarieties of the moduli space of abelian varieties

Let me comment on Felipe Voloch's answer. The supersingular locus (i.e. the locus of $p$-rank zero) is indeed complete: it is a closed subvariety of $overline A_g$ (choose a toroidal compactification) because the $p$-rank is lower semicontinuous, and it does not meet the boundary because a torus has positive $p$-rank. Moreover it will in fact have the largest possible dimension of a complete subvariety of $A_g$ (in positive characteristic); its dimension is $g(g-1)/2$. Indeed if there exists a complete subvariety of dimension $d$ and $eta$ is an ample divisor class, then $eta^d neq 0$. Now $lambda_1$ (1st Chern class of Hodge bundle) is ample and $lambda_1^{1+g(g-1)/2}$ vanishes. For this see

van der Geer, Gerard. Cycles on the moduli space of abelian varieties. Moduli of curves and abelian varieties, 65--89, Aspects Math., E33, Vieweg, Braunschweig, 1999. MR1722539

In characteristic zero there is no complete subvariety of $A_g$ of this dimension. Hence the largest dimension of a compact subvariety depends on the characteristic!

Keel, Sean; Sadun, Lorenzo. Oort's conjecture for $A_ gotimesBbb C$. J. Amer. Math. Soc. 16 (2003), no. 4, 887--900 (electronic). MR1992828

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