Oort's conjecture on supersingular abelian varieties in odd characteristic

Abstract

Oort's conjecture asserts that for any $g\ge 2$ and any prime $p$, every geometric generic member in the supersingular locus ${\mathcal{S}}_g$ has automorphism group $\{\pm 1\}$. This has been proved very recently by Viehmann in full generality, with previously known counterexamples for $p=2$ and $g=2,3$. We construct, for any $g\ge 3$, a closed subvariety of dimension $g-1$ which contains an open dense subset $\mathcal{U}$ of $a$-invariant $g-2$ such that $\mathcal{U}$ meets every irreducible component of ${\mathcal{S}}_g$ and every geometric point in $\mathcal{U}$ has automorphism group $\{\pm 1\}$ if $p>2$. This gives an independent proof of Oort's conjecture for $p>2$. When $g>3$, so $a=g-2>1$, this provides complementary information to the $a=1$ locus investigated in Viehmann's work on how the automorphism groups interact with the geometry.

13 pages, comments welcome