Some generalizations of Oort's conjecture

Abstract: For a prime $p\geq 5$, let $\mathscr S_g$ be the moduli space over $\overline{\mathbb F}<em>p$ of $g$-dimensional principally polarized supersingular abelian varieties. We show that each of the following loci contains an open dense subscheme on which the principally polarized abelian varieties have automorphism group ${\pm1}$: (i) certain supersingular Ekedahl--Oort strata when $g$ is even, (ii) the loci in $\mathscr S_g$ with non-supersingular Ekedahl--Oort invariants of positive Coxeter type when $g\geq 3$, and (iii) the locus in $\mathscr S_g$ with $a$-number at least $2$ when $g\geq 4$. Consequently, for $g\geq 4$, the complement in $\mathscr S_g$ of the open locus where the automorphism group is ${\pm1}$ has codimension at least $2$. These results confirm Oort's conjecture for $p\geq 5$. We reduce them to statements about affine Deligne--Lusztig varieties for $\operatorname{GSp}</em>{2g}$ and prove analogues of (ii) for $\operatorname{GL}<em>{2g}$ and $\operatorname{GSO}</em>{4m}$.