Oort's conjecture for split unitary Shimura varieties

Abstract: We prove that generically on the basic stratum of split unitary Shimura varieties, the universal abelian variety with endomorphism structure and polarization has automorphism group ${\pm 1}$, except for a few degenerate cases. This is a direct analogue of Oort's conjecture on automorphisms of supersingular abelian varieties. On the way, we explicitly compute the generic automorphism group of the universal $p$-divisible group in a given basic isogeny class.