Stability of vector bundles and reduction mod $p$

Abstract

Let $E$ be a vector bundle on a normal complex projective variety. We prove that slope stability of $\mathop{Sym}^2E$ and Zariski density of the reductions of $E$ modulo $p$ that are not strongly geometrically stable imply that the bundle $E$ is projectively unitary flat. This generalizes Shepherd--Barron's result from rank $2$ to arbitrary rank. We also show that if $E$ has rank $\le 3$ we can replace stability of $\mathop{Sym}^2E$ by stability of $E$ but we cannot do so if $E$ has rank $4$. We also generalize these results to reflexive sheaves on some mildly singular normal complex projective varieties.