A variety that cannot be dominated by one that lifts

Abstract

We prove a strong version of a theorem of Siu and Beauville on morphisms to higher genus curves, and we use it to show that if a variety X in characteristic p lifts to characteristic 0, then any morphism $X\to C$ to a curve of genus $g\ge 2$ can be lifted along. We use this to construct, for every prime p, a smooth projective surface X over ${\overline{\mathbf{F}}}_p$ that cannot be rationally dominated by a smooth proper variety Y that lifts to characteristic 0.