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 to a curve of genus can be lifted along. We use this to construct, for every prime p, a smooth projective surface X over that cannot be rationally dominated by a smooth proper variety Y that lifts to characteristic 0.
"A variety that cannot be dominated by one that lifts." Duke Math. J. 170 (7) 1251 - 1289, 15 May 2021. https://doi.org/10.1215/00127094-2020-0055