# A variety that cannot be dominated by one that lifts

## TL;DR

This paper proves a theorem relating to morphisms from varieties to higher genus curves, showing lifting properties in characteristic p and constructing examples of surfaces that cannot be dominated by varieties lifting to characteristic zero.

## Contribution

It establishes a precise version of a theorem on morphisms to higher genus curves and demonstrates the existence of surfaces in characteristic p that cannot be dominated by liftable varieties.

## Key findings

- Proves a precise version of Siu and Beauville's theorem.
- Shows that morphisms to higher genus curves lift when the variety lifts.
- Constructs surfaces in characteristic p not dominated by liftable varieties.

## Abstract

We prove a precise version of a theorem of Siu and Beauville on morphisms to higher genus curves, and 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 $g \geq 2$ can be lifted along. We use this to construct, for every prime $p$, a smooth projective surface $X$ over $\bar{\mathbb F}_p$ that cannot be rationally dominated by a smooth proper variety $Y$ that lifts to characteristic $0$.