An obstruction to lifting to characteristic $0$

TL;DR

This paper introduces a novel obstruction to lifting smooth proper varieties from characteristic p to characteristic 0, utilizing Grothendieck's specialization homomorphism and properties of étale fundamental groups.

Contribution

It presents a new obstruction criterion based on specialization homomorphism and fundamental groups, advancing understanding of lifting problems in algebraic geometry.

Findings

Identifies a new obstruction to lifting varieties in characteristic p

Uses specialization homomorphism to analyze fundamental groups

Provides conditions under which lifting is impossible

Abstract

We introduce a new obstruction to lifting smooth proper varieties in characteristic $p>0$ to characteristic $0$. It is based on Grothendieck's specialization homomorphism and the resulting discrete finiteness properties of \'etale fundamental groups.