The (local) lifting problem for curves

TL;DR

This paper surveys the progress on the local lifting problem for curves in characteristic p, focusing on when automorphisms can be lifted to characteristic zero, especially in the cyclic case related to the Oort conjecture.

Contribution

It provides an overview of recent advances and ongoing research on the local lifting problem, highlighting the case of cyclic automorphism groups and the Oort conjecture.

Findings

Progress made on the local lifting problem for cyclic groups

The Oort conjecture remains a central open question

Current work explores solutions for broader classes of automorphisms

Abstract

The lifting problem that we consider asks: given a smooth curve in characteristic p and a group of automorphisms, can we lift the curve, along with the automorphisms, to characteristic zero? One can reduce this to a local question (the so-called local lifting problem) involving continuous group actions on formal power series rings. In this expository article, we overview much of the progress that has been made toward determining when the local lifting problem has a solution, and we give a taste of the work currently being undertaken. Of particular interest is the case when the group of automorphisms is cyclic. In this case the lifting problem is expected to be solvable---this is the Oort conjecture.