The (local) lifting problem for curves

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.