Abstract If k is an algebraically closed field of characteristic p, and if (Y, H) is a proper, smooth, integral curve over k with a group H of automorphisms, the lifting problem asks whether (Y, H) can be lifted to characteristic zero. It turns out that there is a local-global principle, which states that the lifting problem can be reduced to a local lifting problem in a formal neighborhood of each point of Y where H acts with non-trivial inertia G. Namely, if G acts on k[[z]] by continuous k-automorphisms, does the action lift to characteristic zero?
After giving an introduction to the lifting problem, we will discuss the local lifting problem in the case of a cyclic group G. We prove that the local lifting problem in this case can be solved, subject to a condition on the higher ramification filtration of the action. The lecture of Florian Pop will show how to eliminate this condition, thus solving the local lifting problem for cyclic groups G, also known as the Oort Conjecture.