Let $(C,G)$ be a smooth integral proper curve of genus $g$ over an algebraically closed field $k$ of chararacteristic $p>0$ and $G$ be a finite group of automorphisms of $C.$ In this paper we study the problem of obtaining a smooth galois lifting of $(C,G)$ to characteristic 0. It is well known that over a field of characteristic $p,$ contrary to the characteristic $0$ case, Hurwitz's bound $|G|\leq 84(g-1)$ doesn't hold in general; in such cases this gives an obstruction to obtaining a smooth galois lifting.

We shall give new obstructions of local nature to the lifting problem, even in the case where $G$ is abelian. In the case where the inertia groups are $p^ae$-cyclic with $a\leq 2$ and $(e,p)=1,$ we shall prove that smooth galois liftings exist over $W(k)[\root p^2\of 1].$

Last update: February 4, 1999