This paper has appeared in Compositio Mathematica, 113, no. 3 (July
1998), 239-274, and so the dvi version has been removed. 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].