The local lifting problem for actions of finite groups on curves

Let $k$ be an algebraically closed field of characteristic $p > 0$. We study obstructions to lifting to characteristic 0 the faithful continuous action $\phi$ of a finite group $G$ on $k[[t]]$. To each such $\phi$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $\phi$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic 0 in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H \subset G$. We determine for which $G$ the KGB obstruction of every $\phi$ vanishes. We also consider analogous problems in which one requires only that an obstruction to lifting $\phi$ due to Bertin vanishes for some $\phi$, or for all sufficiently ramified $\phi$. These results provide evidence for a strengthening of Oort's lifting conjecture.