The local lifting problem for actions of finite groups on curves

TL;DR

This paper investigates the obstructions to lifting finite group actions on formal power series rings from characteristic p to characteristic 0, characterizing when such liftings are possible and providing evidence for a strengthened version of Oort's conjecture.

Contribution

It characterizes when the Katz-Gabber-Bertin (KGB) obstruction vanishes for all group actions, advancing understanding of the local lifting problem for finite groups on curves.

Findings

Determines groups for which the KGB obstruction always vanishes.

Analyzes conditions under which Bertin's obstruction vanishes.

Provides evidence supporting a strengthened form of Oort's lifting conjecture.

Abstract

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.