The local lifting problem for actions of finite groups on curves

Ted Chinburg; Robert Guralnick; David Harbater

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arXiv · 0903.0293

The local lifting problem for actions of finite groups on curves

Ted Chinburg·Robert Guralnick·David Harbater

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Abstract

Let $k$ be an algebraically closed field of characteristic $p > 0$. We study obstructions to lifting to characteristic 0 the faithful continuous action $ϕ$ of a finite group $G$ on $k[[t]]$. To each such $ϕ$ 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 $ϕ$ 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 $ϕ$ vanishes. We also consider analogous problems in which one requires only that an obstruction to lifting $ϕ$ due to Bertin vanishes for some $ϕ$, or for all sufficiently ramified $ϕ$. These results provide evidence for a strengthening of Oort's lifting conjecture.

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Ted Chinburg, Robert Guralnick, David Harbater. 2009-10-06. The local lifting problem for actions of finite groups on curves. https://arxiv.org/abs/0903.0293

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