A generalization of the Oort conjecture

The Oort conjecture (now a theorem of Obus–Wewers and Pop) states that if $k$ is an algebraically closed field of characteristic $p$, then any cyclic branched cover of smooth projective $k$-curves lifts to characteristic zero. This is equivalent to the local Oort conjecture, which states that all cy...

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Veröffentlicht in:Commentarii mathematici Helvetici 2017-01, Vol.92 (3), p.551-620
1. Verfasser: Obus, Andrew
Format: Artikel
Sprache:eng
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Zusammenfassung:The Oort conjecture (now a theorem of Obus–Wewers and Pop) states that if $k$ is an algebraically closed field of characteristic $p$, then any cyclic branched cover of smooth projective $k$-curves lifts to characteristic zero. This is equivalent to the local Oort conjecture, which states that all cyclic extensions of $k[[t]]$ lift to characteristic zero. We generalize the local Oort conjecture to the case of Galois extensions with cyclic $p$-Sylow subgroups, reduce the conjecture to a pure characteristic $p$ statement, and prove it in several cases. In particular, we show that $D_9$ is a so-called local Oort group.
ISSN:0010-2571
1420-8946
DOI:10.4171/CMH/419