The Oort conjecture on lifting covers of curves (Q2251005)

This paper completes the proof of the \textit{Oort conjecture}, which states that any cyclic branched cover of smooth projective curves in characteristic \(p\) lifts to characteristic zero, by reducing the general case to a special case that was proven in [the reviewer and \textit{S. Wewers}, Ann. Math. (2) 180, No. 1, 233--284 (2014; Zbl 1307.14042)]. In particular, it is well-known that the Oort conjecture is equivalent to its local version, which states that if \(k\) is an algebraically closed field of characteristic \(p\), then for any cyclic \(G\)-extension \(k[[z]]/k[[t]]\), there is a DVR \(R\) with residue field \(k\) such that \(k[[z]]/k[[t]]\) lifts to a \(G\)-extension \(R[[Z]]/R[[T]]\). The result of Obus-Wewers above proves this conjecture under the assumption of \textit{no essential ramification}, which says that the breaks \((u_1, \ldots, u_n)\) in the higher ramification filtration for the upper numbering of the extension satisfy \(u_i < u_{i-1} + p\) for all \(i\). This paper completes the proof of the (local) Oort conjecture by reducing it to the ``no essential ramification'' case. The key result is a so-called ``characteristic \(p\) Oort conjecture,'' which loosely states that any cyclic extension \(k[[z]]/k[[t]]\) has an \textit{equicharacteristic} deformation to a cyclic extension of \(k[[t]][[\varpi]]\) where the generic fiber has no essential ramification (although it is generally branched at multiple maximal ideals). This deformation is constructed explicitly using Artin-Schreier-Witt theory. The author then proves a global version of this conjecture, stating that a branched Galois cover of \(\mathbb{P}^1_k\) with cyclic inertia groups deforms to a branched cover of \(\mathbb{P}^1_{k((\varpi))}\) with cyclic inertia groups and no essential ramification. The result of Obus-Wewers shows that this (deformed) cover lifts to characteristic zero. A careful algebraic geometry argument of the author now shows that one can obtain a lift of the original cover from this.