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Cyclic extensions and the local lifting problem. (English) Zbl 1307.14042
It was known that a projective, smooth, connected curve \(\bar{Y}\) defined over an algebraically closed field \(k\) of positive characteristic \(p\) can be lifted to characteristic zero, i.e. there is a discrete valuation ring \(R\) of characteristic zero with residue field \(k\) and a relative \(R\) curve \(Y\) with special fibre \(\bar{Y}\).
F. Oort [“Some questions in algebraic geometry” (1995), http://www.math.uu.nl/~oort0109/A-Qnew.ps] proposed a similar problem, on whether a Galois cover \(\bar{f}: \bar{Y} \rightarrow \bar{X}\) with Galois group \(\Gamma\) can be lifted in the same way to a Galois cover of relative \(R\)-curves. For a general group \(\Gamma\) this is not possible, there are obstructions based on the size of \(\Gamma\) with respect to the genus, the Bertin obstruction [J. Bertin, C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 1, 55–58 (1998; Zbl 0952.14018)] and the Hurzitz tree obstruction [L. H. Brewis and S. Wewers, Math. Ann. 345, No. 3, 711–730 (2009; Zbl 1222.14045)]. But for cyclic groups all these obstructions vanish. The Oort conjecture claims that a cyclic group cover of order \(p^n\) can always be lifted to characteristic zero.
The breakthrough result of the authors completes the result of F. Pop [Ann. Math. (2) 180, No. 1, 285–322 (2014; Zbl 1311.12003)] and the two articles together provide a full proof of the Oort conjecture.
The proof uses the local nature of the lifting problem expressed in terms of the local-global lifting property and restates the problem in terms of a lifting problem of formal power series in terms of rigid analytic geometry. This problem is restated in the language of characters, i.e. elements in \(H^1(\mathbb{K},\mathbb{Z}/p^n \mathbb{Z})\), where \(\mathbb{K}\) is the function field of the curve \(X\) in the generic fibre. Such a character corresponds to a branched cover \(Y \rightarrow X\) and several invariants are attached to a character, like three types of Swan conductors, which measure how bad is the reduction of a cover. The proof uses an induction process based on a detailed study of \(\mathbb{Z}/p\mathbb{Z}\)-extensions which are the building blocks of the induction.

MSC:
14H37 Automorphisms of curves
12F10 Separable extensions, Galois theory
11G20 Curves over finite and local fields
12F15 Inseparable field extensions
13B05 Galois theory and commutative ring extensions
13F35 Witt vectors and related rings
14G22 Rigid analytic geometry
14H30 Coverings of curves, fundamental group
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