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Semistable reduction for overconvergent \(F\)-isocrystals. I: Unipotence and logarithmic extensions. (English) Zbl 1144.14012
This article is the first in a series of four (“Semistable reduction for overconvergent \(F\)-isocrystals I, II [Compos. Math. 144, No. 3, 657–672 (2008; Zbl 1153.14015)], III and IV”) in which the author gives a proof of the “semistable reduction theorem” for overconvergent \(F\)-isocrystals (Shiho’s conjecture). Let \(X\) be a smooth variety over a field \(k\) of characteristic \(p\) and let \(\mathcal{E}\) be an overconvergent \(F\)-isocrystal. Shiho’s conjecture states that there always exists an alteration (with good properties) \(f : X_1 \to X\) with \(X_1 \subset \overline{X}_1\) where \(D = \overline{X}_1 \setminus X_1\) is a strict normal crossings divisor, such that \(f^\ast \mathcal{E}\) extends to a convergent \(F\)-log-isocrystal on \((\overline{X}_1,D)\). In the present paper, the author gives a criterion for the existence of a canonical logarithmic extension of \(\mathcal{E}\) to a smooth compactification \(\overline{X}\) of \(X\) whose complement is a strict normal crossings divisor. The author carefully discusses his method and motivation in the introduction of the paper and an outline of the subsequent three papers is given in §7.

14F30 \(p\)-adic cohomology, crystalline cohomology
14F40 de Rham cohomology and algebraic geometry
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