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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.

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14F40 de Rham cohomology and algebraic geometry
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