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Altered local uniformization of Berkovich spaces. (English) Zbl 1419.14032
Let $$k$$ be a complete non-archimedean valued field and $$X$$ be a compact quasi-smooth strictly $$k$$-analytic space. The article under review shows that, after performing a finite extension of the base field and a quasi-étale covering, one may always find a space that admits a strictly semistable formal model. (The word “altered” in the title of the article refers to the fact that one can conjecturally replace the quasi-étale morphism by an embedding of a disjoint union of affinoid domains.) This theorem is used by the author in [in: Nonarchimedean and tropical geometry. Based on two Simons symposia, Island of St. John, March 31 – April 6, 2013 and Puerto Rico, February 1–7, 2015. Cham: Springer. 195–285 (2016; Zbl 1360.32019)] where he investigated pluricanonical forms on quasi-smooth Berkovich spaces.
In the case where the base field $$k$$ is discretely valued, the theorem had formerly been proved by U. T. Hartl [Manuscr. Math. 110, No. 3, 365–380 (2003; Zbl 1099.14010)], building on techniques introduced by A. J. de Jong in [Publ. Math., Inst. Hautes Étud. Sci. 83, 51–93 (1996; Zbl 0916.14005)] and involving moduli spaces of proper curves.
In order to go beyond the discretely valued case, the author uses the stable modification theorem from [J. Algebr. Geom. 19, No. 4, 603–677 (2010; Zbl 1211.14032)] that applies to arbitrary relative curves, with no properness assumption. This allows him to first prove an algebraic version of the result (Section 2), whereas Hartl’s method needed to use analytic methods from the start in order to obtain relative compactifications. Formal and analytic versions of the result are then deduced in Sections 3 and 4.

##### MSC:
 14G22 Rigid analytic geometry
##### Keywords:
local uniformization; Berkovich spaces
Full Text:
##### References:
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