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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
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[1] Abramovich, D.; Karu, K., Weak semistable reduction in characteristic 0, Inventiones Mathematicae, 139, 241-273, (2000) · Zbl 0958.14006
[2] Bosch, S.; Lütkebohmert, W., Formal and rigid geometry. I. rigid spaces, Mathematische Annalen, 295, 291-317, (1993) · Zbl 0808.14017
[3] Bosch, S.; Lütkebohmert, W.; Raynaud, M., Formal and rigid geometry. III. the relative maximum principle, Mathematische Annalen, 302, 1-29, (1995) · Zbl 0839.14013
[4] Conrad, B., Deligne’s notes on Nagata compactifications, Journal of the Ramanujan Mathematical Society, 22, 205-257, (2007) · Zbl 1142.14001
[5] Jong, A. J., Smoothness, semistability and alterations, institut des hautes études scientifiques, Publications Mathématiques, 83, 51-93, (1996) · Zbl 0916.14005
[6] Elkik, R., Solutions d’équations à coefficients dans un anneau hensélien, Annales Scientifiques de l’École Normale Supérieure, 6, 553-603, (1973) · Zbl 0327.14001
[7] A. Grothendieck, Éléments de géométrie algébrique. I-IV, Institut des Hautes Études Scientifiques. Publications Mathématiques (1960-1967). · Zbl 0118.36206
[8] Gubler, W.; Soto, A., Classification of normal toric varieties over a valuation ring of rank one, Documenta Mathematica, 20, 171-198, (2015) · Zbl 1349.14161
[9] Hartl, U. T., Semi-stable models for rigid-analytic spaces, Manuscripta Mathematica, 110, 365-380, (2003) · Zbl 1099.14010
[10] Illusie, L.; Temkin, M., Exposé X: gabber’s modification theorem (log smooth case), Astérisque, 364, 167-212, (2014) · Zbl 1327.14072
[11] G. Kempf, F. F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal Embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer, Berlin-New York, 1973. · Zbl 0271.14017
[12] Orgogozo, F., Exposé II: topologies adaptées à l’uniformisation locale, Astérisque, 364, 21-36, (2014) · Zbl 1320.14004
[13] Raynaud, M.; Gruson, L., Critères de platitude et de projectivité. techniques de “platification” d’un module, Inventiones Mathematicae, 13, 1-89, (1971) · Zbl 0227.14010
[14] Temkin, M., Desingularization of quasi-excellent schemes in characteristic zero, Advances in Mathematics, 219, 488-522, (2008) · Zbl 1146.14009
[15] Temkin, M., Stable modification of relative curves, Journal of Algebraic Geometry, 19, 603-677, (2010) · Zbl 1211.14032
[16] Temkin, M., Relative Riemann-Zariski spaces, Israel Journal of Mathematics, 185, 1-42, (2011) · Zbl 1273.14007
[17] Temkin, M., Metrization of differential pluriforms on berkovich analytic spaces, 195-285, (2016), Simons Symposia · Zbl 1360.32019
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