Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T04:20:03.613Z Has data issue: false hasContentIssue false
  • English
  • Français

Covers in p-adic analytic geometry and log covers II: cospecialization of the (p′)-tempered fundamental group in higher dimensions

Part of: (Co)homology theory Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  15 May 2012

Emmanuel Lepage
Emmanuel Lepage*
Affiliation:
D.M.A., E.N.S., 45 rue d’Ulm, 75005 Paris, France (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The tempered fundamental group of a p-adic variety classifies analytic étale covers that become topological covers for Berkovich topology after pullback by some finite étale cover. This paper constructs cospecialization homomorphisms between the (p′) versions of the tempered fundamental group of the fibers of a smooth morphism with polystable reduction. We study the question for families of curves in another paper. To construct them, we will start by describing the pro-(p′) tempered fundamental group of a smooth and proper variety with polystable reduction in terms of the reduction endowed with its log structure, thus defining tempered fundamental groups for log polystable varieties.

Keywords

fundamental groupsBerkovich spacesspecialization

MSC classification

Secondary: 14G22: Rigid analytic geometry 14F35: Homotopy theory; fundamental groups
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 5 , September 2012 , pp. 1443 - 1482
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[And03]André, Y., On a geometric description of and a p-adic avatar of , Duke Math. J. 119 (2003), 1–39.CrossRefGoogle Scholar
[Ber94]Berkovich, V. G., Vanishing cycles for formal schemes, Invent. Math. 115 (1994), 539–571.CrossRefGoogle Scholar
[Ber99]Berkovich, V. G., Smooth p-adic analytic spaces are locally contractible, Invent. Math. 137 (1999), 1–84.CrossRefGoogle Scholar
[Ber04]Berkovich, V. G., Smooth p-adic analytic spaces are locally contractible. II, in Geometric aspects of Dwork theory (Walter de Gruyter, 2004), 293–370.CrossRefGoogle Scholar
[SGA1]Grothendieck, A. (ed.), Revêtements étales et groupe fondamental (SGA1), Lecture Notes in Mathematics, vol. 224 (Springer, Berlin, 1971).CrossRefGoogle Scholar
[EGA4]Grothendieck, A. and Dieudonné, J., Eléments de geométrie algébrique. IV: Etude locale des schémas et des morphismes de schémas (quatrième partie), Publ. Math. Inst. Hautes Études Sci. 31 (1967), 5–361.Google Scholar
[Ill02]Illusie, L., An overview of the works of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology, in Cohomologies p-adiques et applications arithmétiques (II), Astérisque, vol. 279 (Société Mathématique de France, 2002), 271–322.Google Scholar
[Kat94]Kato, K., Toric singularities, Amer. J. Math. 116 (1994), 1073–1099.CrossRefGoogle Scholar
[Kis00]Kisin, M., Prime to p fundamental groups and tame Galois action, Ann. Inst. Fourier 50 (2000), 1099–1126.CrossRefGoogle Scholar
[Lep09]Lepage, E., Coverings in p-adic analytic geometry and log coverings I: cospecialization of the (p′)-tempered fundamental group for a family of curves, Ann. Inst. Fourier, to appear, available at arXiv:0909.2805.Google Scholar
[Moc06]Mochizuki, S., Semi-graphs of anabelioids, Publ. Res. Inst. Math. Sci. 42 (2006), 221–322.CrossRefGoogle Scholar
[Niz06]Nizioł, W., Toric singularities: log-blow-ups and global resolutions, J. Algebraic Geom. 15 (2006), 1–29.CrossRefGoogle Scholar
[Ogu]Ogus, A., Lectures on logarithmic algebraic geometry, notes préliminaires, http://math.berkeley.edu/∼ogus/preprints/log_book/logbook.pdf.Google Scholar
[Tsu97]Tsuji, T., Saturated morphisms of logarithmic schemes, 1997 (unpublished).Google Scholar