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Hyperbolicity notions for varieties defined over a non-Archimedean field. (English) Zbl 1440.32009
Summary: Firstly, we pursue the work of W. Cherry on the analogue of the Kobayashi semidistance \(d_{\text{CK}} \), which he introduced for analytic spaces defined over a non-Archimedean metrized field \(k\). We prove various characterizations of smooth projective varieties for which \(d_{\text{CK}}\) is an actual distance.
Secondly, we explore several notions of hyperbolicity for a smooth algebraic curve \(X\) defined over \(k\). We prove a non-Archimedean analogue of the equivalence between having a negative Euler characteristic and the normality of certain families of analytic maps taking values in \(X\).

32P05 Non-Archimedean analysis
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
Full Text: DOI Euclid
[1] [ACW08] T. T. H. An, W. Cherry, and J. T.-Y. Wang, Algebraic degeneracy of non-Archimedean analytic maps, Indag. Math. (N.S.) 19 (2008), no. 3, 481-492. · Zbl 1175.32012
[2] [BR10] M. Baker and R. Rumely, Potential theory and dynamics on the Berkovich projective line, Math. Surveys Monogr., 159, American Mathematical Society, Providence, RI, 2010. · Zbl 1196.14002
[3] [Ber90] V. G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Math. Surveys Monogr., 33, American Mathematical Society, Providence, RI, 1990. · Zbl 0715.14013
[4] [Ber94] V. G. Berkovich, Étale cohomology for non-Archimedean analytic spaces, Publ. Math. Inst. Hautes Études Sci. 78 (1994), 5-161, 1993. · Zbl 0804.32019
[5] [Ber06] F. Berteloot, Méthodes de changement d’échelles en analyse complexe, Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), no. 3, 427-483. · Zbl 1123.37019
[6] [Bos77] S. Bosch, Eine bemerkenswerte Eigenschaft der formellen Fasern affinoider Räume, Math. Ann. 229 (1977), no. 1, 25-45. · Zbl 0385.32008
[7] [Bro78] R. Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213-219. · Zbl 0416.32013
[8] [Che93] W. Cherry, Hyperbolic \(p\)-adic analytic spaces, Thesis, Yale University, 1993.
[9] [Che94] W. Cherry, Non-Archimedean analytic curves in Abelian varieties, Math. Ann. 300 (1994), no. 3, 393-404. · Zbl 0808.14019
[10] [Che96] W. Cherry, A non-Archimedean analogue of the Kobayashi semi-distance and its non-degeneracy on Abelian varieties, Illinois J. Math. 40 (1996), no. 1, 123-140. · Zbl 0853.32038
[11] [CTT16] A. Cohen, M. Temkin, and D. Trushin, Morphisms of Berkovich curves and the different function, Adv. Math. 303 (2016), 800-858. · Zbl 1375.14089
[12] [DR] S. Diverio and E. Rousseau, A survey on hyperbolicity of projective hypersurfaces, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2011. [On the title page: A survey on hyperbolicity of projective hypersurfaces].
[13] [Duc14] A. Ducros, La structure des courbes analytiques, preprint, 2014, http://webusers.imj-prg.fr/antoine.ducros/livre.html.
[14] [FKT12] C. Favre, J. Kiwi, and E. Trucco, A non-Archimedean Montel’s theorem, Compos. Math. 148 (2012), no. 3, 966-990. · Zbl 1267.32016
[15] [Gri15] N. Grieve, Diophantine approximation constants for varieties over function fields, Michigan Math. J. 67 (2018), no. 2, 371-404. · Zbl 1408.14087
[16] [JV18] A. Javanpeykar and A. Vezzani, Non-Archimedean hyperbolicity and applications, preprint, 2018.
[17] [Kob67] S. Kobayashi, Invariant distances on complex manifolds and holomorphic mappings, J. Math. Soc. Japan 19 (1967), 460-480. · Zbl 0158.33201
[18] [Kob98] S. Kobayashi, Hyperbolic complex spaces, Grundlehren Math. Wiss., 318, Springer-Verlag, Berlin, 1998.
[19] [KO75] S. Kobayashi and T. Ochiai, Meromorphic mappings onto compact complex spaces of general type, Invent. Math. 31 (1975), no. 1, 7-16. · Zbl 0331.32020
[20] [Lan87] S. Lang, Introduction to complex hyperbolic spaces, Springer-Verlag, 1987.
[21] [Nog92] J. Noguchi, Meromorphic mappings into compact hyperbolic complex spaces and geometric Diophantine problems, Internat. J. Math. 3 (1992), no. 2, 277-289. · Zbl 0759.32016
[22] [Pet09] C. Petsche, Nonarchimedean equidistribution on elliptic curves with global applications, Pacific J. Math. 242 (2009), no. 2, 345-375. · Zbl 1225.11081
[23] [Poi13] J. Poineau, Les espaces de Berkovich sont angéliques, Bull. Soc. Math. France 141 (2013), no. 2, 267-297. · Zbl 1314.14046
[24] [Poi14] J. Poineau, Sur les composantes connexes d’une famille d’espaces analytiques \(p\)-adiques, Forum Math. Sigma 2 (2014), e14. · Zbl 1350.14025
[25] [Rod16] R. Rodríguez Vázquez, Non-Archimedean normal families, preprint, 2016.
[26] [Roy71] H. L. Royden, Remarks on the Kobayashi metric, Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970), Lecture Notes in Math., 185, pp. 125-137, Springer, Berlin, 1971.
[27] [Sam66] P. Samuel, Lectures on old and new results on algebraic curves, Notes by S. Anantharaman. Tata Institute of Fundamental Research Lectures on Mathematics, 36, Tata Institute of Fundamental Research, Bombay, 1966. · Zbl 0165.24102
[28] [Tem15] M. Temkin, Introduction to Berkovich analytic spaces, Berkovich spaces and applications, Lecture Notes in Math., 2119, pp. 3-66, Springer, Cham, 2015. · Zbl 1317.14054
[29] [Thu05] A. Thuillier, Potential theory on curves in non-Archimedean geometry. Applications to Arakelov theory, Thesis, Université Rennes, 1, 2005.
[30] [Tsu79] R. Tsushima, Rational maps to varieties of hyperbolic type, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 95-100. · Zbl 0443.14006
[31] [Voi03] C. Voisin, On some problems of Kobayashi and Lang; algebraic approaches, Current developments in mathematics, 2003, pp. 53-125, Int. Press, Somerville, MA, 2003. · Zbl 1215.32014
[32] [Zal75] L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), no. 8, 813-817. · Zbl 0315.30036
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