Non-archimedean hyperbolicity and applications
-
Ariyan Javanpeykar
and
Alberto Vezzani
Abstract
Inspired by the work of Cherry, we introduce and study a new notion of Brody hyperbolicity for rigid analytic varieties over a non-archimedean field K of characteristic zero. We use this notion of hyperbolicity to show the following algebraic statement: if a projective variety admits a non-constant morphism from an abelian variety, then so does any specialization of it. As an application of this result, we show that the moduli space of abelian varieties is K-analytically Brody hyperbolic in equal characteristic 0. These two results are predicted by the Green–Griffiths–Lang conjecture on hyperbolic varieties and its natural analogues for non-archimedean hyperbolicity. Finally, we use Scholze’s uniformization theorem to prove that the aforementioned moduli space satisfies a non-archimedean analogue of the “Theorem of the Fixed Part” in mixed characteristic.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: SFB/Transregio 45
Funding source: Agence Nationale de la Recherche
Award Identifier / Grant number: ANR-14-CE25-0002
Award Identifier / Grant number: ANR-18-CE40-0017
Funding statement: The first named author gratefully acknowledges support from SFB/Transregio 45, and the second named author from the Agence Nationale de la Recherche, projects ANR-14-CE25-0002 and ANR-18-CE40-0017.
Acknowledgements
We thank Jackson Morrow for very helpful discussions. We thank Peter Scholze for a very helpful and inspiring discussion in Alpbach. We are grateful to Giuseppe Ancona, Johannes Anschütz, Olivier Benoist, Yohan Brunebarbe, Ana Caraiani, Cédric Pepin, Arno Kret, Robert Kucharczyk, Jaclyn Lang, Marco Maculan, Lucia Mocz, Frans Oort, Will Sawin, Benoît Stroh, Yunqing Tang, Jacques Tilouine, Robert Wilms, and Kang Zuo for helpful discussions.
References
[1] D. Abramovich, Uniformity of stably integral points on elliptic curves, Invent. Math. 127 (1997), no. 2, 307–317. 10.1007/s002220050121Search in Google Scholar
[2] 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. 10.1016/S0019-3577(08)80014-6Search in Google Scholar
[3] T. T. H. An, A. Levin and J. T.-Y. Wang, A p-adic Nevanlinna–Diophantine correspondence, Acta Arith. 146 (2011), no. 4, 379–397. 10.4064/aa146-4-5Search in Google Scholar
[4]
Y. André,
Period mappings and differential equations. From
[5] J. Ayoub, Motifs des variétés analytiques rigides, Mém. Soc. Math. Fr. (N. S.) 140-141 (2015), 1–386. 10.24033/msmf.449Search in Google Scholar
[6] M. Baker, S. Payne and J. Rabinoff, Nonarchimedean geometry, tropicalization, and metrics on curves, Algebr. Geom. 3 (2016), no. 1, 63–105. 10.14231/AG-2016-004Search in Google Scholar
[7] P. Berthelot, Cohomologie rigide et cohomologie rigide à supports propres, première partie, Prépublication 96-03, Université de Rennes 1, 1996. Search in Google Scholar
[8] C. Blakestad, D. Gvirtz, B. Heuer, D. Shchedrina, K. Shimizu, P. Wear and Z. Yao, Perfectoid covers of abelian varieties, preprint (2018), https://arxiv.org/abs/1804.04455; to appear in. Math. Res. Lett. 10.4310/MRL.2022.v29.n3.a2Search in Google Scholar
[9] F. A. Bogomolov, L. Kamenova and M. Verbitsky, Algebraically hyperbolic manifolds have finite automorphism groups, Commun. Contemp. Math. 22 (2020), no. 2, Article ID 1950003. 10.1142/S0219199719500032Search in Google Scholar
[10] R. van Bommel, A. Javanpeykar and L. Kamenova, Boundedness in families with applications to arithmetic hyperbolicity, preprint (2019), https://arxiv.org/abs/1907.11225. Search in Google Scholar
[11] S. Bosch, U. Güntzer and R. Remmert, Non-Archimedean analysis. A systematic approach to rigid analytic geometry, Grundlehren Math. Wiss. 261, Springer, Berlin 1984. 10.1007/978-3-642-52229-1Search in Google Scholar
[12] S. Bosch and W. Lütkebohmert, Stable reduction and uniformization of abelian varieties. II, Invent. Math. 78 (1984), no. 2, 257–297. 10.1007/BF01388596Search in Google Scholar
[13] M. Brion and T. Szamuely, Prime-to-p étale covers of algebraic groups and homogeneous spaces, Bull. Lond. Math. Soc. 45 (2013), no. 3, 602–612. 10.1112/blms/bds110Search in Google Scholar
[14] Y. Brunebarbe, Symmetric differentials and variations of Hodge structures, J. reine angew. Math. 743 (2018), 133–161. 10.1515/crelle-2015-0109Search in Google Scholar
[15] K. Buzzard and A. Verberkmoes, Stably uniform affinoids are sheafy, J. reine angew. Math. 740 (2018), 25–39. 10.1515/crelle-2015-0089Search in Google Scholar
[16] W. Cherry, Non-Archimedean analytic curves in abelian varieties, Math. Ann. 300 (1994), no. 3, 393–404. 10.1007/BF01450493Search in Google Scholar
[17] 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. 10.1215/ijm/1255986193Search in Google Scholar
[18] W. Cherry and M. Ru, Rigid analytic Picard theorems, Amer. J. Math. 126 (2004), no. 4, 873–889. 10.1353/ajm.2004.0025Search in Google Scholar
[19] B. Conrad, Irreducible components of rigid spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 2, 473–541. 10.5802/aif.1681Search in Google Scholar
[20] B. Conrad, A modern proof of Chevalley’s theorem on algebraic groups, J. Ramanujan Math. Soc. 17 (2002), no. 1, 1–18. Search in Google Scholar
[21] B. Conrad, Relative ampleness in rigid geometry, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, 1049–1126. 10.5802/aif.2207Search in Google Scholar
[22] G. Cornell and J. H. Silverman, Arithmetic geometry, Springer, New York 1986. 10.1007/978-1-4613-8655-1Search in Google Scholar
[23] A. J. de Jong, Étale fundamental groups of non-Archimedean analytic spaces, Compos. Math. 97 (1995), no. 1–2, 89–118. Search in Google Scholar
[24] A. J. de Jong, Smoothness, semi-stability and alterations, Publ. Math. Inst. Hautes Études Sci. 83 (1996), 51–93. 10.1007/BF02698644Search in Google Scholar
[25] J.-P. Demailly, Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Algebraic geometry, Proc. Sympos. Pure Math. 62, American Mathematical Society, Providence (1997), 285–360. 10.1090/pspum/062.2/1492539Search in Google Scholar
[26] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366. 10.1007/BF01388432Search in Google Scholar
[27] G. Faltings, Complements to Mordell, Rational points (Bonn 1983/1984), Aspects Math. E6, Friedrich Vieweg, Braunschweig (1984), 203–227. 10.1007/978-3-322-83918-3_6Search in Google Scholar
[28] G. Faltings, Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), no. 3, 549–576. 10.2307/2944319Search in Google Scholar
[29] G. Faltings, The general case of S. Lang’s conjecture, Barsotti Symposium in Algebraic Geometry (Abano Terme 1991), Perspect. Math. 15, Academic Press, San Diego (1994), 175–182. 10.1016/B978-0-12-197270-7.50012-7Search in Google Scholar
[30] O. Gabber, Q. Liu and D. Lorenzini, Hypersurfaces in projective schemes and a moving lemma, Duke Math. J. 164 (2015), no. 7, 1187–1270. 10.1215/00127094-2877293Search in Google Scholar
[31] M. Green and P. Griffiths, Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979, Springer, New York (1980), 41–74. 10.1007/978-1-4613-8109-9_4Search in Google Scholar
[32] A. Grothendieck, Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957–1962], Secrétariat mathématique, Paris 1962. Search in Google Scholar
[33] A. Grothendieck, Revêtements étales et groupe fondamental (SGA I) Fasc. II: Exposés 6, 8 à 11, volume 1960/61 of Séminaire de Géométrie Algébrique, Institut des Hautes Études Scientifiques, Paris 1963. Search in Google Scholar
[34] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43–60. 10.1090/S0002-9947-1969-0251026-XSearch in Google Scholar
[35] F. Hu, S. Meng and D.-Q. Zhang, Ampleness of canonical divisors of hyperbolic normal projective varieties, Math. Z. 278 (2014), no. 3–4, 1179–1193. 10.1007/s00209-014-1351-1Search in Google Scholar
[36] R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), no. 4, 513–551. 10.1007/BF02571959Search in Google Scholar
[37] R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Math. E30, Friedrich Vieweg & Sohn, Braunschweig 1996. 10.1007/978-3-663-09991-8Search in Google Scholar
[38] A. Javanpeykar, Arithmetic hyperbolicity: Automorphisms and persistence, preprint (2018), https://arxiv.org/abs/1809.06818; to appear in Math. Ann. 10.1007/s00208-021-02155-0Search in Google Scholar
[39] A. Javanpeykar, The Lang–Vojta conjectures on projective pseudo-hyperbolic varieties, Arithmetic geometry of logarithmic pairs and hyperbolicity of moduli spaces, CRM Short Courses, Springer, Cham (2020), 135–196. 10.1007/978-3-030-49864-1_3Search in Google Scholar
[40] A. Javanpeykar and L. Kamenova, Demailly’s notion of algebraic hyperbolicity: Geometricity, boundedness, moduli of maps, Math. Z. 296 (2020), no. 3–4, 1645–1672. 10.1007/s00209-020-02489-6Search in Google Scholar
[41] A. Javanpeykar and R. Kucharczyk, Algebraicity of analytic maps to a hyperbolic variety, Math. Nachr. 293 (2020), no. 8, 1490–1504. 10.1002/mana.201900098Search in Google Scholar
[42] A. Javanpeykar and D. Loughran, Complete intersections: Moduli, Torelli, and good reduction, Math. Ann. 368 (2017), no. 3–4, 1191–1225. 10.1007/s00208-016-1455-5Search in Google Scholar
[43] A. Javanpeykar and D. Loughran, Arithmetic hyperbolicity and a stacky Chevalley–Weil theorem, J. Lond. Math. Soc. (2) 103 (2021), no. 3, 846–869. 10.1112/jlms.12394Search in Google Scholar
[44] A. Javanpeykar and J. Xie, Finiteness properties of pseudo-hyperbolic varieties, preprint (2019), https://arxiv.org/abs/1909.12187; to appear in Int. Math. Res. Not. IMRN. 10.1093/imrn/rnaa168Search in Google Scholar
[45] Y. Kawamata, On Bloch’s conjecture, Invent. Math. 57 (1980), no. 1, 97–100. 10.1007/BF01389820Search in Google Scholar
[46] S. Kobayashi, Hyperbolic complex spaces, Grundlehren Math. Wiss. 318, Springer, Berlin 1998. 10.1007/978-3-662-03582-5Search in Google Scholar
[47] U. Köpf, Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen, Schr. Math. Inst. Univ. Münster (2) 7 (1974), 1–72. Search in Google Scholar
[48] S. J. Kovács, Subvarieties of moduli stacks of canonically polarized varieties: Generalizations of Shafarevich’s conjecture, Algebraic geometry—Seattle 2005. Part 2, Proc. Sympos. Pure Math. 80, American Mathematical Society, Providence (2009), 685–709. 10.1090/pspum/080.2/2483952Search in Google Scholar
[49] S. Lang, Hyperbolic and Diophantine analysis, Bull. Amer. Math. Soc. (N. S.) 14 (1986), no. 2, 159–205. 10.1090/S0273-0979-1986-15426-1Search in Google Scholar
[50] A. Levin, Integral points of bounded degree on affine curves, Compos. Math. 152 (2016), no. 4, 754–768. 10.1112/S0010437X15007708Search in Google Scholar
[51] A. Levin and J. T.-Y. Wang, On non-Archimedean curves omitting few components and their arithmetic analogues, Canad. J. Math. 69 (2017), no. 1, 130–142. 10.4153/CJM-2015-030-1Search in Google Scholar
[52] C.-W. Lin and J. T.-Y. Wang, Generalizations of rigid analytic Picard theorems, Proc. Amer. Math. Soc. 138 (2010), no. 1, 133–139. 10.1090/S0002-9939-09-10038-2Search in Google Scholar
[53] J. Ludwig, A quotient of the Lubin–Tate tower, Forum Math. Sigma 5 (2017), Paper No. e17. 10.1017/fms.2017.15Search in Google Scholar
[54] W. Lütkebohmert, Der Satz von Remmert–Stein in der nichtarchimedischen Funktionentheorie, Math. Z. 139 (1974), 69–84. 10.1007/BF01194146Search in Google Scholar
[55] W. Lütkebohmert, Riemann’s existence problem for a p-adic field, Invent. Math. 111 (1993), no. 2, 309–330. 10.1007/BF01231290Search in Google Scholar
[56] L. Moret-Bailly, Pinceaux de variétés abéliennes, Astérisque 129, Société Mathématique de France, Paris 1985. Search in Google Scholar
[57] A. M. Nadel, The nonexistence of certain level structures on abelian varieties over complex function fields, Ann. of Math. (2) 129 (1989), no. 1, 161–178. 10.2307/1971489Search in Google Scholar
[58]
M. C. Olsson,
[59] R. Rodríguez Vázquez, Hyperbolicity notions for varieties defined over a non-Archimedean field, Michigan Math. J. 69 (2020), no. 1, 41–78. 10.1307/mmj/1574326880Search in Google Scholar
[60] P. Scholze, Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 245–313. 10.1007/s10240-012-0042-xSearch in Google Scholar
[61] P. Scholze, p-adic Hodge theory for rigid-analytic varieties, Forum Math. Pi 1 (2013), Article ID e1. 10.1017/fmp.2013.1Search in Google Scholar
[62] P. Scholze, On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), no. 3, 945–1066. 10.4007/annals.2015.182.3.3Search in Google Scholar
[63] P. Scholze, Etale cohomology of diamonds, preprint (2017), https://arxiv.org/abs/1709.07343. 10.2139/ssrn.3542189Search in Google Scholar
[64] P. Scholze and J. Weinstein, Moduli of p-divisible groups, Camb. J. Math. 1 (2013), no. 2, 145–237. 10.4310/CJM.2013.v1.n2.a1Search in Google Scholar
[65] R. Sun, Non-archimedean hyperbolicity of the moduli space of curves, preprint (2020), https://arxiv.org/abs/2009.13096. Search in Google Scholar
[66] K. Ueno, Classification of algebraic varieties. I, Compos. Math. 27 (1973), 277–342. Search in Google Scholar
[67] G. van der Geer, Points of degree d on curves over number fields, Diophantine approximation and abelian varieties (Soesterberg 1992), Lecture Notes in Math. 1566, Springer, Berlin (1993), 111–116. 10.1007/978-3-540-48208-6_12Search in Google Scholar
[68] C. Voisin, Hodge theory and complex algebraic geometry. II, Cambridge Stud. Adv. Math. 77, Cambridge University, Cambridge 2003. 10.1017/CBO9780511615177Search in Google Scholar
[69] P. Vojta, A higher-dimensional Mordell conjecture, Arithmetic geometry (Storrs 1984), Springer, New York (1986), 341–353. 10.1007/978-1-4613-8655-1_15Search in Google Scholar
[70] J. G. Zarhin, A remark on endomorphisms of abelian varieties over function fields of finite characteristic, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 471–474. 10.1070/IM1974v008n03ABEH002115Search in Google Scholar
[71] K. Zuo, On the negativity of kernels of Kodaira–Spencer maps on Hodge bundles and applications, Asian J. Math. 4 (2000), 279–301. 10.4310/AJM.2000.v4.n1.a17Search in Google Scholar
[72] The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2015. Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Non-archimedean hyperbolicity and applications
- Deligne–Lusztig duality on the stack of local systems
- Motivic decompositions for the Hilbert scheme of points of a K3 surface
- Extending meromorphic connections to coadmissible ̑𝒟-modules
- Brauer’s Height Zero Conjecture for principal blocks
- Four-dimensional complete gradient shrinking Ricci solitons
- Siegel domains over Finsler symmetric cones
- Basis divisors and balanced metrics
- Erratum to The braided Thompson's groups are of type F∞ (J. reine angew. Math. 718 (2016), 59–101)
Articles in the same Issue
- Frontmatter
- Non-archimedean hyperbolicity and applications
- Deligne–Lusztig duality on the stack of local systems
- Motivic decompositions for the Hilbert scheme of points of a K3 surface
- Extending meromorphic connections to coadmissible ̑𝒟-modules
- Brauer’s Height Zero Conjecture for principal blocks
- Four-dimensional complete gradient shrinking Ricci solitons
- Siegel domains over Finsler symmetric cones
- Basis divisors and balanced metrics
- Erratum to The braided Thompson's groups are of type F∞ (J. reine angew. Math. 718 (2016), 59–101)
