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Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry


Author: Sam Payne
Journal: Bull. Amer. Math. Soc. 52 (2015), 223-247
MSC (2010): Primary 32K10; Secondary 14B05, 14T05, 32J05, 32S45, 32S50
DOI: https://doi.org/10.1090/S0273-0979-2014-01469-7
Published electronically: August 18, 2014
MathSciNet review: 3312632
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Abstract: This note surveys basic topological properties of nonarchimedean analytic spaces, in the sense of Berkovich, including the recent tameness results of Hrushovski and Loeser. We also discuss interactions between the topology of nonarchimedean analytic spaces and classical algebraic geometry.


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  • [AB12] O. Amini and M. Baker, Linear series on metrized complexes of algebraic curves, preprint arXiv:1204.3508, 2012.
  • [AB14] K. Adiprasito and A. Björner, Filtered geometric lattices and Lefschetz section theorems over the tropical semiring, preprint arXiv:1401.7301, 2014.
  • [ABBR13] O. Amini, M. Baker, E. Brugallé, and J. Rabinoff, Lifting harmonic morphisms of tropical curves, metrized complexes, and Berkovich skeleta, preprint arXiv:1303.4812, 2013.
  • [Abh56] Shreeram Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321-348. MR 0082477 (18,556b)
  • [AC13] Omid Amini and Lucia Caporaso, Riemann-Roch theory for weighted graphs and tropical curves, Adv. Math. 240 (2013), 1-23. MR 3046301, https://doi.org/10.1016/j.aim.2013.03.003
  • [ACP12] D. Abramovich, L. Caporaso, and S. Payne, The tropicalization of the moduli space of curves, preprint arXiv:1212.0373, 2012.
  • [AM69] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802 (39 #4129)
  • [Bak08a] Matthew Baker, An introduction to Berkovich analytic spaces and non-Archimedean potential theory on curves, $ p$-adic geometry, Univ. Lecture Ser., vol. 45, Amer. Math. Soc., Providence, RI, 2008, pp. 123-174. MR 2482347 (2010g:14029)
  • [Bak08b] Matthew Baker, Specialization of linear systems from curves to graphs, Algebra Number Theory 2 (2008), no. 6, 613-653. With an appendix by Brian Conrad. MR 2448666 (2010a:14012), https://doi.org/10.2140/ant.2008.2.613
  • [Ber90] Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990. MR 1070709 (91k:32038)
  • [Ber93] Vladimir G. Berkovich, Étale cohomology for non-Archimedean analytic spaces, Inst. Hautes Études Sci. Publ. Math. (1993), no. 78, 5-161 (1994).
  • [Ber99] Vladimir G. Berkovich, Smooth $ p$-adic analytic spaces are locally contractible, Invent. Math. 137 (1999), no. 1, 1-84. MR 1702143 (2000i:14028), https://doi.org/10.1007/s002220050323
  • [Ber00] Vladimir G. Berkovich, An analog of Tate's conjecture over local and finitely generated fields, Internat. Math. Res. Notices 13 (2000), 665-680. MR 1772523 (2001h:14022), https://doi.org/10.1155/S1073792800000362
  • [Ber04] Vladimir G. Berkovich, Smooth $ p$-adic analytic spaces are locally contractible. II, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 293-370. MR 2023293 (2005h:14057)
  • [BGR84] S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften, vol. 261, Springer-Verlag, Berlin, 1984.
  • [BL93a] Siegfried Bosch and Werner Lütkebohmert, Formal and rigid geometry. I. Rigid spaces, Math. Ann. 295 (1993), no. 2, 291-317. MR 1202394 (94a:11090), https://doi.org/10.1007/BF01444889
  • [BL93b] Siegfried Bosch and Werner Lütkebohmert, Formal and rigid geometry. II. Flattening techniques, Math. Ann. 296 (1993), no. 3, 403-429. MR 1225983 (94e:11070), https://doi.org/10.1007/BF01445112
  • [BLR95a] Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Formal and rigid geometry. III. The relative maximum principle, Math. Ann. 302 (1995), no. 1, 1-29. MR 1329445 (97e:11074), https://doi.org/10.1007/BF01444485
  • [BLR95b] Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Formal and rigid geometry. IV. The reduced fibre theorem, Invent. Math. 119 (1995), no. 2, 361-398. MR 1312505 (97e:11075), https://doi.org/10.1007/BF01245187
  • [BN07] Matthew Baker and Serguei Norine, Riemann-Roch and Abel-Jacobi theory on a finite graph, Adv. Math. 215 (2007), no. 2, 766-788. MR 2355607 (2008m:05167), https://doi.org/10.1016/j.aim.2007.04.012
  • [BN09] Matthew Baker and Serguei Norine, Harmonic morphisms and hyperelliptic graphs, Int. Math. Res. Not. IMRN 15 (2009), 2914-2955. MR 2525845 (2010e:14031), https://doi.org/10.1093/imrn/rnp037
  • [Bot59] Raoul Bott, On a theorem of Lefschetz, Michigan Math. J. 6 (1959), 211-216. MR 0215323 (35 #6164)
  • [BPR11] M. Baker, S. Payne, and J. Rabinoff, Nonarchimedean geometry, tropicalization, and metrics on curves, preprint, arXiv:1104.0320v1, 2011.
  • [BR13] M. Baker and J. Rabinoff, The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves, To appear in Int. Math. Res. Not. arXiv:1308.3864, 2013.
  • [Cap12] L. Caporaso, Gonality of algebraic curves and graphs, preprint arXiv:1201.6246v3, 2012.
  • [CC12] Wouter Castryck and Filip Cools, Newton polygons and curve gonalities, J. Algebraic Combin. 35 (2012), no. 3, 345-366. MR 2892979, https://doi.org/10.1007/s10801-011-0304-6
  • [CDPR12] Filip Cools, Jan Draisma, Sam Payne, and Elina Robeva, A tropical proof of the Brill-Noether theorem, Adv. Math. 230 (2012), no. 2, 759-776. MR 2914965, https://doi.org/10.1016/j.aim.2012.02.019
  • [CGP14] M. Chan, S. Galatius, and S. Payne, in preparation, 2014.
  • [CKK12] G. Cornelissen, F. Kato, and J. Kool, A combinatorial Li-Yau inequality and rational points on curves, preprint arXiv:1211.2681, 2012.
  • [Dan75] V. I. Danilov, Polyhedra of schemes and algebraic varieties, Mat. Sb. (N.S.) 139 (1975), no. 1, 146-158, 160 (Russian); Russian transl., Math. USSR-Sb. 26 (1975), no. 1, 137-149 (1976). MR 0441970 (56 #359)
  • [Del80] Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137-252 (French). MR 601520 (83c:14017)
  • [dFKX13] T. de Fernex, J. Kollár, and C. Xu, The dual complex of singularities, To appear in proceedings of the conference in honor of Yujiro Kawamata's 60th birthday, Adv. Stud. Pure Math. arXiv:1212.1675v2, 2013.
  • [dJ96] A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51-93. MR 1423020 (98e:14011)
  • [DL98] Jan Denef and François Loeser, Motivic Igusa zeta functions, J. Algebraic Geom. 7 (1998), no. 3, 505-537. MR 1618144 (99j:14021)
  • [Duc13] Antoine Ducros, Les espaces de Berkovich sont modérés (d'après Ehud Hrushovski et François Loeser), Astérisque 352 (2013), Exp. No. 1056, x, 459-507 (French, with French summary). Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043-1058. MR 3087354
  • [FGP12] T. Foster, P. Gross, and S. Payne, Limits of tropicalizations, To appear in Israel J. Math. arXiv:1211.2718, 2012.
  • [GK08] Andreas Gathmann and Michael Kerber, A Riemann-Roch theorem in tropical geometry, Math. Z. 259 (2008), no. 1, 217-230. MR 2377750 (2009a:14014), https://doi.org/10.1007/s00209-007-0222-4
  • [Gub13] Walter Gubler, A guide to tropicalizations, Algebraic and combinatorial aspects of tropical geometry, Contemp. Math., vol. 589, Amer. Math. Soc., Providence, RI, 2013, pp. 125-189. MR 3088913, https://doi.org/10.1090/conm/589/11745
  • [Hac08] Paul Hacking, The homology of tropical varieties, Collect. Math. 59 (2008), no. 3, 263-273. MR 2452307 (2010c:14076), https://doi.org/10.1007/BF03191187
  • [Har77] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157 (57 #3116)
  • [HHM08] Deirdre Haskell, Ehud Hrushovski, and Dugald Macpherson, Stable domination and independence in algebraically closed valued fields, Lecture Notes in Logic, vol. 30, Association for Symbolic Logic, Chicago, IL; Cambridge University Press, Cambridge, 2008. MR 2369946 (2010c:03002)
  • [HK06] Ehud Hrushovski and David Kazhdan, Integration in valued fields, Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhäuser Boston, Boston, MA, 2006, pp. 261-405. MR 2263194 (2007k:03094), https://doi.org/10.1007/978-0-8176-4532-8_4
  • [HK12] David Helm and Eric Katz, Monodromy filtrations and the topology of tropical varieties, Canad. J. Math. 64 (2012), no. 4, 845-868. MR 2957233, https://doi.org/10.4153/CJM-2011-067-9
  • [HL11] E. Hrushovski and F. Loeser, Monodromy and the Lefschetz fixed point formula, preprint, arXiv:1111.1954, 2011.
  • [HL12] E. Hrushovski and F. Loeser, Nonarchimedean topology and stably dominated types, preprint, arXiv:1009.0252v3, 2012.
  • [HLP12] E. Hrushovski, F. Loeser, and B. Poonen, Berkovich spaces embed in euclidean spaces, preprint, arXiv:1210.6485, 2012.
  • [Igu75] Jun-ichi Igusa, Complex powers and asymptotic expansions. II. Asymptotic expansions, J. Reine Angew. Math. 278/279 (1975), 307-321. MR 0404215 (53 #8018)
  • [Jon12] M Jonsson, Dynamics on Berkovich spaces of low dimensions, To appear in Berkovich spaces and applications. Séminaires et Congrès. arXiv:1201.1944v1, 2012.
  • [JP14] D. Jensen and S. Payne, Tropical multiplication maps and the Gieseker-Petri Theorem, preprint, arXiv:1401.2584, 2014.
  • [KK11] Michael Kapovich and János Kollár, Fundamental groups of links of isolated singularities, J. Amer. Math. Soc. 27 (2014), no. 4, 929-952. MR 3230815, https://doi.org/10.1090/S0894-0347-2014-00807-9
  • [KKMSD73] G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR 0335518 (49 #299)
  • [Kol13] János Kollár, Links of complex analytic singularities, Surveys in differential geometry. Geometry and topology, Surv. Differ. Geom., vol. 18, Int. Press, Somerville, MA, 2013, pp. 157-193. MR 3087919, https://doi.org/10.4310/SDG.2013.v18.n1.a4
  • [KS06] Maxim Kontsevich and Yan Soibelman, Affine structures and non-Archimedean analytic spaces, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 321-385. MR 2181810 (2006j:14054), https://doi.org/10.1007/0-8176-4467-9_9
  • [KS12a] Eric Katz and Alan Stapledon, Tropical geometry and the motivic nearby fiber, Compos. Math. 148 (2012), no. 1, 269-294. MR 2881316, https://doi.org/10.1112/S0010437X11005446
  • [KS12b] Moritz Kerz and Shuji Saito, Cohomological Hasse principle and motivic cohomology for arithmetic schemes, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 123-183. MR 2929729, https://doi.org/10.1007/s10240-011-0038-y
  • [KT02] M. Kontsevich and Y. Tschinkel, Nonarchimedean Kähler geometry, unpublished, 2002.
  • [LS03] François Loeser and Julien Sebag, Motivic integration on smooth rigid varieties and invariants of degenerations, Duke Math. J. 119 (2003), no. 2, 315-344. MR 1997948 (2004g:14026), https://doi.org/10.1215/S0012-7094-03-11924-9
  • [Mar02] David Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002. An introduction. MR 1924282 (2003e:03060)
  • [Mil68] John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612 (39 #969)
  • [MN12] M. Mustaţă and J. Nicaise, Weight functions on non-archimedean analytic spaces and the Kontsevich-Soibelman skeleton, preprint, arXiv:1212.6328, 2012.
  • [MZ08] Grigory Mikhalkin and Ilia Zharkov, Tropical curves, their Jacobians and theta functions, Curves and abelian varieties, Contemp. Math., vol. 465, Amer. Math. Soc., Providence, RI, 2008, pp. 203-230. MR 2457739 (2011c:14163), https://doi.org/10.1090/conm/465/09104
  • [Nic11] Johannes Nicaise, Singular cohomology of the analytic Milnor fiber, and mixed Hodge structure on the nearby cohomology, J. Algebraic Geom. 20 (2011), no. 2, 199-237. MR 2762990 (2012i:14028), https://doi.org/10.1090/S1056-3911-10-00526-6
  • [NS07] Johannes Nicaise and Julien Sebag, Motivic Serre invariants, ramification, and the analytic Milnor fiber, Invent. Math. 168 (2007), no. 1, 133-173. MR 2285749 (2009c:14040), https://doi.org/10.1007/s00222-006-0029-7
  • [NX13] J. Nicaise and C. Xu, The essential skeleton of a degeneration of algebraic varieties, arXiv:1307.4041, 2013.
  • [Pay09] Sam Payne, Analytification is the limit of all tropicalizations, Math. Res. Lett. 16 (2009), no. 3, 543-556. MR 2511632 (2010j:14104), https://doi.org/10.4310/MRL.2009.v16.n3.a13
  • [Pay13] Sam Payne, Boundary complexes and weight filtrations, Michigan Math. J. 62 (2013), no. 2, 293-322. MR 3079265, https://doi.org/10.1307/mmj/1370870374
  • [Ray74] Michel Raynaud, Géométrie analytique rigide d'après Tate, Kiehl,$ \cdots $, Table Ronde d'Analyse non archimédienne (Paris, 1972) Soc. Math. France, Paris, 1974, pp. 319-327. Bull. Soc. Math. France, Mém. No. 39-40 (French). MR 0470254 (57 #10012)
  • [Smi14] G. Smith, Brill-Noether theory of curves in toric surfaces, preprint arXiv:1403.2317. To appear in J. Pure Appl. Alg., 2014.
  • [Ste06] D. A. Stepanov, A remark on the dual complex of a resolution of singularities, Uspekhi Mat. Nauk 61 (2006), no. 1(367), 185-186 (Russian); English transl., Russian Math. Surveys 61 (2006), no. 1, 181-183. MR 2239783 (2007c:14010), https://doi.org/10.1070/RM2006v061n01ABEH004309
  • [Thu05] A. Thuillier, Théorie du potentiel sur ler courbes en géométrie analytique non archimédienne. Applications à la theéorie d'Arakelov, Ph.D. thesis, University of Rennes, 2005.
  • [Thu07] Amaury Thuillier, Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d'homotopie de certains schémas formels, Manuscripta Math. 123 (2007), no. 4, 381-451 (French, with English summary). MR 2320738 (2008g:14038), https://doi.org/10.1007/s00229-007-0094-2

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Additional Information

Sam Payne
Affiliation: Department of Mathematics, Yale University, 10 Hillhouse Ave., New Haven, Connecticut 06511
Email: sam.payne@yale.edu

DOI: https://doi.org/10.1090/S0273-0979-2014-01469-7
Received by editor(s): September 30, 2013
Published electronically: August 18, 2014
Additional Notes: The author was partially supported by NSF DMS-1068689 and NSF CAREER DMS-1149054.
Article copyright: © Copyright 2014 American Mathematical Society

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