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Degenerations of amoebae and Berkovich spaces

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Abstract

We prove a continuity result for the fibers of the Berkovich analytification of a complex algebraic variety with respect to the maximum of the Archimedean norm and the trivial norm. As a consequence, we obtain generalizations of a result of Mikhalkin and Rullgård about degenerations of amoebae onto tropical varieties.

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Notes

  1. We use negative signs to match the standard convention for valuations. All logarithms are natural logarithms.

  2. The superscript “\(h\)” stands for “holomorphic”.

  3. While we shall only consider the analytification as a topological space, one can also equip it with a structure sheaf.

  4. They are good \(k\)-analytic spaces without boundary.

  5. As with the case of the analytification, the tropicalization \({Y^\mathrm {trop}}\) will only be considered as a topological space (together with an action by \({\mathbf {R}}_+^*\)) and not equipped with a structure sheaf.

References

  1. Amini, O., Baker, M., Brugallé, E., Rabinoff, J.: Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta. Res. Math. Sci. (to appear). arXiv:1303.4812

  2. Amini, O., Baker, M., Brugallé, E., Rabinoff, J.: Lifting harmonic morphisms II: tropical curves and metrized complexes. Algebra Number Theory 9, 267–315 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ash, A., Mumford, D., Rapoport, M., Tai, Y.-S.: Smooth Compactifications of Locally Symmetric Varieties. With the Collaboration of Peter Scholze, 2nd edn. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  4. Avendaño, M., Kogan, R., Nisse, M., Rojas, J.M.: Metric estimates and membership complexity for Archimedean amoebae and tropical hypersurfaces. arXiv:1307.3681

  5. Baker, M., Norine, S.: Riemann–Roch and Abel–Jacobi theory on a finite graph. Adv. Math. 215, 766–788 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baker, M., Payne, S., Rabinoff, J.: Nonarchimedean geometry, tropicalization, and metrics on curves. arXiv:1104.0320

  7. Bergman, G.M.: The logarithmic limit-set of an algebraic variety. Trans. Am. Math. Soc. 157, 459–469 (1971)

    Article  MathSciNet  Google Scholar 

  8. Berkovich, V.: Spectral Theory and Analytic Geometry over Non-Archimedean Fields. Mathematical Surveys and Monographs, vol. 33. American Mathematical Society, Providence (1990)

    Google Scholar 

  9. Berkovich, V.G.: Étale cohomology for non-Archimedean analytic spaces. Publ. Math. Inst. Hautes Études Sci. 78, 5–161 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Berkovich, V.G.: A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures. In: Algebra, Arithmetic, and Geometry in Honor of Yu. I. Manin. Progress in Mathematics, vol. 269, pp. 49–67. Birkhäuser, Boston (2009)

  11. Bieri, R., Groves, J.R.R.: The geometry of the set of characters induced by valuations. J. Reine Angew. Math. 347, 168–195 (1984)

    MathSciNet  MATH  Google Scholar 

  12. Bogart, T., Jensen, A.N., Speyer, D., Sturmfels, B., Thomas, R.R.: Computing tropical varieties. J. Symbolic Comput. 42, 54–73 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Boucksom, S., Favre, C., Jonsson, M.: Valuations and plurisubharmonic singularities. Publ. Res. Inst. Math. Sci. 44, 449–494 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Boucksom, S., Favre, C., Jonsson, M.: Singular semipositive metrics in non-Archimedean geometry. J. Algebraic Geom. arXiv:1201.0187 (to appear)

  15. Cartwright, D., Payne, S.: Connectivity of tropicalizations. Math. Res. Lett. 19, 1089–1095 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Conrad, B.: Deligne’s notes on Nagata compactifications. J. Ramanujan Math. Soc. 22, 205–257 (2007)

    MathSciNet  MATH  Google Scholar 

  17. Cools, F., Draisma, J., Payne, S., Robeva, E.: A tropical proof of the Brill–Noether Theorem. Adv. Math. 230, 759–776 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Douady, A.: Flatness and privilege. Ens. Math. 14, 47–74 (1968)

    MathSciNet  MATH  Google Scholar 

  19. Draisma, J.: A tropical approach to secant dimensions. J. Pure Appl. Algebra 212, 349–363 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ducros, A.: Families of Berkovich spaces. arXiv:1107.4259v3

  21. Ducros, A.: Espaces de Berkovich, polytopes, squelettes et théorie des modèles. Confluentes Math. 4(4), 1250007 (2010)

  22. Ein, L., Lazarsfeld, R., Smith, K.: Uniform approximation of Abhyankar valuation ideals in smooth function fields. Am. J. Math. 125, 409–440 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Einsiedler, M., Kapranov, M., Lind, D.: Non-archimedean amoebas and tropical varieties. J. Reine Angew. Math. 601, 139–157 (2006)

    MathSciNet  MATH  Google Scholar 

  24. Favre, C.: Compactifications of affine varieties with non-Archimedean techniques and dynamical applications. Preliminary manuscript (2012)

  25. Favre, C., Jonsson, M.: The Valuative Tree. Lecture Notes in Mathematics, vol. 1853. Springer, Berlin (2004)

    Book  Google Scholar 

  26. Favre, C., Jonsson, M.: Valuative analysis of planar plurisubharmonic functions. Invent. Math. 162(2), 271–311 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Favre, C., Jonsson, M.: Valuations and multiplier ideals. J. Am. Math. Soc 18, 655–684 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Favre, C., Jonsson, M.: Eigenvaluations. Ann. Sci. École Norm. Sup. 40, 309–349 (2007)

    MathSciNet  MATH  Google Scholar 

  29. Favre, C., Jonsson, M.: Dynamical compactifications of \({\mathbf{C}}^2\). Ann. Math. 173, 211–249 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Forsberg, M., Passare, M., Tsikh, A.: Laurent determinants and arrangements of hyperplane amoebas. Adv. Math. 151, 45–70 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Foster, T., Gross, P., Payne, S.: Limits of tropicalizations. Israeli J. Math. 201, 835–846 (2014)

  32. Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, 131. Princeton University Press, Princeton (1993)

    Google Scholar 

  33. Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser Boston Inc., Boston (1994)

    Book  MATH  Google Scholar 

  34. Gubler, W.: Tropical varieties for non-archimedean analytic spaces. Invent. Math. 169, 321–376 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Gubler, W.: A guide to tropicalizations. In: Algebraic and Combinatorial Aspects of Tropical Geometry. Contemporary Mathematics, vol. 589, pp. 125–189. American Mathematical Society Providence (2013)

  36. Gubler, W., Rabinoff, J., Werner, A.: Skeletons and tropicalizations. arXiv:1404.7044

  37. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, No. 52. Springer, New York, Heidelberg (1977)

    Google Scholar 

  38. Henriques, A.: An analogue of convexity for complements of amoebas of varieties of higher codimension, an answer to a question asked by B. Sturmfels. Adv. Geom. 4, 61–73 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  39. Itenberg, I.: Amibes de variétés algébriques et dénombrement de courbes (d’après G. Mikhalkin). Astérisque 294, 335–361 (2004)

  40. Itenberg, I., Mikhalkin, G.: Geometry in the tropical limit. Math. Semesterber. 59, 57–73 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  41. Itenberg, I., Mikhalkin, G., Shustin, E.: Tropical Algebraic Geometry. Oberwolfach Seminars, 35, 2nd edn. Birkhäuser, Basel (2009)

    Book  Google Scholar 

  42. Jensen, D., Payne, S.: Tropical independence I: Shapes of divisors and a proof of the Gieseker-Petri Theorem. Algebra Number Theory (to appear). arXiv:1401.2584

  43. Jonsson, M.: Dynamics on Berkovich spaces in low dimensions. In: Ducros, A., Favre, C., Nicaise, J. (eds.) Berkovich Spaces and Applications. Lecture Notes in Mathematics, 2119. Springer (2015). arXiv:1201.1944

  44. Jonsson, M., Mustaţǎ, M.: Valuations and asymptotic invariants for sequences of ideals. Ann. Inst. Fourier 62, 2145–2209 (2012)

    Article  MATH  Google Scholar 

  45. Kajiwara, T.: Tropical toric geometry. In: Toric Topology. Contemporary Mathematics, vol. 460, pp. 197–207. American Mathematical Society, Providence (2008)

  46. Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. Math. 163, 1019–1056 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kiwi, J.: Puiseux series polynomial dynamics and iteration of complex cubic polynomials. Ann. Inst. Fourier 56, 1337–1404 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  48. Knaf, H., Kuhlmann, F.-V.: Abhyankar places admit local uniformization in any characteristic. Ann. Sci. École Norm. Sup. (4) 38, 833–846 (2005)

    MathSciNet  MATH  Google Scholar 

  49. Kuratowski, K.: Topology, vol. I–II. Academic Press, New York, London (1966)

    Google Scholar 

  50. Litvinov, G.L.: The Maslov dequantization, idempotent and tropical mathematics: a very brief introduction. In: Idempotent Mathematics and Mathematical Physics. Contemporary Mathematics, vol. 377, pp. 1–17. American Mathematical Society, Providence (2005)

  51. Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. Graduate Studies in Mathematics, vol. 161. American Mathematical Society, Providence (2015)

    Google Scholar 

  52. Madani, F., Nisse, M.: Generalized logarithmic Gauss map and its relation to (co)amoebas. Math. Nachr. 286, 1510–1513 (2013)

    MathSciNet  MATH  Google Scholar 

  53. Madani, F., Nisse, M.: On the volume of complex amoebas. Proc. Am. Math. Soc. 141, 1113–1123 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  54. Mikhalkin, G.: Real algebraic curves, moment map and amoebas. Ann. Math. 151, 309–326 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  55. Mikhalkin, G.: Decomposition into pairs-of-pants for complex algebraic hypersurfaces. Topology 43, 1035–1065 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  56. Mikhalkin, G.: Amoebas of algebraic varieties and tropical geometry. In: Different Faces of Geometry. International Mathematical Series (New York), vol. 3, pp. 257–300. Kluwer/Plenum, New York (2004)

  57. Mikhalkin, G.: Enumerative tropical geometry in \(\mathbb{R}^2\). J. Am. Math. Soc. 18, 313–377 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  58. Mikhalkin, G.: Tropical Geometry and its Applications. International Congress of Mathematicians, vol. II, pp. 827–852. European Mathematical Society, Zürich (2006)

    Google Scholar 

  59. Mikhalkin, G., Rullgård, H.: Amoebas of maximal area. Int. Math. Res. Notices 9, 441–451 (2001)

    Article  Google Scholar 

  60. Morgan, J.W., Shalen, P.B.: Valuations, trees, and degenerations of hyperbolic structures I. Ann. Math. 120, 401–476 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  61. Nisse, M., Sottile, F.: Higher convexity for complements of tropical varieties. arXiv:1411.7363

  62. Osserman, B., Payne, S.: Lifting tropical intersections. Doc. Math. 18, 121–175 (2013)

    MathSciNet  MATH  Google Scholar 

  63. Passare, M., Rullgård, H.: Multiple Laurent series and polynomial amoebas. Actes des Rencontres d’Analyse Complexe (Poitiers-Futuroscope, 1999), pp. 123–129. Atlantique, Poitiers (2002)

  64. Passare, M., Rullgård, H.: Amoebas, Monge–Ampère measure, and triangulations of the Newton polytope. Duke Math. J. 121, 481–507 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  65. Passare, M., Potchekutov, D., Tsikh, A.: Amoebas of complex hypersurfaces in statistical thermodynamics. Math. Phys. Anal. Geom. 16, 89–108 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  66. Passare, M., Tsikh, A.: Amoebas: their spines and their contours. In: Idempotent Mathematics and Mathematical Physics. Contemporary Mathematics, vol. 377, pp. 275–288. American Mathematical Society, Providence (2005)

  67. Payne, S.: Adelic amoebas disjoint from open halfspaces. J. Reine Angew. Math. 625, 115–123 (2008)

    MathSciNet  MATH  Google Scholar 

  68. Payne, S.: Analytification is the limit of all tropicalizations. Math. Res. Lett. 16, 543–556 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  69. Poineau, J.: La droite de Berkovich sur \(\mathbf{Z}\). Astérisque 334 (2010)

  70. Poineau, J.: Espaces de Berkovich sur \(\mathbf{Z}\): étude locale. Invent. Math. 194, 535–590 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  71. Poineau, J.: Les espaces de Berkovich sont angélique. Bull. Soc. Math. France 141, 267–297 (2013)

    MathSciNet  MATH  Google Scholar 

  72. Purbhoo, K.: A Nullstellensatz for amoebas. Duke Math. J. 141, 407–445 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  73. Rabinoff, J.: Tropical analytic geometry, Newton polygons, and tropical intersections. Adv. Math. 229, 3192–3255 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  74. Rashkovskii, A.: A remark on amoebas in higher codimension. In: Gustafsson, B., Vasil’ev, A. (eds.) Analysis and mathematical physics, pp. 465–471. Birkhäuser Verlag, Basel (2009)

    Chapter  Google Scholar 

  75. H. Rullgård. Stratification des espaces de polynômes de Laurent et la structure de leurs amibes. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 355–358

  76. Rullgård, H. Polynomial Amoebas and Convexity. Research Reports in Mathematics, Number 8. Stockholm University (2001)

  77. Schroeter, F., de Wolff, T. The boundary of amoebas. arXiv:1310.7363

  78. Speyer, D.: Tropical geometry. Thesis, University of California, Berkeley (2005)

  79. Speyer, D., Sturmfels, B.: The tropical Grassmannian. Adv. Geom. 4, 389–411 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  80. Sturmfels, B., Tevelev, J.: Elimination theory for tropical varieties. Math. Res. Lett. 15, 543–562 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  81. Teissier, B.: Amibes non archimédiennes. In: Géométrie Tropicale, pp. 85–114. Ed. Éc. Polytech., Palaiseau (2008)

  82. Theobald, T.: Computing amoebas. Exp. Math. 11, 513–526 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  83. Theobald, T., de Wolff, T.: Approximating amoebas and coamoebas by sums of squares. Math. Comp. (to appear). arXiv:1101.4114

  84. Theobald, T., de Wolff, T.: Amoebas of genus at most one. Adv. Math. 239, 190–213 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  85. Thuillier, A.: Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels. Manuscr. Math. 123, 381–451 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  86. Viro, O.: Patchworking real algebraic varieties. arXiv:0611382

  87. de Wolff, T.: On the geometry, topology and approximation of amoebas. Thesis, Goethe Universität, Frankfurt am Main (2013)

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Acknowledgments

I thank V. Berkovich for the proof of Lemma 3.2, and M. Baker, S. Boucksom, A. Ducros, W. Gubler, S. Payne and A. Werner, for comments on a preliminary version of this manuscript. I have also benefitted from discussions with E. Brugallé, C. Favre, G. Mikhalkin and B. Teissier.

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  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1043, USA

    Mattias Jonsson

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Correspondence to Mattias Jonsson.

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Partially supported by NSF Grants DMS-1001740 and DMS-1266207.

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Jonsson, M. Degenerations of amoebae and Berkovich spaces. Math. Ann. 364, 293–311 (2016). https://doi.org/10.1007/s00208-015-1210-3

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