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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


IMPACT FACTOR 2018: 1.859

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Volume 2018, Issue 741

Issues

Gromov compactness in non-archimedean analytic geometry

Tony Yue Yu
  • Institut de Mathématiques de Jussieu – Paris Rive Gauche, CNRS-UMR 7586, Case 7012, Université Paris Diderot – Paris 7, Bâtiment Sophie Germain, 75205 Paris Cedex 13, France
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Published Online: 2016-01-14 | DOI: https://doi.org/10.1515/crelle-2015-0077

Abstract

Gromov’s compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov’s compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin’s representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry.

References

  • [1]

    D. Abramovich, L. Caporaso and S. Payne, The tropicalization of the moduli space of curves, preprint (2012), http://arxiv.org/abs/1212.0373.

  • [2]

    D. Abramovich and F. Oort, Alterations and resolution of singularities, Resolution of singularities (Obergurgl 1997), Progr. Math. 181, Birkhäuser, Basel (2000), 39–108. Google Scholar

  • [3]

    D. Abramovich and F. Oort, Stable maps and Hurwitz schemes in mixed characteristics, Advances in algebraic geometry motivated by physics (Lowell 2000), Contemp. Math. 276, American Mathematical Society, Providence (2001), 89–100. Google Scholar

  • [4]

    D. Abramovich and A. Vistoli, Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002), no. 1, 27–75, (electronic). CrossrefGoogle Scholar

  • [5]

    M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189. CrossrefGoogle Scholar

  • [6]

    M. Baker, S. Payne and J. Rabinoff, Nonarchimedean geometry, tropicalization, and metrics on curves, preprint (2011), http://arxiv.org/abs/1104.0320.

  • [7]

    K. Behrend and Y. Manin, Stacks of stable maps and Gromov–Witten invariants, Duke Math. J. 85 (1996), no. 1, 1–60. CrossrefGoogle Scholar

  • [8]

    V. G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Math. Surveys Monogr. 33, American Mathematical Society, Providence 1990. Google Scholar

  • [9]

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

  • [10]

    V. G. Berkovich, Vanishing cycles for formal schemes, Invent. Math. 115 (1994), no. 3, 539–571. CrossrefGoogle Scholar

  • [11]

    V. G. Berkovich, Vanishing cycles for formal schemes. II, Invent. Math. 125 (1996), no. 2, 367–390. CrossrefGoogle Scholar

  • [12]

    V. G. Berkovich, Smooth p-adic analytic spaces are locally contractible, Invent. Math. 137 (1999), no. 1, 1–84. CrossrefGoogle Scholar

  • [13]

    S. Bosch, U. Güntzer and R. Remmert, Non-Archimedean analysis, Grundlehren Math. Wiss. 261, Springer, Berlin 1984. Google Scholar

  • [14]

    S. Bosch and W. Lütkebohmert, Formal and rigid geometry. I: Rigid spaces, Math. Ann. 295 (1993), no. 2, 291–317. CrossrefGoogle Scholar

  • [15]

    S. Boucksom, C. Favre and M. Jonsson, Singular semipositive metrics in non-archimedean geometry, preprint (2011), http://arxiv.org/abs/1201.0187.

  • [16]

    R. Cavalieri, H. Markwig and D. Ranganathan, Tropical compactification and the Gromov–Witten theory of 1, preprint (2014), http://arxiv.org/abs/1410.2837.

  • [17]

    R. Cavalieri, H. Markwig and D. Ranganathan, Tropicalizing the space of admissible covers, preprint (2014), http://arxiv.org/abs/1401.4626.

  • [18]

    A. Chambert-Loir, Mesures et équidistribution sur les espaces de Berkovich, J. reine angew. Math. 595 (2006), 215–235. Google Scholar

  • [19]

    A. Chambert-Loir, Heights and measures on analytic spaces. A survey of recent results, and some remarks, Motivic integration and its interactions with model theory and non-Archimedean geometry. Volume II, London Math. Soc. Lecture Note Ser. 384, Cambridge University Press, Cambridge (2011), 1–50. Google Scholar

  • [20]

    B. Conrad, Descent for coherent sheaves on rigid-analytic spaces, preprint (2003), http://math.stanford.edu/~conrad/papers/cohdescent.pdf.

  • [21]

    B. Conrad, Relative ampleness in rigid geometry, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, 1049–1126. CrossrefGoogle Scholar

  • [22]

    B. Conrad and M. Temkin, Descent for non-archimedean analytic spaces, preprint (2010).

  • [23]

    A. Ducros, Families of Berkovich spaces, preprint (2011), http://arxiv.org/abs/1107.4259.

  • [24]

    W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry (Santa Cruz 1995), Proc. Sympos. Pure Math. 62, American Mathematical Society, Providence (1997), 45–96. Google Scholar

  • [25]

    M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. CrossrefGoogle Scholar

  • [26]

    M. Gross, Tropical geometry and mirror symmetry, CBMS Reg. Conf. Ser. Math. 114, American Mathematical Society, Providence 2011. Google Scholar

  • [27]

    M. Gross, P. Hacking and S. Keel, Mirror symmetry for log Calabi–Yau surfaces I, preprint (2011), http://arxiv.org/abs/1106.4977v1.

  • [28]

    M. Gross and B. Siebert, From real affine geometry to complex geometry, Ann. of Math. (2) 174 (2011), no. 3, 1301–1428. CrossrefGoogle Scholar

  • [29]

    A. Grothendieck, Éléments de géométrie algébrique. I: Le langage des schémas, Publ. Math. Inst. Hautes Études Sci. 4 (1960), 1–228. Google Scholar

  • [30]

    A. Grothendieck, Éléments de géométrie algébrique. III: Étude cohomologique des faisceaux cohérents. I, Publ. Math. Inst. Hautes Études Sci. 11 (1961), 1–167. Google Scholar

  • [31]

    A. Grothendieck, Séminaire de géométrie algébrique du Bois Marie 1960–61. Revêtements étales et groupe fondamental. Fasc. II: Exposés 6, 8 à 11, Institut des Hautes Études Scientifiques, Paris 1963. Google Scholar

  • [32]

    A. Grothendieck, Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. III, Publ. Math. Inst. Hautes Études Sci. 28 (1966), 1–255. Google Scholar

  • [33]

    W. Gubler, Local heights of subvarieties over non-Archimedean fields, J. reine angew. Math. 498 (1998), 61–113. Google Scholar

  • [34]

    W. Gubler, J. Rabinoff and A. Werner, Skeletons and tropicalizations, preprint (2014), http://arxiv.org/abs/1404.7044v1.

  • [35]

    U. Hartl and W. Lütkebohmert, On rigid-analytic Picard varieties, J. reine angew. Math. 528 (2000), 101–148. Google Scholar

  • [36]

    C. Hummel, Gromov’s compactness theorem for pseudo-holomorphic curves, Progr. Math. 151, Birkhäuser, Basel 1997. Google Scholar

  • [37]

    L. Illusie, Complexe cotangent et déformations. I and II, Lecture Notes in Math. 239 and 283, Springer, Berlin 1971 and 1972. Google Scholar

  • [38]

    A. J. de Jong, Smoothness, semi-stability and alterations, Publ. Math. Inst. Hautes Études Sci. 83 (1996), 51–93. CrossrefGoogle Scholar

  • [39]

    A. J. de Jong, Families of curves and alterations, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 2, 599–621. CrossrefGoogle Scholar

  • [40]

    D. Knutson, Algebraic spaces, Lecture Notes in Math. 203, Springer, Berlin 1971. Google Scholar

  • [41]

    M. Kontsevich, Enumeration of rational curves via torus actions, The moduli space of curves (Texel Island 1994), Progr. Math. 129, Birkhäuser, Boston (1995), 335–368. Google Scholar

  • [42]

    M. Kontsevich and Y. Soibelman, Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry (Seoul 2000), World Scientific Publishing, River Edge (2001), 203–263. Google Scholar

  • [43]

    M. Kontsevich and Y. Soibelman, Affine structures and non-Archimedean analytic spaces, The unity of mathematics, Progr. Math. 244, Birkhäuser, Boston (2006), 321–385. Google Scholar

  • [44]

    M. Kontsevich and Y. Tschinkel, Non-archimedean Kähler geometry, in preparation.

  • [45]

    U. Köpf, Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen, Schr. Math. Inst. Univ. Münster (2) 7, Universität Münster, Münster 1974. Google Scholar

  • [46]

    G. Laumon and L. Moret-Bailly, Champs algébriques, Ergeb. Math. Grenzgeb. (3) 39, Springer, Berlin 2000. Google Scholar

  • [47]

    W. Lütkebohmert, Formal-algebraic and rigid-analytic geometry, Math. Ann. 286 (1990), no. 1–3, 341–371. CrossrefGoogle Scholar

  • [48]

    D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory, 3rd ed., Ergeb. Math. Grenzgeb. (2) 34, Springer, Berlin 1994. Google Scholar

  • [49]

    J. Nicaise and C. Xu, The essential skeleton of a degeneration of algebraic varieties, preprint (2013), http://arxiv.org/abs/1307.4041.

  • [50]

    M. C. Olsson, On proper coverings of Artin stacks, Adv. Math. 198 (2005), no. 1, 93–106. CrossrefGoogle Scholar

  • [51]

    P. Pansu, Compactness, Holomorphic curves in symplectic geometry, Progr. Math. 117, Birkhäuser, Basel (1994), 233–249. Google Scholar

  • [52]

    M. Porta and T. Y. Yu, Higher analytic stacks and GAGA theorems, preprint (2014), http://arxiv.org/abs/1412.5166.

  • [53]

    D. Ranganathan, Moduli of rational curves in toric varieties and non-archimedean geometry, preprint (2015), http://arxiv.org/abs/1506.03754.

  • [54]

    M. Raynaud, Géométrie analytique rigide d’après Tate, Kiehl, …, Table ronde d’analyse non archimédienne (Paris 1972), Bull. Soc. Math. France, Mém. 39/40, Société Mathématique de France, Paris (1974), 319–327. Google Scholar

  • [55]

    M. Temkin, On local properties of non-Archimedean analytic spaces, Math. Ann. 318 (2000), no. 3, 585–607. CrossrefGoogle Scholar

  • [56]

    M. Temkin, Desingularization of quasi-excellent schemes in characteristic zero, Adv. Math. 219 (2008), no. 2, 488–522. CrossrefGoogle Scholar

  • [57]

    M. Ulirsch, A geometric theory of non-archimedean analytic stacks, preprint (2014), http://arxiv.org/abs/1410.2216.

  • [58]

    R. Ye, Gromov’s compactness theorem for pseudo holomorphic curves, Trans. Amer. Math. Soc. 342 (1994), no. 2, 671–694. Google Scholar

  • [59]

    T. Y. Yu, Balancing conditions in global tropical geometry, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 4, 1647–1667.CrossrefGoogle Scholar

  • [60]

    T. Y. Yu, Enumeration of holomorphic cylinders in log Calabi–Yau surfaces, preprint (2015), http://arxiv.org/abs/1504.01722.

  • [61]

    T. Y. Yu, Tropicalization of the moduli space of stable maps, Math. Z. 281 (2015), no. 3–4, Article ID 1519, 1035–1059. CrossrefGoogle Scholar

  • [62]

    S. Zhang, Positive line bundles on arithmetic varieties, J. Amer. Math. Soc. 8 (1995), no. 1, 187–221. CrossrefGoogle Scholar

  • [63]

    The Stacks Project Authors, Stacks project, http://stacks.math.columbia.edu, 2013.

About the article

Received: 2014-07-30

Revised: 2015-07-01

Published Online: 2016-01-14

Published in Print: 2018-08-01


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018, Issue 741, Pages 179–210, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2015-0077.

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