Advertisement

Links of sandwiched surface singularities and self-similarity

  • Lorenzo Fantini
  • Charles Favre
  • Matteo RuggieroEmail author
Article
  • 13 Downloads

Abstract

We characterize sandwiched singularities in terms of their link in two different settings. We first prove that such singularities are precisely the normal surface singularities having self-similar non-archimedean links. We describe this self-similarity both in terms of Berkovich analytic geometry and of the combinatorics of weighted dual graphs. We then show that a complex surface singularity is sandwiched if and only if its complex link can be embedded in a Kato surface in such a way that its complement remains connected.

Mathematics Subject Classification

14B05 32P05 32S05 

Notes

Acknowledgements

We would like to warmly thank B. Teissier, who asked us about a characterization of singularities having self-similar Riemann-Zariski spaces. This paper is a tentative answer to his question in the framework of normalized Berkovich analytic spaces. We also thank the referee for their very careful reading of a first version of this paper and for their constructive comments.

References

  1. 1.
    Abhyankar, S.: On the valuations centered in a local domain. Am. J. Math. 78(2), 321–348 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Artin, M.: Some numerical criteria for contractability of curves on algebraic surfaces. Am. J. Math. 84, 485–496 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Artin, M.: On the solutions of analytic equations. Invent. Math. 5, 277–291 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Artin, M.: Algebraization of formal moduli. II. Existence of modifications. Ann. Math. 91(2), 88–135 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, second edition (2004)Google Scholar
  6. 6.
    Berkovich, V.G.: Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33. American Mathematical Society, Providence, RI (1990)Google Scholar
  7. 7.
    Brieskorn, E.: Rationale Singularitäten komplexer Flächen. Invent. Math. 4, 336–358, (1967/1968)Google Scholar
  8. 8.
    Caubel, C., Némethi, A., Popescu-Pampu, P.: Milnor open books and Milnor fillable contact 3-manifolds. Topology 45(3), 673–689 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    de Felipe, A.B.: Topology of spaces of valuations and geometry of singularities. Trans. Amer. Math. Soc. 371(5), 3593–3626 (2019)Google Scholar
  10. 10.
    de Fernex, T., Kollár, J., Xu, C.: The dual complex of singularities. 74, 103–129 (2017)Google Scholar
  11. 11.
    de Jong, T., van Straten, D.: Deformation theory of sandwiched singularities. Duke Math. J. 95(3), 451–522 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dloussky, G.: Structure des surfaces de Kato. Mém. Soc. Math. France (N.S.), (14):ii+120, (1984)Google Scholar
  13. 13.
    Ducros, A.: La structure des courbes analytiques. Book in preparation. The numbering in the text refers to the preliminary version of 12/02/2014, available at http://www.math.jussieu.fr/~ducros/livre.html (2014)
  14. 14.
    Fantini, L.: Normalized Berkovich spaces and surface singularities. Trans. Am. Math. Soc. 370(11), 7815–7859 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fantini, L., Turchetti, D.: Galois descent of semi-affinoid spaces. Math. Z. 290(3–4), 1085–1114 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Favre, C.: Holomorphic self-maps of singular rational surfaces. Publ. Mat. 54(2), 389–432 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Favre, C., Jonsson, M.: The valuative tree, Lecture Notes in Mathematics, vol. 1853. Springer, Berlin (2004)Google Scholar
  18. 18.
    Favre, C., Ruggiero, M.: Normal surface singularities admitting contracting automorphisms. Ann. Fac. Sci. Toulouse Math. (6) 23(4), 797–828 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    GarcíaBarroso, E.R., González Pérez, P.D., Popescu-Pampu, P., Ruggiero, M.: Ultrametric properties for valuation spaces ofnormal surface singularities. Transact. AMS. (to appear). Preprint available at arXiv:1802.01165
  20. 20.
    Gignac W., Ruggiero, M.: Local dynamics of non-invertible maps near normal surface singularities. Mem. AMS. (to appear). Preprintavailable at arXiv:1704.04726
  21. 21.
    Grauert, H., Peternell, Th., Remmert, R. (eds). Several complex variables. VII, volume 74 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 1994. Sheaf-theoretical methods in complex analysis, A reprint of ıt Current problems in mathematics. Fundamental directions. Vol. 74 (Russian), Vseross. Inst. Nauchn. i Tekhn. Inform. (VINITI), MoscowGoogle Scholar
  22. 22.
    Grauert, H.: Über Modifikationen und exzeptionelle analytische Mengen. Math. Ann. 146, 331–368 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hrushovski, E., Loeser, F., Poonen, B.: Berkovich spaces embed in Euclidean spaces. Enseign. Math. 60(3–4), 273–292 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Jonsson, M.: Dynamics of Berkovich spaces in low dimensions. In: Berkovich Spaces and Applications, Lecture Notes in Math., vol. 2119, pp. 205–366. Springer, Cham (2015)Google Scholar
  25. 25.
    Kato, M.: Compact complex manifolds containing “global” spherical shells. I. In: Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), pp. 45–84. Tokyo. Kinokuniya Book Store (1978)Google Scholar
  26. 26.
    Kato, M.: Compact complex surfaces containing global strongly pseudoconvex hypersurfaces. Tôhoku Math. J. (2) 31(4), 537–547 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kollár, J.: Links of complex analytic singularities. In: Surveys in differential geometry. Geometry and topology, Surv. Differ. Geom., vol. 18, pp. 157–193. Int. Press, Somerville, MA (2013)Google Scholar
  28. 28.
    Laufer, H.B.: Normal Two-Dimensional Singularities. Princeton University Press, Princeton, N.J. Annals of Mathematics Studies, No. 71 (1971)Google Scholar
  29. 29.
    Lee, Y., Nakayama, N.: Simply connected surfaces of general type in positive characteristic via deformation theory. Proc. Lond. Math. Soc. (3) 106(2), 225–286 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Looijenga, E.J.N.: Isolated singular points on complete intersections, Surveys of Modern Mathematics, 2nd edn, vol. 5. International Press, Somerville, MA; Higher Education Press, Beijing (2013)Google Scholar
  31. 31.
    McLean, M.: Reeb orbits and the minimal discrepancy of an isolated singularity. Inventiones mathematicae 204, 1–90 (2015)MathSciNetGoogle Scholar
  32. 32.
    Némethi, A., Popescu-Pampu, P.: On the Milnor fibers of sandwiched singularities. Int. Math. Res. Not. IMRN 6, 1041–1061 (2010)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Nicaise, J., Sebag, J.: Motivic Serre invariants, ramification, and the analytic Milnor fiber. Invent. Math. 168(1), 133–173 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Rossi, H.: Vector fields on analytic spaces. Ann. Math. 2(78), 455–467 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Spivakovsky, M.: Sandwiched singularities and desingularization of surfaces by normalized Nash transformations. Ann. Math. (2) 131(3), 411–491 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Teleman, A.: Instantons and curves on class VII surfaces. Ann. Math. (2) 172(3), 1749–1804 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Thuillier, A.: Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels. Manuscripta Math 123(4), 381–451 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Viehweg, E.: Quasi-projective moduli for polarized manifolds, volume 30 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin (1995)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut de Mathématiques de MarseilleUniversité Aix-MarseilleMarseilleFrance
  2. 2.CNRS - Centre de Mathématiques Laurent Schwartz, École polytechniquePalaiseau CedexFrance
  3. 3.Institut Mathématique de JussieuUniversité Paris Diderot - Université de ParisParis Cedex 13France

Personalised recommendations