Inventiones mathematicae

, Volume 204, Issue 2, pp 505–594 | Cite as

Reeb orbits and the minimal discrepancy of an isolated singularity

  • Mark McLeanEmail author


Let A be an affine variety inside a complex N dimensional vector space which has an isolated singularity at the origin. The intersection of A with a very small sphere turns out to be a contact manifold called the link of A. Any contact manifold contactomorphic to the link of A is said to be Milnor fillable by A. If the first Chern class of our link is torsion then we can assign an invariant of our singularity called the minimal discrepancy, which is an important invariant in birational geometry. We define an invariant of the link up to contactomorphism using Conley–Zehnder indices of Reeb orbits and then we relate this invariant with the minimal discrepancy. As a result we show that the standard contact five dimensional sphere has a unique Milnor filling up to normalization proving a conjecture by Seidel.



I would like to thank Chris Wendl, Ivan Smith, Cheuk Yu Mak and Paul Seidel (who enabled me to generalize the result from numerically Gorenstein to numerically \({\mathbb {Q}}\)-Gorenstein singularities). I would also like to thank the referees for numerous helpful comments which have improved this paper. The main work for this paper was done while I was at the University of Aberdeen and therefore I would like to thank the people at the mathematics department for providing a great working environment.


  1. 1.
    Ambro, F.: The minimal log discrepancy. In: Proceedings of the Workshop, Arc Spaces and Multiplier Ideal Sheaves, vol. 1550, pp. 121–130 (2006)Google Scholar
  2. 2.
    Arnol’d, V.: On a characteristic class entering into conditions of quantization. Funkcional. Anal. i Priložen. 1, 1–14 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Abouzaid, M., Seidel, P.: An open string analogue of Viterbo functoriality. Geom. Topol. 14(2), 627–718 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Boucksom, S., de Fernex, T., Favre, C., Urbinati, S.: Valuation spaces and multiplier ideals on singular varieties. In: Trends in Singularities. London Mathematical Society Lecture Note Series, vol. 417, pp. 29–51. Cambridge University Press, Cambridge (2014)Google Scholar
  5. 5.
    Behrend, K.: Gromov–Witten invariants in algebraic geometry. Invent. Math. 127(3), 601–617 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bourgeois, F., Eliashberg, Y., Hofer, H., Wysocki, K., Zehnder, E.: Compactness results in symplectic field theory. Geom. Topol. 7, 799–888 (2003)Google Scholar
  7. 7.
    Behrend, K., Fantechi, B.: The intrinsic normal cone. Invent. Math. 128(1), 45–88 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bao, E., Honda, K.: Definition of cylindrical contact homology in dimension three. pp. 1–53 (2014). arXiv:1412.0276
  9. 9.
    Bourgeois, F.: A Morse-Bott approach to contact homology. Ph.D. Thesis, Stanford University, ProQuest LLC, Ann Arbor (2002)Google Scholar
  10. 10.
    Brieskorn, E.: Beispiele zur Differentialtopologie von Singularitäten. Invent. Math. 2, 1–14 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Brieskorn, E.: Examples of singular normal complex spaces which are topological manifolds. Proc. Nat. Acad. Sci. USA 55, 1395–1397 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Brieskorn, E.: Singularities in the work of Friedrich Hirzebruch. In: Survey of Differential Geometry, vol. 7. pp. 17–60. Int. Press, Somerville (2000)Google Scholar
  13. 13.
    Cieliebak, K., Mohnke, K.: Compactness for punctured holomorphic curves. J. Symp. Geom. 4, 589–654 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cieliebak, K., Mohnke, K.: Symplectic hypersurfaces and transversality in Gromov–Witten theory. J. Symplectic Geom. 5(3), 281–356 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Caubel, C., Némethi, A., Popescu-Pampu, P.: Milnor open books and Milnor fillable contact 3-manifolds. Topology 45(3), 673–689 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cieliebak, K., Volkov, E.: First steps in stable Hamiltonian topology. J. Eur. Math. Soc. 17, 321–404 (2015). doi: 10.4171/JEMS/505
  17. 17.
    Conley, C., Zehnder, E.: Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Commun. Pure Appl. Math. 37(2), 207–253 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dragnev, D.: Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations. Commun. Pure Appl. Math. 57(6), 726–763 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Durfee, A.: The signature of smoothings of complex surface singularities. Math. Ann. 232(1), 85–98 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Eliashberg, Y., Givental, A., Hofer, H.: Introduction to symplectic field theory. In: Geometric Functional Analysis, (Special Volume, Part II), pp. 560–673. GAFA 2000 (Tel Aviv, 1999) (2000)Google Scholar
  21. 21.
    Eisenbud, D.: Commutative Algebra—With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)zbMATHGoogle Scholar
  22. 22.
    Eliashberg, Y.: Filling by holomorphic discs and its applications. In: Geometry of Low-Dimensional Manifolds, 2 (Durham, 1989), London Mathematical Society, Lecture Note Series. pp. 45–67, vol. 151. Cambridge University Press, Cambridge (1990)Google Scholar
  23. 23.
    Ein, L., Mustaţă, M., Yasuda, T.: Jet schemes, log discrepancies and inversion of adjunction. Invent. Math. 153(3), 519–535 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Fish, J.: Target-local Gromov compactness. Geom. Topol. 15(5), 765–826 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Fukaya, K., Ono, K.: Arnold conjecture and Gromov–Witten invariant. Topology 38(5), 933–1048 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library. Wiley, New York (1994) (reprint of the 1978 original)Google Scholar
  27. 27.
    Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82(2), 307–347 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gutt, J.: Generalized Conley–Zehnder index. Annales de la faculté des sciences de Toulouse 23(4), 907–932 (2014)Google Scholar
  29. 29.
    Gutt, J.: The Conley–Zehnder Index for a Path of Symplectic Matrices, pp. 1–60 (2013). arXiv:1201.3728
  30. 30.
    Heegaard, P.: Sur l”’Analysis situs. Bull. Soc. Math. France 44, 161–242 (1916)MathSciNetGoogle Scholar
  31. 31.
    Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. Math. (2) 79, 109–203 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. Math. (2) 79, 205–326 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Hirzebruch, F.: Singularities and exotic spheres. Séminaire N. Bourbaki. Exp. No. 314, 13–32 (1966–1968)Google Scholar
  34. 34.
    Hofer, H., Wysocki, K., Zehnder, E.: Applications of Polyfold Theory I: The Polyfolds of Gromov–Witten Theory. pp. 1–205 (2011). arXiv:1107.2097
  35. 35.
    Hacon, C., Xu, C.: On finiteness of b-representation and semi-log canonical abundance, pp. 1–13 (2011). arXiv:1107.4149
  36. 36.
    Hofer, H., Zehnder, E.: Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel (1994)Google Scholar
  37. 37.
    Kawamata, Y.: The minimal discrepancy of a 3-fold terminal singularity, Appendix to Shukorov, V.V. 3-fold log flips. Russ. Acad. Sci. Izv. Math 40(3), 93–202 (1993)Google Scholar
  38. 38.
    Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998) (with the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original)Google Scholar
  39. 39.
    Kollár, J., Némethi, A.: Holomorphic arcs on singularities. Invent. Math. (2014). doi: 10.1007/s00222-014-0530-3
  40. 40.
    Kollár, J.: Flips and Abundance for Algebraic Threefolds: 2nd Algebraic Geometry Summer Seminar: Papers. Asterisque—Societe Mathematique de France. Societe Mathematique de France (1992)Google Scholar
  41. 41.
    Kwon, M., van Koert, O.: Brieskorn manifolds in contact topology, pp. 1–34 (2013). arXiv:1310.0343
  42. 42.
    Li, J., Tian, G.: Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties. J. Am. Math. Soc. 11(1), 119–174 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Li, J., Tian, G.: Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds. In: Topics in Symplectic \(4\)-Manifolds (Irvine, CA, 1996), First Int. Press Lect. Ser., I, pp. 47–83. Int. Press, Cambridge (1998)Google Scholar
  44. 44.
    Markushevich, D.: Minimal discrepancy for a terminal cdv singularity is 1. J. Math. Sci. Univ. Tokyo 3, 445–456 (1996)MathSciNetzbMATHGoogle Scholar
  45. 45.
    McLean, M.: The growth rate of symplectic homology and affine varieties. Geom. Funct. Anal. 22(2), 369–442 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Milnor, J.: Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton, University of Tokyo Press, Tokyo (1968)Google Scholar
  47. 47.
    McDuff, D., Salamon, D.: Introduction to Symplectic Topology. Oxford Mathematical Monographs, 2nd edn. The Clarendon Press, Oxford University Press, New York (1998)Google Scholar
  48. 48.
    McDuff, D., Salamon, D.: \(J\)-Holomorphic Curves and Symplectic Topology, American Mathematical Society Colloquium Publications, vol. 52. American Mathematical Society, Providence (2004)zbMATHGoogle Scholar
  49. 49.
    McLean, M., Tehrani, M.F., Zinger, A.: Normal Crossings Divisors and Configurations for Symplectic Topology, pp. 1–66, (2014). arXiv:1410.0609
  50. 50.
    Mumford, D.: The topology of normal singularities of an algebraic surface and a criterion for simplicity. Inst. Hautes Études Sci. Publ. Math. 9, 5–22 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Nash, J.: Arc structure of singularities. Duke Math. J. 81(1), 31–38 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Nelson, J.: Automatic transversality in contact homology I: regularity. pp. 1–68 (2014). arXiv:1407.3993
  53. 53.
    Nelson, J., Hutchings, M.: Cylindrical contact homology for dynamically convex contact forms in three dimensions, pp. 1–27 (2014). arXiv:1407.2898
  54. 54.
    Reid, M.: Canonical 3-folds. Journées de géométrie algébrique d’Angers, pp. 273–310 (1979)Google Scholar
  55. 55.
    Reid, M.: Minimal models of canonical 3-folds. In: Algebraic varieties and analytic varieties (Tokyo, 1981), Adv. Stud. Pure Math., pp. 131–180, vol. 1. North-Holland, Amsterdam (1983)Google Scholar
  56. 56.
    Ritter, A.: Floer theory for negative line bundles via Gromov–Witten invariants. Adv. Math. 262, 1035–1106 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Robbin, J., Salamon, D.: The Maslov index for paths. Topology 32(4), 827–844 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Ruan, Y., Tian, G.: A mathematical theory of quantum cohomology. J. Differ. Geom. 42(2), 259–367 (1995)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Ruan, Y.: Symplectic topology on algebraic \(3\)-folds. J. Differ. Geom. 39(1), 215–227 (1994)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Ruan, Y.: Topological sigma model and Donaldson-type invariants in Gromov theory. Duke Math. J. 83(2), 461–500 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Seidel, P.: Topics in Geometric Topology (2007).
  62. 62.
    Seidel, P.: A biased view of symplectic cohomology. In: Current Developments in Mathematics, vol. 2006, pp. 211–253. Int. Press, Somerville (2008)Google Scholar
  63. 63.
    Shukorov, V.V.: Problems about fano varieties. In: Birational Geometry of Algebraic Varieties, Open Problems. pp. 30–32 (1988)Google Scholar
  64. 64.
    Shukorov, V.V.: Letters of a Bi-Rationalist IV: Geometry of log flips (2002). arXiv:math/0206004
  65. 65.
    Salamon, D., Zehnder, E.: Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Commun. Pure Appl. Math. 45(10), 1303–1360 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Ustilovsky, I.: Infinitely many contact structures on \(S^{4m+1}\). Int. Math. Res. Not. 14, 781–791 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Varchenko, A.: Contact structures and isolated singularities. Mosc. Univ. Math. Bull. 35(2), 18–22 (1982)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Vilms, J.: Totally geodesic maps. J. Differ. Geom. 4, 73–79 (1970)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Zariski, O.: The connectedness theorem for birational transformations. Algebraic geometry and topology. In: A Symposium in Honor of S. Lefschetz, pp. 182–188. Princeton University Press, Princeton (1957)Google Scholar

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Authors and Affiliations

  1. 1.Stony Brook UniversityStony BrookUSA

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