Mathematische Annalen

, Volume 334, Issue 3, pp 679–691 | Cite as

Symplectic tori in rational elliptic surfaces

  • Tolga Etgü
  • B. Doug ParkEmail author


Let E(1) p denote the rational elliptic surface with a single multiple fiber f p of multiplicity p. We construct an infinite family of homologous non-isotopic symplectic tori representing the primitive 2-dimensional homology class [f p ] in E(1) p when p>1. As a consequence, we get infinitely many non-isotopic symplectic tori in the fiber class of the rational elliptic surface Open image in new window . We also show how these tori can be non-isotopically embedded as homologous symplectic submanifolds in other symplectic 4-manifolds.

Mathematics Subject Classification (2000)

Primary 57R17 57R57 Secondary 32J27 53D35 57R95 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Buchdahl, N.: On compact Kähler surfaces. Ann. Inst. Fourier, Grenoble 49, 287–302 (1999)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Demailly, J.-P.: E-mail communication to the second author on 25 July 2005Google Scholar
  3. 3.
    Demailly, J.-P., Paun, M.: Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. Math. 159, 1247–1274 (2004)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Etgü, T., Park, B.D.: Non-isotopic symplectic tori in the same homology class. Trans. Amer. Math. Soc. 356, 3739–3750 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Etgü, T., Park, B.D.: Homologous non-isotopic symplectic tori in a K3 surface. Commun. Contemp. Math. 7, 325–340 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Etgü, T., Park, B.D.: Homologous non-isotopic symplectic tori in homotopy rational elliptic surfaces. Math. Proc. Cambridge Philos. Soc. (to appear)Google Scholar
  7. 7.
    Fintushel, R., Stern, R.J.: Rational blowdown of smooth 4-manifolds. J. Differential Geom. 46, 181–235 (1997)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Fintushel, R., Stern, R.J.: Knots, links and 4-manifolds. Invent. Math. 134, 363–400 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Fintushel, R., Stern, R.J.: Symplectic surfaces in a fixed homology class. J. Differential Geom. 52, 203–222 (1999)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Freedman, M.H., Quinn, F.: Topology of 4-Manifolds. Princeton University Press, Princeton, NJ, 1990Google Scholar
  11. 11.
    Friedman, R.: Vector bundles and SO(3)-invariants for elliptic surfaces. J. Amer. Math. Soc. 8, 29–139 (1995)CrossRefGoogle Scholar
  12. 12.
    Friedman, R., Morgan, J.W.: Smooth Four-Manifolds and Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) vol. 27, Springer-Verlag, Berlin, 1994Google Scholar
  13. 13.
    Gompf, R.E., Stipsicz, A.I.: 4-Manifolds and Kirby Calculus. Graduate Studies in Mathematics vol. 20, Amer. Math. Soc., Providence, RI, 1999Google Scholar
  14. 14.
    Kodaira, K., Spencer, D.C.: On deformations of complex analytic structures, III. Stability theorems for complex structures. Ann. Math. 71, 43–76 (1960)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Lamari, A.: Le cône kählérien d'une surface. J. Math. Pures Appl. 78, 249–263 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Li, T.J., Liu, A.: Symplectic structure on ruled surfaces and a generalized adjunction formula. Math. Res. Lett. 2, 453–471 (1995)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Matumoto, T.: Extension problem of diffeomorphisms of a 3-torus over some 4-manifolds. Hiroshima Math. J. 14, 189–201 (1984)zbMATHMathSciNetGoogle Scholar
  18. 18.
    McDuff, D.: From symplectic deformation to isotopy. In: Topics in Symplectic 4-Manifolds (Irvine, CA, 1996), pp. 85–99, Int. Press Lect. Ser. vol. 1, International Press, Cambridge, MA, 1998Google Scholar
  19. 19.
    McMullen, C.T., Taubes, C.H.: 4-manifolds with inequivalent symplectic forms and 3-manifolds with inequivalent fibrations. Math. Res. Lett. 6, 681–696 (1999)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Morton, H.R.: The multivariable Alexander polynomial for a closed braid. In: Low-dimensional Topology (Funchal, 1998), pp. 167–172, Contemp. Math. vol. 233, Amer. Math. Soc., Providence, RI, 1999Google Scholar
  21. 21.
    Park, B. D.: A gluing formula for the Seiberg-Witten invariant along T 3. Michigan Math. J. 50, 593–611 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Ruan, Y.: Symplectic topology and complex surfaces. In: Geometry and Analysis on Complex Manifolds, pp. 171–197, World Sci. Publishing, River Edge, NJ, 1994Google Scholar
  23. 23.
    Siebert, B., Tian, G.: On the holomorphicity of genus two Lefschetz fibrations. Ann. Math. 161, 955–1016 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sikorav, J.-C.: The gluing construction for normally generic J-holomorphic curves. In: Symplectic and Contact Topology: Interactions and Perspectives (Toronto, ON/Montréal, QC, 2001), pp. 175–199, Fields Inst. Commun. vol. 35, Amer. Math. Soc., Providence, RI, 2003Google Scholar
  25. 25.
    Symington, M.: Symplectic rational blowdowns. J. Differential Geom. 50, 505–518 (1998)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Taubes, C.H.: The Seiberg-Witten invariants and 4-manifolds with essential tori. Geom. Topol. 5, 441–519 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Vidussi, S.: Smooth structure of some symplectic surfaces. Michigan Math. J. 49, 325–330 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Vidussi, S.: Nonisotopic symplectic tori in the fiber class of elliptic surfaces. J. Symplectic Geom. 2, 207–218 (2004)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Vidussi, S.: Symplectic tori in homotopy E(1)'s. Proc. Amer. Math. Soc. 133, 2477–2481 (2005)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Department of MathematicsKoç UniversityIstanbulTurkey
  2. 2.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada

Personalised recommendations