Advertisement

Mathematische Annalen

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

Symplectic tori in rational elliptic surfaces

  • Tolga Etgü
  • B. Doug ParkEmail author
Article

Abstract

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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