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Acta Mathematica Vietnamica

, Volume 38, Issue 1, pp 123–144 | Cite as

Uniqueness of symplectic structures

  • Dietmar Salamon
Article

Abstract

This survey paper discusses some uniqueness questions for symplectic forms on compact manifolds without boundary.

Keywords

Symplectic structures Uniqueness Donaldson’s geometric flow 

Mathematics Subject Classification

53D35 

Notes

Acknowledgement

Thanks to Paul Biran, Simon Donaldson, Yael Karshon, Janko Latschev, Dusa McDuff, and Stefano Vidussi for many helpful comments and suggestions.

D.M. partially supported by the Swiss National Science Foundation Grant 200021-127136.

References

  1. 1.
    Abreu, M.: Topology of symplectomorphism groups of S 2×S 2. Invent. Math. 131, 1–23 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Abreu, M., McDuff, D.: Topology of symplectomorphism groups of rational ruled surfaces. J. Am. Math. Soc. 13, 971–1009 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bates, L., Peschke, G.: A remarkable symplectic structure. J. Differ. Geom. 32, 33–38 (1990) MathSciNetGoogle Scholar
  4. 4.
    Biran, P.: Connectedness of spaces of symplectic embeddings. IMRN, 487–491 (1996) Google Scholar
  5. 5.
    Biran, P.: Symplectic packing in dimension 4. Geom. Funct. Anal. 7, 420–437 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Biran, P.: A stability property of symplectic packing. Invent. Math. 136, 123–155 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Biran, P.: From symplectic packing to algebraic geometry and back In: European Congress of Mathematics, Barcelone (2000). Progress in Math., vol. 202, pp. 507–524. Birkhäuser, Basel (2001) CrossRefGoogle Scholar
  8. 8.
    Conolly, F., Lê, H.V., Ono, K.: Almost complex structures which are compatible with Kähler or symplectic structures. Ann. Glob. Anal. Geom. 15, 325–334 (1997) CrossRefGoogle Scholar
  9. 9.
    Demazure, M.: Surfaces de del Pezzo II–V. In: Séminar sur les singularités de surfaces (1976–1977). Lecture Notes in Mathematics, vol. 777. Springer, Berlin (1980) CrossRefGoogle Scholar
  10. 10.
    Donaldson, S.K.: The orientation of Yang–Mills moduli spaces and four-manifold topology. J. Differ. Geom. 26, 397–428 (1987) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Donaldson, S.K.: Polynomial invariants of smooth four-manifolds. J. Differ. Geom. 29, 257–315 (1990) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Donaldson, S.K.: Moment maps and diffeomorphisms. Asian J. Math. 3, 1–16 (1999) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Donaldson, S.K.: Two-forms on four-manifolds and elliptic equations. Inspired by S.S. Chern. In: Griffits, P.A. (ed.) Nankai Tracts Mathematics, vol. 11, pp. 153–172. World Scientific, Singapore (2006) Google Scholar
  14. 14.
    Eliashberg, Y., Mishachev, N.: Introduction to the h-Principle. Graduate Studies in Mathematics, vol. 48. AMS, New York (2002) zbMATHGoogle Scholar
  15. 15.
    Fintushel, R., Stern, R.J.: Knots, links, and four-manifolds. Invent. Math. 134, 363–400 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Gompf, R., Mrowka, T.: Irreducible four-manifolds need not be complex. Ann. Math. 138, 61–111 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Gromov, M.: Stable mappings of foliations into manifolds. Izv. Akad. Nauk SSSR, Ser. Mat. 33, 707–734 (1969) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Gromov, M.: Partial Differential Relations. Springer, Berlin (1986) zbMATHCrossRefGoogle Scholar
  20. 20.
    Ionel, E.-N., Parker, T.H.: Gromov invariants and symplectic maps. Math. Ann. 314, 127–158 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Karshon, Y., Kessler, L., Pinsonnault, M.: Symplectic blowups of the complex projective plane and counting toric actions. Preprint. http://www.math.toronto.edu/karshon/, October (2012)
  22. 22.
    Krom, R.: Ph.D. Thesis, in preparation Google Scholar
  23. 23.
    Kronheimer, P.B.: Some nontrivial families of symplectic structures. http://www.math.harvard.edu/~kronheim/papers.html. Preprint, Harvard (1997)
  24. 24.
    Lalonde, F.: Isotopy of symplectic balls, Gromov’s radius, and the structure of symplectic manifolds. Math. Ann. 300, 273–296 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Lalonde, F., McDuff, D.: The classification of ruled symplectic 4-manifolds. Math. Res. Lett. 3, 769–778 (1996) MathSciNetzbMATHGoogle Scholar
  26. 26.
    Lalonde, F., McDuff, D.: J-curves and the classification of rational and ruled symplectic 4-manifolds. In: Thomas, C. (ed.) Symplectic and Contact Geometry, pp. 3–42. Cambridge University Press, Cambridge (1996) Google Scholar
  27. 27.
    Latschev, J., McDuff, D., Schlenk, F.: The Gromov-width of four-dimensional tori. Preprint, November (2011) Google Scholar
  28. 28.
    Lawson, H.B., Michelsohn, M.-L.: Spin Geometry. Princeton Mathematical Series, vol. 58. Princeton University Press, Princeton (1989) zbMATHGoogle Scholar
  29. 29.
    Li, T.J., Liu, A.: General wallcrossing formula. Math. Res. Lett. 2, 797–810 (1995) MathSciNetzbMATHGoogle Scholar
  30. 30.
    Li, T.J., Liu, A.: Symplectic structure Liu on ruled surfaces and generalized adjunction formula. Math. Res. Lett. 2, 453–471 (1995) MathSciNetzbMATHGoogle Scholar
  31. 31.
    Li, T.J., Liu, A.: Uniqueness of symplectic canonical class, surface cone and symplectic cone of 4-manifolds with b +=1. J. Differ. Geom. 58, 331–370 (2001) zbMATHGoogle Scholar
  32. 32.
    McDuff, D.: Examples of symplectic structures. Invent. Math. 89, 13–36 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    McDuff, D.: The structure of rational and ruled symplectic 4-manifolds. J. Am. Math. Soc. 3, 679–712 (1990). Erratum: J. Amer. Math. Soc, 5, 987–988 (1992) MathSciNetzbMATHGoogle Scholar
  34. 34.
    McDuff, D.: Blowing up and symplectic embeddings. Topology 30, 409–421 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    McDuff, D.: Remarks on the uniqueness of symplectic blowing up. In: Salamon, D. (ed.) Symplectic Geometry. LMS Lecture Note Series, vol. 192, pp. 157–168. Cambridge University Press, Cambridge (1993) Google Scholar
  36. 36.
    McDuff, D.: From symplectic deformation to isotopy. In: Stern, R.J. (ed.) Topics in Symplectic Four-Manifolds. International Press Lecture Series, vol. 1, pp. 85–100. Intl. Press, Cambridge (1998) Google Scholar
  37. 37.
    McDuff, D., Polterovich, L.: Symplectic pac: kings and algebraic geometry. Invent. Math. 115, 405–429 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    McDuff, D., Salamon, D.: Introduction to Symplectic Topology, 2nd edn. Oxford University Press, Oxford (2005) Google Scholar
  39. 39.
    McDuff, D., Salamon, D.: J-Holomorphic Curves and Symplectic Topology, 2nd edn. AMS Colloquium Publications, vol. 52 (2012) zbMATHGoogle Scholar
  40. 40.
    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) MathSciNetzbMATHGoogle Scholar
  41. 41.
    Milnor, J.W.: Topology from the Differential Viewpoint. University Press of Virginia, Charlottesville (1965) Google Scholar
  42. 42.
    Ruan, Y.: Symplectic topology on algebraic 3-folds. J. Differ. Geom. 39, 215–227 (1994) zbMATHGoogle Scholar
  43. 43.
    Ruan, Y., Tian, G.: Higher genus symplectic invariants and sigma models coupled with gravity. Invent. Math. 130, 455–516 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Seidel, P.: Floer homology and the symplectic isotopy problem. Ph.D. Thesis, Oxford University (1997) Google Scholar
  45. 45.
    Seidel, P.: A long exact sequence for symplectic Floer cohomology. Topology 42, 1003–1063 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Seidel, P.: Lectures on four-dimensional Dehn twists. In: Symplectic Four-Manifolds and Algebraic Surfaces. Lecture Notes in Mathematics, vol. 1938, pp. 231–267. Springer, Berlin (2008) CrossRefGoogle Scholar
  47. 47.
    Smith, I.: On moduli spaces of symplectic forms. Math. Res. Lett. 7, 779–788 (2000) MathSciNetzbMATHGoogle Scholar
  48. 48.
    Smith, I.: Torus fibrations on symplectic 4-manifolds. Turk. J. Math. 25(1), 69–95 (2001) zbMATHGoogle Scholar
  49. 49.
    Taubes, C.H.: The Seiberg–Witten invariants and symplectic forms. Math. Res. Lett. 1, 809–822 (1994) MathSciNetzbMATHGoogle Scholar
  50. 50.
    Taubes, C.H.: More constraints on symplectic forms from Seiberg–Witten invariants. Math. Res. Lett. 2, 9–14 (1995) MathSciNetzbMATHGoogle Scholar
  51. 51.
    Taubes, C.H.: The Seiberg–Witten and the Gromov invariants. Math. Res. Lett. 2, 221–238 (1995) MathSciNetzbMATHGoogle Scholar
  52. 52.
    Taubes, C.H.: Counting pseudoholomorphic submanifolds in dimension four. J. Differ. Geom. 44, 818–893 (1996) (republished in [53]) MathSciNetzbMATHGoogle Scholar
  53. 53.
    Taubes, C.H.: Seiberg–Witten and Gromov invariants for symplectic 4-manifolds. In: First Internat. Press Lecture Series, vol. 2. Internat. Press, Cambridge (2000) Google Scholar
  54. 54.
    Vidussi, S.: Homotopy K3’s with several symplectic structures. Geom. Topol. 5, 267–285 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Vidussi, S.: Smooth structure of some symplectic surfaces. Mich. Math. J. 49, 325–330 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Wall, C.T.C.: On the orthogonal groups of unimodula. Math. Ann. 147, 328–338 (1962) MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Wall, C.T.C.: On the orthogonal groups of unimodular quadratic forms II. J. Reine Angew. Math. 213, 122–136 (1963) MathSciNetzbMATHGoogle Scholar
  58. 58.
    Wall, C.T.C.: Diffeomorphisms of 4-manifolds. J. Lond. Math. Soc. 39, 131–140 (1963) CrossRefGoogle Scholar
  59. 59.
    Wall, C.T.C.: On simply connected 4-manifolds. J. Lond. Math. Soc. 39, 141–149 (1964) zbMATHCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013

Authors and Affiliations

  1. 1.ETH ZürichZürichGermany

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