Immersed spheres in symplectic 4-manifolds

Dusa McDuff

Annales de l'institut Fourier (1992)

  • Volume: 42, Issue: 1-2, page 369-392
  • ISSN: 0373-0956

Abstract

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We discuss conditions under which a symplectic 4-manifold has a compatible Kähler structure. The theory of J -holomorphic embedded spheres is extended to the immersed case. As a consequence, it is shown that a symplectic 4-manifold which has two different minimal reductions must be the blow-up of a rational or ruled surface.

How to cite

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McDuff, Dusa. "Immersed spheres in symplectic 4-manifolds." Annales de l'institut Fourier 42.1-2 (1992): 369-392. <http://eudml.org/doc/74958>.

@article{McDuff1992,
abstract = {We discuss conditions under which a symplectic 4-manifold has a compatible Kähler structure. The theory of $J$-holomorphic embedded spheres is extended to the immersed case. As a consequence, it is shown that a symplectic 4-manifold which has two different minimal reductions must be the blow-up of a rational or ruled surface.},
author = {McDuff, Dusa},
journal = {Annales de l'institut Fourier},
keywords = {Kähler surfaces; symplectic 4-manifold; Kähler structure; - holomorphic embedded spheres; blow-up},
language = {eng},
number = {1-2},
pages = {369-392},
publisher = {Association des Annales de l'Institut Fourier},
title = {Immersed spheres in symplectic 4-manifolds},
url = {http://eudml.org/doc/74958},
volume = {42},
year = {1992},
}

TY - JOUR
AU - McDuff, Dusa
TI - Immersed spheres in symplectic 4-manifolds
JO - Annales de l'institut Fourier
PY - 1992
PB - Association des Annales de l'Institut Fourier
VL - 42
IS - 1-2
SP - 369
EP - 392
AB - We discuss conditions under which a symplectic 4-manifold has a compatible Kähler structure. The theory of $J$-holomorphic embedded spheres is extended to the immersed case. As a consequence, it is shown that a symplectic 4-manifold which has two different minimal reductions must be the blow-up of a rational or ruled surface.
LA - eng
KW - Kähler surfaces; symplectic 4-manifold; Kähler structure; - holomorphic embedded spheres; blow-up
UR - http://eudml.org/doc/74958
ER -

References

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  1. [BPV] W. BARTH, C. PETERS & A. VAN de VEN, Complex Surfaces, Springer Verlag, 1984. Zbl0718.14023MR86c:32026
  2. [U] H. CLEMENS, J. KOLLAR, S. MORI, Higher dimensional complex geometry, Astérisque, 166 (1989). 
  3. [FGG] M. FERNANDEZ, M. GOTAY, A. GRAY, Four dimensional parallelizable symplectic and complex manifolds, Proc. Amer. Math. Soc., 103 (1988), 1209-1212. Zbl0656.53034MR90a:53039
  4. [FM] FRIEDMAN and MORGAN, Diffeomorphism types of 4-manifolds, Journ. Diff. Geo., (1988). 
  5. [GR] M. GROMOV, Pseudo-holomorphic curves on almost-complex manifolds, Invent. Math., 82 (1985), 307-347. Zbl0592.53025MR87j:53053
  6. [EX] D. MCDUFF, Examples of symplectic structures, Invent. Math., 89 (1987), 13-36. Zbl0625.53040MR88m:58061
  7. [RR] D. MCDUFF, The Structure of Rational and Ruled Symplectic 4-manifolds, Journ. Amer. Math. Soc., 3 (1990), 679-712. Zbl0723.53019MR91k:58042
  8. [EL] D. MCDUFF, Elliptic methods in symplectic geometry, Bull. Amer. Math. Soc., 23 (1990), 311-358. Zbl0723.53018MR91i:58046
  9. [BL] D. MCDUFF, Blow ups and symplectic embeddings in dimension 4, Topology, 30 (1991), 409-421. Zbl0731.53035MR92m:57039
  10. [LB] D. MCDUFF, The Local Behaviour of holomorphic curves in almost complex 4-manifolds, Journ. Diff. Geom., 34 (1991), 143-164. Zbl0736.53038MR93e:53050
  11. [KY] D. MCDUFF, Symplectic 4-manifolds, to appear in Proceedings of I.C.M., Kyoto, 1990. Zbl0732.57012MR94b:57042
  12. [UB] D. MCDUFF, Remarks on the uniqueness of symplectic blowing up, preprint, 1990. 
  13. [RU] D. MCDUFF, Notes on Ruled Symplectic 4-manifolds, preprint, 1992. Zbl0810.53020
  14. [PW] T. PARKER and J. WOLFSON, A compactness theorem for Gromov's moduli space, preprint, 1991. 
  15. [TH] W. THURSTON, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc., 55 (1976), 467-468. Zbl0324.53031MR53 #6578
  16. [WO] J. WOLFSON, Gromov's compactness of pseudo-holomorphic curves and symplectic geometry, J. Diff. Geom., 28 (1988), 383-405. Zbl0661.53024MR89m:53058
  17. [YE] R. YE, Gromov's Compactness Theorem for Pseudo-holomorphic Curves, preprint, UCSB, 1991. Zbl0810.53024

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