Immersed spheres in symplectic 4-manifolds
Annales de l'Institut Fourier, …, Tome 42 (1992) no. 1-2, pp. 369-392.

Nous étudions des conditions sous lesquelles une variété symplectique de dimension 4 admet une structure kählérienne compatible. La théorie des sphères plongées J-holomorphes est généralisée au cas immergé. Nous démontrons comme conséquence qu’une variété symplectique de dimension 4 qui a deux réductions minimales, est nécessairement l’éclatement d’une surface rationnelle ou réglée.

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.

@article{AIF_1992__42_1-2_369_0,
     author = {Duff, Dusa Mc},
     title = {Immersed spheres in symplectic 4-manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {369--392},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {42},
     number = {1-2},
     year = {1992},
     doi = {10.5802/aif.1296},
     mrnumber = {93k:53030},
     zbl = {0756.53021},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1296/}
}
TY  - JOUR
AU  - Duff, Dusa Mc
TI  - Immersed spheres in symplectic 4-manifolds
JO  - Annales de l'Institut Fourier
PY  - 1992
SP  - 369
EP  - 392
VL  - 42
IS  - 1-2
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1296/
DO  - 10.5802/aif.1296
LA  - en
ID  - AIF_1992__42_1-2_369_0
ER  - 
%0 Journal Article
%A Duff, Dusa Mc
%T Immersed spheres in symplectic 4-manifolds
%J Annales de l'Institut Fourier
%D 1992
%P 369-392
%V 42
%N 1-2
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1296/
%R 10.5802/aif.1296
%G en
%F AIF_1992__42_1-2_369_0
Duff, Dusa Mc. Immersed spheres in symplectic 4-manifolds. Annales de l'Institut Fourier, …, Tome 42 (1992) no. 1-2, pp. 369-392. doi : 10.5802/aif.1296. http://www.numdam.org/articles/10.5802/aif.1296/

[BPV] W. Barth, C. Peters & A. Van De Ven, Complex Surfaces, Springer Verlag, 1984. | MR | Zbl

[U] H. Clemens, J. Kollar, S. Mori, Higher dimensional complex geometry, Astérisque, 166 (1989).

[FGG] M. Fernandez, M. Gotay, A. Gray, Four dimensional parallelizable symplectic and complex manifolds, Proc. Amer. Math. Soc., 103 (1988), 1209-1212. | MR | Zbl

[FM] Friedman and Morgan, Diffeomorphism types of 4-manifolds, Journ. Diff. Geo., (1988).

[GR] M. Gromov, Pseudo-holomorphic curves on almost-complex manifolds, Invent. Math., 82 (1985), 307-347. | MR | Zbl

[EX] D. Mcduff, Examples of symplectic structures, Invent. Math., 89 (1987), 13-36. | MR | Zbl

[RR] D. Mcduff, The Structure of Rational and Ruled Symplectic 4-manifolds, Journ. Amer. Math. Soc., 3 (1990), 679-712. | MR | Zbl

[EL] D. Mcduff, Elliptic methods in symplectic geometry, Bull. Amer. Math. Soc., 23 (1990), 311-358. | MR | Zbl

[BL] D. Mcduff, Blow ups and symplectic embeddings in dimension 4, Topology, 30 (1991), 409-421. | MR | Zbl

[LB] D. Mcduff, The Local Behaviour of holomorphic curves in almost complex 4-manifolds, Journ. Diff. Geom., 34 (1991), 143-164. | MR | Zbl

[KY] D. Mcduff, Symplectic 4-manifolds, to appear in Proceedings of I.C.M., Kyoto, 1990. | MR | Zbl

[UB] D. Mcduff, Remarks on the uniqueness of symplectic blowing up, preprint, 1990.

[RU] D. Mcduff, Notes on Ruled Symplectic 4-manifolds, preprint, 1992. | Zbl

[PW] T. Parker and J. Wolfson, A compactness theorem for Gromov's moduli space, preprint, 1991.

[TH] W. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc., 55 (1976), 467-468. | MR | Zbl

[WO] J. Wolfson, Gromov's compactness of pseudo-holomorphic curves and symplectic geometry, J. Diff. Geom., 28 (1988), 383-405. | MR | Zbl

[YE] R. Ye, Gromov's Compactness Theorem for Pseudo-holomorphic Curves, preprint, UCSB, 1991. | Zbl

Cité par Sources :