Nous étudions des conditions sous lesquelles une variété symplectique de dimension 4 admet une structure kählérienne compatible. La théorie des sphères plongées -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 -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}, volume = {42}, year = {1992}, pages = {369-392}, doi = {10.5802/aif.1296}, mrnumber = {93k:53030}, zbl = {0756.53021}, language = {en}, url = {http://dml.mathdoc.fr/item/AIF_1992__42_1-2_369_0} }
Duff, Dusa Mc. Immersed spheres in symplectic 4-manifolds. Annales de l'Institut Fourier, Tome 42 (1992) pp. 369-392. doi : 10.5802/aif.1296. http://gdmltest.u-ga.fr/item/AIF_1992__42_1-2_369_0/
[BPV] Complex Surfaces, Springer Verlag, 1984. | MR 86c:32026 | Zbl 0718.14023
, & ,[U] Higher dimensional complex geometry, Astérisque, 166 (1989).
, , ,[FGG] Four dimensional parallelizable symplectic and complex manifolds, Proc. Amer. Math. Soc., 103 (1988), 1209-1212. | MR 90a:53039 | Zbl 0656.53034
, , ,[FM] Diffeomorphism types of 4-manifolds, Journ. Diff. Geo., (1988).
and ,[GR] Pseudo-holomorphic curves on almost-complex manifolds, Invent. Math., 82 (1985), 307-347. | MR 87j:53053 | Zbl 0592.53025
,[EX] Examples of symplectic structures, Invent. Math., 89 (1987), 13-36. | MR 88m:58061 | Zbl 0625.53040
,[RR] The Structure of Rational and Ruled Symplectic 4-manifolds, Journ. Amer. Math. Soc., 3 (1990), 679-712. | MR 91k:58042 | Zbl 0723.53019
,[EL] Elliptic methods in symplectic geometry, Bull. Amer. Math. Soc., 23 (1990), 311-358. | MR 91i:58046 | Zbl 0723.53018
,[BL] Blow ups and symplectic embeddings in dimension 4, Topology, 30 (1991), 409-421. | MR 92m:57039 | Zbl 0731.53035
,[LB] The Local Behaviour of holomorphic curves in almost complex 4-manifolds, Journ. Diff. Geom., 34 (1991), 143-164. | MR 93e:53050 | Zbl 0736.53038
,[KY] Symplectic 4-manifolds, to appear in Proceedings of I.C.M., Kyoto, 1990. | MR 94b:57042 | Zbl 0732.57012
,[UB] Remarks on the uniqueness of symplectic blowing up, preprint, 1990.
,[RU] Notes on Ruled Symplectic 4-manifolds, preprint, 1992. | Zbl 0810.53020
,[PW] A compactness theorem for Gromov's moduli space, preprint, 1991.
and ,[TH] Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc., 55 (1976), 467-468. | MR 53 #6578 | Zbl 0324.53031
,[WO] Gromov's compactness of pseudo-holomorphic curves and symplectic geometry, J. Diff. Geom., 28 (1988), 383-405. | MR 89m:53058 | Zbl 0661.53024
,[YE] Gromov's Compactness Theorem for Pseudo-holomorphic Curves, preprint, UCSB, 1991. | Zbl 0810.53024
,