McDuff, Dusa Immersed spheres in symplectic 4-manifolds. (English) Zbl 0756.53021 Ann. Inst. Fourier 42, No. 1-2, 369-392 (1992). 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. Reviewer: D.McDuff Cited in 1 ReviewCited in 11 Documents MSC: 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 53C40 Global submanifolds Keywords:Kähler surfaces; symplectic 4-manifold; Kähler structure; \(J\)- holomorphic embedded spheres; blow-up PDFBibTeX XMLCite \textit{D. McDuff}, Ann. Inst. Fourier 42, No. 1--2, 369--392 (1992; Zbl 0756.53021) Full Text: DOI Numdam EuDML References: [1] [BPV] , & , Complex Surfaces, Springer Verlag, 1984. · Zbl 0718.14023 [2] [U] , , , Higher dimensional complex geometry, Astérisque, 166 (1989). · Zbl 0689.14016 [3] [FGG] , , , Four dimensional parallelizable symplectic and complex manifolds, Proc. Amer. Math. Soc., 103 (1988), 1209-1212. · Zbl 0656.53034 [4] [E6] , New invariants of open symplectic and contact manifolds, J. Amer. Math. Soc., 4 (1991), 513-520. · Zbl 0733.58011 [5] [GR] , Pseudo-holomorphic curves on almost-complex manifolds, Invent. Math., 82 (1985), 307-347. · Zbl 0592.53025 [6] [EX] , Examples of symplectic structures, Invent. Math., 89 (1987), 13-36. · Zbl 0625.53040 [7] [RR] , The Structure of Rational and Ruled Symplectic 4-manifolds, Journ. Amer. Math. Soc., 3 (1990), 679-712. · Zbl 0723.53019 [8] [EL] , Elliptic methods in symplectic geometry, Bull. Amer. Math. Soc., 23 (1990), 311-358. · Zbl 0723.53018 [9] [BL] , Blow ups and symplectic embeddings in dimension 4, Topology, 30 (1991), 409-421. · Zbl 0731.53035 [10] [LB] , The Local Behaviour of holomorphic curves in almost complex 4-manifolds, Journ. Diff. Geom., 34 (1991), 143-164. · Zbl 0736.53038 [11] [KY] , Symplectic 4-manifolds, to appear in Proceedings of I.C.M., Kyoto, 1990. · Zbl 0732.57012 [12] [6] , An introduction to complex Analysis in several variables, New York, Van Nostrand Co., · Zbl 0138.06203 [13] [RU] , Notes on Ruled Symplectic 4-manifolds, preprint, 1992. · Zbl 0810.53020 [14] [PW] and , A compactness theorem for Gromov’s moduli space, preprint, 1991. [15] [TH] , Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc., 55 (1976), 467-468. · Zbl 0324.53031 [16] [WO] , Gromov’s compactness of pseudo-holomorphic curves and symplectic geometry, J. Diff. Geom., 28 (1988), 383-405. · Zbl 0661.53024 [17] [YE] , Gromov’s Compactness Theorem for Pseudo-holomorphic Curves, preprint, UCSB, 1991. · Zbl 0810.53024 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.