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The structure of rational and ruled symplectic 4-manifolds. (English) Zbl 0723.53019

From the abstract: “This paper investigates the structure of compact symplectic 4-manifolds \((V,\omega)\) which contain a symplectically embedded copy C of \(S^ 2\) with nonnegative self-intersection number. Such a pair \((V,C,\omega)\) is called minimal if, in addition, the open manifold V-C contains no exceptional curves (i.e., symplectically embedded 2-spheres with self-intersection -1). We show that every such pair \((V,C,\omega)\) covers a minimal pair \((\bar V,C,\bar\omega)\) which may be obtained from V by blowing down a finite number of disjoint exceptional curves in V-C. Further, the family of manifold pairs \((V,C,\omega)\) under consideration is closed under blowing up and down. We next give a complete list of the possible minimal pairs. We show that \(\bar V\) is symplectomorphic either to \(CP^ 2\) with its standard form, or to an \(S^ 2\)-bundle over a compact surface with a symplectic structure which is uniquely determined by its cohomology class. Moreover, this symplectomorphism may be chosen so that it takes C either to a complex line or quadric in \(CP^ 2\), or, in the case when \(\bar V\) is a bundle, to a fiber or section of the bundle.”

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57R99 Differential topology
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