## 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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