Geodesic
The structure of rational and ruled symplectic 4-manifolds
Journal of the American Mathematical Society, Tome 03 (1990) no. 3, pp. 679-712

Voir la notice de l'article provenant de la source American Mathematical Society

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 $(\overline V ,C,\overline \omega )$ 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 $\overline V$ is symplectomorphic either to $\mathbb {C}{P^2}$ 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 $\mathbb {C}{P^2}$, or, in the case when $\overline V$ is a bundle, to a fiber or section of the bundle. 
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McDuff, Dusa. The structure of rational and ruled symplectic 4-manifolds. Journal of the American Mathematical Society, Tome 03 (1990) no. 3, pp. 679-712. doi: 10.1090/S0894-0347-1990-1049697-8

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