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. 
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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[1] Audin, Michèle Hamiltoniens périodiques sur les variétés symplectiques compactes de dimension 4 1990 1 25

[2] Eliashberg, Yakov On symplectic manifolds with some contact properties J. Differential Geom. 1991 233 238

[3] Griffiths, Phillip, Harris, Joseph Principles of algebraic geometry 1978

[4] Guillemin, V., Sternberg, S. Birational equivalence in the symplectic category Invent. Math. 1989 485 522

[5] Mcduff, Dusa Examples of symplectic structures Invent. Math. 1987 13 36

[6] Mcduff, Dusa Blow ups and symplectic embeddings in dimension 4 Topology 1991 409 421

[7] Wolfson, J. G. Gromov’s compactness of pseudo-holomorphic curves and symplectic geometry J. Differential Geom. 1988 383 405

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