## The structure of rational and ruled symplectic $4$-manifolds

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- by Dusa McDuff
- J. Amer. Math. Soc.
**3**(1990), 679-712 - DOI: https://doi.org/10.1090/S0894-0347-1990-1049697-8
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Erratum: J. Amer. Math. Soc.

**5**(1992), 987-988.

## 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 $(\overline V ,C,\overline \omega )$ 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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## Bibliographic Information

- © Copyright 1990 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**3**(1990), 679-712 - MSC: Primary 58F05; Secondary 53C15, 57R50, 58C10
- DOI: https://doi.org/10.1090/S0894-0347-1990-1049697-8
- MathSciNet review: 1049697