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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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The structure of rational and ruled symplectic $4$-manifolds
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by Dusa McDuff
J. Amer. Math. Soc. 3 (1990), 679-712

Erratum: J. Amer. Math. Soc. 5 (1992), 987-988.


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:
  • MathSciNet review: 1049697