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


Author: Dusa McDuff
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
Erratum: J. Amer. Math. Soc. 5 (1992), 987-988.
MathSciNet review: 1049697
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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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Keywords: Symplectic manifold, <IMG WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$4$">-manifold, contact structures, pseudo-holomorphic curves, almost complex manifold, blowing up
Article copyright: © Copyright 1990 American Mathematical Society