The structure of rational and ruled symplectic -manifolds

Author:
Dusa McDuff

Journal:
J. Amer. Math. Soc. **3** (1990), 679-712

MSC:
Primary 58F05; Secondary 53C15, 57R50, 58C10

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

MathSciNet review:
1049697

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper investigates the structure of compact symplectic -manifolds which contain a symplectically embedded copy of with nonnegative self-intersection number. Such a pair is called *minimal* if, in addition, the open manifold contains no exceptional curves (i.e., symplectically embedded -spheres with self-intersection -1). We show that every such pair covers a minimal pair which may be obtained from by blowing down a finite number of disjoint exceptional curves in . Further, the family of manifold pairs under consideration is closed under blowing up and down. We next give a complete list of the possible minimal pairs. We show that is symplectomorphic either to with its standard form, or to an -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 either to a complex line or quadric in , or, in the case when is a bundle, to a fiber or section of the bundle.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0894-0347-1990-1049697-8

Keywords:
Symplectic manifold,
-manifold,
contact structures,
pseudo-holomorphic curves,
almost complex manifold,
blowing up

Article copyright:
© Copyright 1990
American Mathematical Society