Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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
Erratum: J. Amer. Math. Soc. 5 (1992), 987-988.
MathSciNet review: 1049697
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC: 58F05, 53C15, 57R50, 58C10

Retrieve articles in all journals with MSC: 58F05, 53C15, 57R50, 58C10


Additional Information

DOI: http://dx.doi.org/10.1090/S0894-0347-1990-1049697-8
PII: S 0894-0347(1990)1049697-8
Keywords: Symplectic manifold, $ 4$-manifold, contact structures, pseudo-holomorphic curves, almost complex manifold, blowing up
Article copyright: © Copyright 1990 American Mathematical Society