The structure of rational and ruled symplectic $4$-manifolds
HTML articles powered by AMS MathViewer
- by Dusa McDuff
- J. Amer. Math. Soc. 3 (1990), 679-712
- DOI: https://doi.org/10.1090/S0894-0347-1990-1049697-8
- PDF | Request permission
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.References
- Michèle Audin, Hamiltoniens périodiques sur les variétés symplectiques compactes de dimension $4$, Géométrie symplectique et mécanique (La Grande Motte, 1988) Lecture Notes in Math., vol. 1416, Springer, Berlin, 1990, pp. 1–25 (French). MR 1047474, DOI 10.1007/BFb0097462
- Yakov Eliashberg, On symplectic manifolds with some contact properties, J. Differential Geom. 33 (1991), no. 1, 233–238. MR 1085141 —, private communication.
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725 M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347. V. Guillemin and E. Lerman, Fiber bundles with symplectic fibers, preprint, MIT, Cambridge, MA, 1989.
- V. Guillemin and S. Sternberg, Birational equivalence in the symplectic category, Invent. Math. 97 (1989), no. 3, 485–522. MR 1005004, DOI 10.1007/BF01388888
- Dusa McDuff, Examples of symplectic structures, Invent. Math. 89 (1987), no. 1, 13–36. MR 892186, DOI 10.1007/BF01404672
- Dusa McDuff, Blow ups and symplectic embeddings in dimension $4$, Topology 30 (1991), no. 3, 409–421. MR 1113685, DOI 10.1016/0040-9383(91)90021-U —, Elliptic methods in symplectic geometry, Bull. Amer. Math. Soc. (to appear). —, The local behaviour of holomorphic curves in almost complex $4$-manifolds, preprint, SUNY, Stony Brook, NY, 1989. —, Symplectic manifolds with contact-type boundaries, preprint, 1990. —, Rational and ruled symplectic $4$-manifolds, Proc. Conf. at Durham, Summer 1989 (S. Donaldson and C. Thomas, eds.), Pitman, Oxford (to appear).
- J. G. Wolfson, Gromov’s compactness of pseudo-holomorphic curves and symplectic geometry, J. Differential Geom. 28 (1988), no. 3, 383–405. MR 965221
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