Notes on ruled symplectic -manifolds

Author:
Dusa McDuff

Journal:
Trans. Amer. Math. Soc. **345** (1994), 623-639

MSC:
Primary 57R15; Secondary 53C15, 57R52, 58F05

MathSciNet review:
1188638

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Abstract: A symplectic 4-manifold is said to be ruled if it is the total space of a fibration whose fibers are 2-spheres on which the symplectic form does not vanish. This paper develops geometric methods for analysing the symplectic structure of these manifolds, and shows how this structure is related to that of a generic complex structure on *V*. It is shown that each *V* admits a unique ruled symplectic form up to pseudo-isotopy (or deformation). Moreover, if the base is a sphere or if *V* is the trivial bundle over the torus, all ruled cohomologous forms are isotopic. For base manfolds of higher genus this remains true provided that a cohomological conditon on the form is satisfied: one needs the fiber to be "small" relative to the base. These results correct the statement of Theorem 1.3 in *The structure of rational and ruled symplectic manifolds*, J. Amer. Math. Soc. **3** (1990), 679-712, and give more details of some of the proofs.

**[1]**M. F. Atiyah,*Complex fibre bundles and ruled surfaces*, Proc. London Math. Soc. (3)**5**(1955), 407–434. MR**0076409****[2]**M. F. Atiyah,*Vector bundles over an elliptic curve*, Proc. London Math. Soc. (3)**7**(1957), 414–452. MR**0131423****[3]**Phillip Griffiths and Joseph Harris,*Principles of algebraic geometry*, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR**507725****[4]**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR**0463157****[5]**François Lalonde,*Isotopy of symplectic balls, Gromov’s radius and the structure of ruled symplectic 4-manifolds*, Math. Ann.**300**(1994), no. 2, 273–296. MR**1299063**, 10.1007/BF01450487**[6]**Dusa McDuff,*Examples of symplectic structures*, Invent. Math.**89**(1987), no. 1, 13–36. MR**892186**, 10.1007/BF01404672**[7]**Dusa McDuff,*The moment map for circle actions on symplectic manifolds*, J. Geom. Phys.**5**(1988), no. 2, 149–160. MR**1029424**, 10.1016/0393-0440(88)90001-0**[8]**Dusa McDuff,*The structure of rational and ruled symplectic 4-manifolds*, J. Amer. Math. Soc.**3**(1990), no. 3, 679–712. MR**1049697**, 10.1090/S0894-0347-1990-1049697-8**[9]**Dusa McDuff,*Immersed spheres in symplectic 4-manifolds*, Ann. Inst. Fourier (Grenoble)**42**(1992), no. 1-2, 369–392 (English, with French summary). MR**1162567**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1994-1188638-X

Article copyright:
© Copyright 1994
American Mathematical Society