Notes on ruled symplectic $4$-manifolds

Abstract:

A symplectic 4-manifold $(V,\omega )$ 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.