An irrational ruled symplectic 4-manifold

Abstract

Ruled surfaces form an important and fairly well understood class of symplectic 4-manifolds. They are orientable 2-sphere bundles V over a Riemann surface M which have a symplectic form ω that is non-degenerate on each fiber of the projection π: V → M and is compatible with the obvious orientations on the base and fiber. Such forms are said to be π-compatible. One of the main results of [3] is that every minimal symplectic 4-manifold that contains a symplectic 2-sphere with trivial normal bundle is π-compatible for a suitable fibration π. (A symplectic manifold is said to be minimal if it contains no symplectic 2-spheres with self-intersection - 1.) We showed in [4] that when V is rational (that is, if the base M is a 2-sphere) or if V is the trivial bundle over T 2, then in each possible cohomology class the symplectic structure is unique up to isotopy. What happens in the general case is not yet clear, though one can show that uniqueness continues to hold in a certain range of cohomology classes.