Symplectic Lefschetz Fibrations on S ×m

A remarkable theorem of Donaldson [D] says that a symplectic 4manifold admits a Lefschetz pencil by symplectic surfaces. Given a Lefschetz pencil on a 4-manifold X, one can blow up points at the base locus to get a Lefschetz fibration of X over S, which admits a nice handlebody decomposition and can be described by geometric monodromy representation into the mapping class group of a regular fiber. On the other hand, Gompf observed that if an oriented 4-manifoldX admits a Lefschetz fibration over an oriented surface such that the fiber class is non-zero inH2(X;R), then X carries a symplectic form with respect to which the fibers of the Lefschetz fibration are symplectic surfaces in X. Such a singular fibration is called a symplectic Lefschetz fibration with respect to the symplectic form. Suppose M is a closed oriented 3-manifold fibered over S, then the 4-manifold X = S × M fibers over S × S such that the fiber class is non-zero in H2(X;R). There is a canonical up to deformation equivalence symplectic structure on X with respect to which each fiber is a symplectic surface. An interesting question motivated by Taubes’s fundamental research on symplectic 4-manifolds [T] asks whether the converse is true: Does every symplectic structure (up to deformation equivalence) on X = S ×M come from a fibration of M over S, in particular, is it true that M must be fibered? In this paper we prove the following