The classification of ruled symplectic $4$-manifolds

Let M be an oriented S 2 -bundle over a compact Riemann surface . We show that up to diffeomorphism there is at most one symplectic form on M in each cohomology class. Since the possible cohomology classes of symplectic forms on M are known, this completes the classification of symplectic forms on these manifolds. Our proof relies on a simplification of our previous arguments and on the equivalence between Gromov and Seiberg-Witten invariants that we apply twice.