The classification of ruled symplectic $4$-manifolds

Let $M$ be an oriented $S^2$-bundle over a compact Riemann surface $\Sigma$. 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.

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Published 1 January 1996