POLARIZED 4-MANIFOLDS, EXTREMAL KAHLER METRICS, AND SEIBERG-WITTEN THEORY

Using Seiberg-Witten theory, it is shown that any Kahler metric of constant negative scalar curvature on a compact 4-manifold M minimizes the L-2-norm of scalar curvature among Riemannian metrics compatible with a fixed decomposition H-2(M) = H+ circle plus H-. This implies, for example, that any such metric on a minimal ruled surface must be locally symmetric.