Kodaira Dimension and the Yamabe Problem

The Yamabe invariant Y (M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar curvature Riemannian metrics g on M . (To be absolutely precise, one only considers constant-scalar-curvature metrics which are Yamabe minimizers, but this does not affect the sign of the answer.) If M is the underlying smooth 4-manifold of a complex algebraic surface (M, J), it is shown that the sign of Y (M) is completely determined by the Kodaira dimension Kod(M,J). More precisely, Y (M) < 0 iff Kod(M, J) = 2; Y (M) = 0 iff Kod(M,J) = 0 or 1; and Y (M) > 0 iff Kod(M, J) = −∞.