  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)=
-infinity.
