Kodaira Dimension and the Yamabe Problem, II

For compact complex surfaces (M^4, J) of Kaehler type, it was previously shown that the sign of the Yamabe invariant Y(M) only depends on the Kodaira dimension Kod (M, J). In this paper, we prove that this pattern in fact extends to all compact complex surfaces except those of class VII. In the process, we give a simplified proof of a result that explains why the exclusion of class VII is essential here.