Kodaira dimension & the Yamabe problem, II

Abstract

For compact complex surfaces $(M^{4}, J)$ of Kähler type, it was previously shown [30] that the sign of the Yamabe invariant $\mathscr{Y}(M)$ only depends on the Kodaira dimension $\text{Kod} (M, J)$. 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 also reprove a result from [2] that explains why the exclusion of class VII is essential here.