Kodaira Dimension and the Yamabe Problem, II

TL;DR

This paper extends the understanding of the Yamabe invariant's dependence on Kodaira dimension from Kaehler surfaces to all compact complex surfaces except class VII, providing a simplified proof and broadening the scope of previous results.

Contribution

It generalizes the relationship between Yamabe invariant sign and Kodaira dimension to all compact complex surfaces except class VII, with a simplified proof of the key result.

Findings

Yamabe invariant sign depends on Kodaira dimension for most complex surfaces.

The pattern extends beyond Kaehler surfaces to all but class VII.

A simplified proof clarifies the role of class VII surfaces.

Abstract

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.