The Yamabe invariants of Inoue surfaces, Kodaira surfaces, and their blowups

TL;DR

This paper demonstrates that the Yamabe invariants of certain non-K"ahler surfaces, including Inoue and Kodaira surfaces, as well as their blowups, are all zero, contrasting with the behavior of K"ahler surfaces.

Contribution

It shows that LeBrun's theorem relating Yamabe invariants to Kodaira dimension does not extend to non-K"ahler surfaces, providing new results for Inoue and Kodaira surfaces.

Findings

Yamabe invariants of Inoue surfaces are zero

Yamabe invariants of their blowups are zero

Yamabe invariants of Kodaira surfaces are zero

Abstract

Shortly after the introduction of Seiberg-Witten theory, LeBrun showed that the sign of the Yamabe invariant of a compact K\"ahler surface is determined by its Kodaira dimension. In this paper, we show that LeBrun's Theorem is no longer true for non-K\"ahler surfaces. In particular, we show that the Yamabe invariants of Inoue surfaces and their blowups are all zero. We also take this opportunity to record a proof that the Yamabe invariants of Kodaira surfaces and their blowups are all zero, as previously indicated by LeBrun.