Shortly after the introduction of Seiberg-Witten theory, LeBrun showed that the sign of the Yamabe invariant of a compact Kahler surface is determined by its Kodaira dimension. In this paper, we show that LeBrun's Theorem is no longer true for non-Kahler 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.

中文翻译：

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Inoue 曲面、Kodaira 曲面的 Yamabe 不变量及其爆炸

在引入 Seiberg-Witten 理论后不久，LeBrun 表明紧凑 Kahler 表面的 Yamabe 不变量的符号由其 Kodaira 维数决定。在本文中，我们表明勒布朗定理不再适用于非 Kahler 曲面。特别是，我们证明了井上表面的 Yamabe 不变量及其爆炸都为零。我们还借此机会记录了一个证明，证明 Kodaira 表面的 Yamabe 不变量及其膨胀都为零，正如 LeBrun 先前所指出的。