Communications in Analysis and Geometry

Volume 31 (2023)

Number 10

Kodaira dimension and the Yamabe problem, II

Pages: 2387 – 2411

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n10.a3

Authors

Michael Albanese (School of Computer and Mathematical Sciences, University of Adelaide, SA, Australia)

Claude LeBrun (Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.)

Abstract

For compact complex surfaces $(M^4, J)$ of Kähler type, it was previously shown $\href{https://doi.org/10.48550/arXiv.dg-ga/9702012}{[30]}$ that the sign of the Yamabe invariant $\mathscr{Y} (M)$ only depends on the Kodaira dimension $\operatorname{Kod}(M, J)$. 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 $\href{https://doi.org/10.1007/s10455-020-09744-3}{[2]}$ that explains why the exclusion of class VII is essential here.

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The authors were supported in part by NSF grant DMS-1906267.

Received 5 July 2021

Accepted 29 November 2021

Published 13 August 2024