Apostolov, V.; Tønnesen-Friedman, C. A remark on Kähler metrics of constant scalar curvature on ruled complex surfaces. (English) Zbl 1122.53042 Bull. Lond. Math. Soc. 38, No. 3, 494-500 (2006). Recent results by S. Donaldson, X.X. Chen and G. Tian, H. Luo, T. Mabuchi and others, which relate the existence and uniqueness issue for extremal Kähler metrics in Hodge Kähler classes to various notions of stability of polarized projective varieties, have led to the following conjecture.Let \((M,J)=P(E)\) be a geometrically ruled complex manifold over a compact complex curve. Then \((M,J)\) admits a constant scalar curvature (CSC) Kähler metric in some (and hence in any) Kähler class if and only if the vector bundle \(E\) is polystable.In this paper the authors combine results by Chen and Tian, A. Lichnerowicz, Y. Mutsushima, the Futaki obstruction theory, and a deformation argument by A. Fujiki and C. LeBrun and S. Simanca to prove the conjecture in the case when the rank of the vector bundle is two. Reviewer: Abdelkrim Brania (Atlanta) Cited in 2 ReviewsCited in 12 Documents MSC: 53C55 Global differential geometry of Hermitian and Kählerian manifolds 58E11 Critical metrics 32Q15 Kähler manifolds Keywords:Kähler metrics; constant scalar curvature; ruled complex surface PDFBibTeX XMLCite \textit{V. Apostolov} and \textit{C. Tønnesen-Friedman}, Bull. Lond. Math. Soc. 38, No. 3, 494--500 (2006; Zbl 1122.53042) Full Text: DOI arXiv