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A remark on Kähler metrics of constant scalar curvature on ruled complex surfaces. (English) Zbl 1122.53042

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.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
58E11 Critical metrics
32Q15 Kähler manifolds
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