Extremal Kähler metrics on projective bundles over a curve

• Published in: Advances in mathematics (New York. 1965) 2011, Vol.227 (6), p.2385-2424
• Main Authors: Apostolov, Vestislav, Calderbank, David M.J., Gauduchon, Paul, Tønnesen-Friedman, Christina W.
• Format: Article
• Language: English
• Subjects: Algebraic Geometry; Differential Geometry; Extremal and constant scalar curvature Kähler metrics; Mathematics; Projective bundles; Stable vector bundles; Toric fibrations
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2011.05.006

Let M = P ( E ) be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle E → Σ over a compact complex curve Σ of genus ⩾2. Building on ideas of Fujiki (1992) [27], we prove that M admits a Kähler metric of constant scalar curvature if and only if E is polystable. We also address the more general existence problem of extremal Kähler metrics on such bundles and prove that the splitting of E as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal Kähler metrics in Kähler classes sufficiently far from the boundary of the Kähler cone. The methods used to prove the above results apply to a wider class of manifolds, called rigid toric bundles over a semisimple base, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature Kähler manifolds with fibres isomorphic to a given toric Kähler variety. We discuss various ramifications of our approach to this class of manifolds.