Extremal Kähler metrics on projective bundles over a curve
By V. Apostolov, D. Calderbank, P. Gauduchon, Christina W. Tønnesen-Friedman. 2009
69 citations recorded by the source corpus.
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 Kahler metric of constant scalar curvature if and only if E is polystable. We also address the more general existence problem of extremal Kahler 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 Kahler metrics in Kahler classes sufficiently far from the boundary of the Kahler 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 Kahler manifolds with fibres isomorphic to a given toric Kahler variety. We discuss various ramifications of our approach to this class of manifolds.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/b49e92f890e4a14e9dfab282461642e0f5ed3b1c
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Kähler metrics with constant weighted scalar curvature and weighted K‐stability (2018, methodology): “Such Kähler metrics, parametrized by u ∈ S(P,L) and the real constants cj , are referred to in [5] as given by the generalized Calabi ansatz in reference to the well-known construction of Calabi [16] of extremal Kähler metrics on P(1)-bundles.”