Abstract
Assuming the existence of a real torus acting through holomorphic isometries on a Kähler manifold, we construct an ansatz for Kähler-Einstein metrics and an ansatz for Kähler metrics with constant scalar curvature. Using this Hamiltonian approach we solve the differential equations in special cases and find, in particular, a family of constant scalar curvature Kähler metrics describing a non-linear superposition of the Bergman metric, the Calabi metric and a higher dimensional generalization of the LeBrun Kähler metric. The superposition contains Kähler-Einstein metrics and all the geometries are complete on the open disk bundle of some line bundle over the complex projective spaceP n. We also build such Kähler geometries on Kähler quotients of higher cohomogeneity.
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Communicated by S.-T. Yau
Partially supported by the NSF Under Grant No. DMS 8906809
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Pedersen, H., Poon, Y.S. Hamiltonian constructions of Kähler-Einstein metrics and Kähler metrics of constant scalar curvature. Commun.Math. Phys. 136, 309–326 (1991). https://doi.org/10.1007/BF02100027
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DOI: https://doi.org/10.1007/BF02100027