Hamiltonian constructions of Kähler-Einstein metrics and Kähler metrics of constant scalar curvature

Abstract

Assuming the existence of a real torus acting through holomorphic isometries on a Kahler manifold, we construct an ansatz for Kahler-Einstein metrics and an ansatz for Kahler 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 Kahler metrics describing a non-linear superposition of the Bergman metric, the Calabi metric and a higher dimensional generalization of the LeBrun Kahler metric. The superposition contains Kahler-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 Kahler geometries on Kahler quotients of higher cohomogeneity.