Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one. II
The manifolds \(M_n\) constructed by blowing up the diagonal of two copies of a complex projective space are Fano. The author proves the uniqueness of Kähler metric with constant scalar curvature for an open subset of all Kähler classes. He proves the existence for a Kähler class with which a certain integral is nonnegative and he expects that if the considered integral is nonnegative then there is no extremal metric in the class, that is suggested from the generalized Futaki invariants. In fact the boundary of Kähler classes which do not satisfy some special condition corresponds to the zero set of the Futaki invariants. He also gives a complete proof for the existence of constant scalar curvature on Kähler almost homogeneous manifolds with two ends and reductive automorphism group for the case of hypersurface ends. The article contains definitions of the Kähler metrics, the volume form, computations such as the Ricci curvature and scalar curvature (Thm 3.2). In section 4 the author determines the equation for metrics with constant scalar curvature and proves partially the existence, which is completed in sections 5 and 6. Finally he proves the uniqueness making use of previous results.
scientific article; zbMATH DE number 1777195