Abstract
For certain compact complex Fano manifolds $M$ with reductive Lie algebras of holomorphic vector fields, we determine the analytic subvariety of the second cohomology group of $M$ consisting of Kahler classes whose Bando-Calabi-Futaki character vanishes. Then a Kahler class contains a Kahler metric of constant scalar curvature if and only if the Kahler class is contained in the analytic subvariety. On examination of the analytic subvariety, it is shown that $M$ admits infinitely many nonhomothetic Kahler classes containing Kahler metrics of constant scalar curvature but does not admit any Kahler-Einstein metric.
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