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 Kähler classes whose Bando-Calabi-Futaki character vanishes. Then a Kähler class contains a Kähler metric of constant scalar curvature if and only if the Kähler class is contained in the analytic subvariety. On examination of the analytic subvariety, it is shown that $M$ admits infinitely many nonhomothetic Kähler classes containing Kähler metrics of constant scalar curvature but does not admit any Kähler-Einstein metric.