On the existence of K\"ahler metrics of constant scalar curvature

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