  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.
