  We consider the problem of existence of constant scalar curvature Kaehler
metrics on complete intersections of sections of vector bundles. In particular
we give general formulas relating the Futaki invariant of such a manifold to
the weight of sections defining it and to the Futaki invariant of the ambient
manifold. As applications we give a new Mukai-Umemura-Tian like example of Fano
5-fold admitting no Kaehler-Einstein metric and a strong evidence of
K-stability of complete intersections on Grassmannians.
