We consider the problem of existence of constant scalar curvature Kähler 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 Kähler–Einstein metric, and a strong evidence of K-stability of complete intersections in Grassmannians.
On the K-stability of complete intersections in polarized manifolds / C. Arezzo; A. Della Vedova. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - 226:6(2011), pp. 4796-4815. [10.1016/j.aim.2010.12.018]