On the K-stability of complete intersections in polarized manifolds

摘要

We consider the problem of existence of constant scalar curvature K?hler 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 K?hler-Einstein metric, and a strong evidence of K-stability of complete intersections in Grassmannians.