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

Abstract

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.