On the Futaki Invariants of Complete Intersections

In 1983, Futaki [2] introduced his invariants which generalize the obstruction of Kazdan-Warner to prescribe Gauss curvature on S. The Futaki invariants are defined for any compact Kähler manifold with positive first Chern class that has nontrivial holomorphic vector fields. Their vanishing are necessary conditions to the existence of Kähler-Einstein metric on the underlying manifold. Let M be a compact Kähler manifold with positive first Chern class c1(M) > 0. Choosing an arbitrary positive (1, 1) form ω in c1(M) as a Kähler metric on M , we can find a smooth function f on M , determined up to a constant, such that the following