A new holomorphic invariant and uniqueness of Kähler-Ricci solitons

Abstract

In this paper, a new holomorphic invariant is defined on a compact Kahler manifold with positive first Chern class and nontrivial holomorphic vector fields. This invariant generalizes the Futaki invariant. We prove that this invariant is an obstruction to the existence of Kahler-Ricci solitons. In particular, using this invariant together with the main result in [TZ1], we solve completely the uniqueness problem of Kahler-Ricci solitons. Two functionals associated to the new holomorphic invariant are also discussed. The main result here was announced in [TZ2].