Kähler-Ricci soliton typed equations on compact complex manifolds withC 1(M) > 0

Abstract

As a generalization of Calabi’s conjecture for Kahler-Ricci forms, which was solved by Yau in 1977, we discuss the existence of Kahler-Ricci soliton typed equation on a compact Kahler manifold (M, g) with positive first Chem C1 (M) > 0 as well as the uniqueness. For a given positively definite (1,1)-form Ω ∈ C1 (M) of M and a holomorphic vector field X on M, we prove that there is a Kahler form ω in the Kahler class [ωg] solving the Kahler-Ricci soliton typed equation if and only if, i) X is belonged to a reductive subalgebra of holomorphic vector fields and the imaginary part of X generates a compact one-parameter transformations subgroup of M; and ii) LX Ω is a real-valued (1,1)-form. Moreover, the solution ω is unique in the class [ωg].