Abstract
As a generalization of Calabi’s conjecture for Kähler-Ricci forms, which was solved by Yau in 1977, we discuss the existence of Kähler-Ricci soliton typed equation on a compact Kähler 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 Kähler form ω in the Kähler class [ωg] solving the Kähler-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].
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Communicated by Peter Li
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Zhu, X. Kähler-Ricci soliton typed equations on compact complex manifolds withC 1(M) > 0. J Geom Anal 10, 759–774 (2000). https://doi.org/10.1007/BF02921996
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DOI: https://doi.org/10.1007/BF02921996