Remarks on Kähler–Ricci solitons

Abstract

Abstract We prove that a compact complex manifold endowed with a Kähler-Ricci soliton cannot be isometrically embedded in a complex projective space ℂℙn in such a way that the Gauss map is rational, unless the metric is Einstein. This applies to hypersurfaces of complex compact homogeneous spaces canonically embedded in ℂℙn.We moreover obtain two curvature constraints for invariant Kähler-Ricci solitons on complex manifolds acted on by a compact Lie group with cohomogeneity one.