Remarks on Kähler–Ricci solitons

We prove that a complex compact manifold endowed with a Ka ̈hler Ricci soliton cannot be isometrically embedded in a complex projective space CP^n 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 CP^n. We moreover obtain two curvature constraints for invariant Ka ̈hler Ricci solitons on complex manifolds acted on by a compact Lie group with cohomogeneity one.