Kähler Ricci solitons and deformation of complex structures

Given a compact Fano Kähler manifold (M,J) with a Kähler Ricci soliton g, we consider smooth families {J_t} of complex deformations of (M,J) which are invariant under the action of a maximal torus T in the full isometry group of (M,g). We prove that, under a certain condition on the spectrum of the Laplacian of g, there exists a smooth family of T-invariant Kähler Ricci solitons g_t on every complex manifold (M, J_t) with J_t sufficiently close to J. The result extends a theorem by Koiso on complex deformations of Kähler Einstein manifolds.

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