K\"ahler Ricci solitons and deformation of complex structures

TL;DR

This paper studies how K"ahler Ricci solitons on Fano manifolds can be smoothly deformed along certain complex structure variations, extending Koiso's theorem to solitons.

Contribution

It proves the existence of smooth families of T-invariant K"ahler Ricci solitons under spectral conditions, generalizing known results for Einstein manifolds.

Findings

Existence of T-invariant K"ahler Ricci solitons on deformed complex structures.

Extension of Koiso's theorem to Ricci solitons.

Spectral condition ensures stability of the soliton under deformation.

Abstract

Given a compact Fano K\"ahler manifold (M,J) with a K\"ahler 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 K\"ahler 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 K\"ahler Einstein manifolds.