Rigidity of four-dimensional Kähler-Ricci solitons

In this article, we investigate four-dimensional gradient shrinking Ricci solitons close to a Kähler model. The first theorem could be considered as a rigidity result for the Kähler-Ricci soliton structure on S²× R² (in the sense of Remark 1). Moreover, we show that if the quotient of the norm of the self-dual Weyl tensor and the scalar curvature is close to that on a Kähler metric in a specific sense, then the gradient Ricci soliton must be either half-conformally flat or locally Kähler.