Rigidity of four-dimensional K\"ahler-Ricci solitons

In this article, we investigate four-dimensional gradient shrinking Ricci solitons close to a K\"ahler model. The first theorem could be considered as a rigidity result for the K\"ahler-Ricci soliton structure on $\mathbb{S}^2\times \mathbb{R}^2$ (in the sense of Remark 1). Moreover, we show that if the quotient of norm of the self-dual Weyl tensor and scalar curvature is close to that on a K\"ahler metric in a specific sense, then the gradient Ricci soliton must be either half-conformally flat or locally K\"ahler.