Gradient shrinking Ricci solitons and modified sectional curvature

We investigate four-dimensional gradient shrinking Ricci solitons with positive modified sectional curvature. Our first main result shows that if the norm of the self-dual Weyl tensor and the scalar curvature satisfy a certain sharp pinching condition — closely related, in a precise sense, to the Kähler case — then the soliton is necessarily locally Kähler. We further obtain a characterization theorem and a weighted integral gap result for compact gradient shrinking Ricci solitons with modified sectional curvature bounded from below. In addition, we establish a Hitchin–Thorpe type inequality for compact four-dimensional Ricci solitons, providing new topological constraints on such manifolds.