Gradient shrinking Ricci solitons of half harmonic Weyl curvature

We prove that a four-dimensional gradient shrinking Ricci soliton with $\delta W^{\pm}=0$ is either Einstein, or a finite quotient of $S^3\times\mathbb{R}$, $S^2\times\mathbb{R}^2$ or $\mathbb{R}^4$. We also prove that a four-dimensional cscK gradient Ricci soliton is either K\"ahler-Einstein, or a finite quotient of $M\times\mathbb{C}$, where $M$ is a Riemann surface. The main arguments are curvature decompositions, the Weitzenb\"ock formula for half Weyl curvature, and the maximum principle.