Classification of Gradient Ricci solitons with harmonic Weyl curvature

We make classifications of gradient Ricci solitons $(M, g, f)$ with harmonic Weyl curvature. As a local classification, we prove that the soliton metric $g$ is locally isometric to one of the following four types: an Einstein manifold, the Riemannian product of a Ricci flat manifold and an Einstein manifold, a warped product of $\mathbb{R}$ and an Einstein manifold, and a singular warped product of $\mathbb{R}^2$ and a Ricci flat manifold. Compared with the previous four-dimensional study in \cite{Ki}, we have developed a novel method of {\it refined adapted frame fields} and overcome the main difficulty arising from a large number of Riemmannian connection components in dimension$\geq 5$. Next we have obtained a classification of {\it complete} gradient Ricci solitons with harmonic Weyl curvature. For the proof, using the real analytic nature of $g$ and $f$, we elaborate geometric arguments to fit together local regions.