Linear stability and instability of K\"ahler Ricci solitons

We show that the recently discovered BCCD shrinking soliton is linearly unstable, by extending the approach of \cite{chi04} and \cite{hm11}, via recent work the \cite{cm21} on gradient shrinking Ricci solitons. On the other hand, we prove that the weighted $L^2$-spectra of the weighted Lichnerowicz Laplacians of steady and expanding K\"ahler Ricci solitons are nonpositive in real dimension $4$. We additionally determine the linear stability of the orbifold singularities of K\"ahler solitons: shrinkers are unstable, steadies are neutrally stable and expanders are strictly stable. All of these results follow from new Weitzenb\"ock formulae for the weighted Lichnerowicz Laplacian specialized to K\"ahler metrics.