Fano-Ricci limit spaces and spectral convergence

We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted $\barpartial$-Laplacian on compact K\"ahler manifolds. This situation typically occurs for a sequence of Fano manifolds with anticanonical K\"ahler class. We apply it to show that, if an almost smooth Fano-Ricci limit space admits a K\"ahler-Ricci limit soliton and the space of all $L^2$ holomorphic vector fields with smooth potentials is a Lie algebra with respect to the Lie bracket, then the Lie algebra has the same structure as smooth K\"ahler-Ricci solitons. In particular if a $\Q$-Fano variety admits a K\"ahler-Ricci limit soliton and all holomorphic vector fields are $L^2$ with smooth potentials then the Lie algebra has the same structure as smooth K\"ahler-Ricci solitons. If the sequence consists of K\"ahler-Ricci solitons then the Ricci limit space is a weak K\"ahler-Ricci soliton on a $\mathbb{Q}$-Fano variety and the space of limits of $1$ eigenfunctions for the weighted $\barpartial$-Laplacian forms a Lie algebra with respect to the Poisson bracket and admits a similar decomposition as smooth K\"ahler-Ricci solitons.