Fano-Ricci limit spaces and spectral convergence

We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted ∂-Laplacian on compact Kähler manifolds. This situation typically occurs for a sequence of Fano manifolds with anticanonical Kähler class. We apply it to show that, if an almost smooth Fano-Ricci limit space admits a Kähler-Ricci limit soliton and the space of all L² 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 Kähler-Ricci solitons. In particular if a ℚ-Fano variety admits a Kähler-Ricci limit soliton and all holomorphic vector fields are L² with smooth potentials then the Lie algebra has the same structure as smooth Kähler-Ricci solitons. If the sequence consists of Kähler-Ricci solitons then the Ricci limit space is a weak Kähler-Ricci soliton on a ℚ-Fano variety and the space of limits of 1-eigenfunctions for the weighted ∂-Laplacian forms a Lie algebra with respect to the Poisson bracket and admits a similar decomposition as smooth Kähler-Ricci solitons.