2015/09/13 by Futaki, Akito, Honda, Shouhei, Saito, Shunsuke
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1509.03862
We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted \barpartial-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 \barpartial-Laplacian forms a Lie algebra with respect to the Poisson bracket and admits a similar decomposition as smooth Kähler-Ricci solitons.