Fano-Ricci limit spaces and spectral convergence

Abstract

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 $L^2$ 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 Kähler-Ricci solitons. In particular if a $\Q$-Fano variety admits a Kähler-Ricci limit soliton and all holomorphic vector fields are $L^2$ 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 $\mathbb{Q}$-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 Kähler-Ricci solitons.

52 pages

References in corpus (5)

• Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry, II
• Nonlinear diffusion equations and curvature conditions in metric measure spaces
• Quasi-projectivity of the moduli space of smooth Kahler-Einstein Fano manifolds
• Compact moduli space of Kahler-Einstein Fano varieties
• Remark on a diameter bound for complete manifolds with positive Bakry-Émery Ricci curvature

Cited by in corpus (1)

• Spectral convergence under bounded Ricci curvature