Fano–Ricci limit spaces and spectral convergence

We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted $\overline{\partial}$-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 $\mathbb{Q}$-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 $\overline{\partial}$-Laplacian forms a Lie algebra with respect to the Poisson bracket and admits a similar decomposition as smooth Kähler–Ricci solitons.

Keywords

Gromov–Hausdorff convergence, Fano manifold, Kähler–Ricci soliton

2010 Mathematics Subject Classification

53C23, 58C40, 58J50

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The first author was supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (A) 25247003 and Grant-in-Aid for Challenging Exploratory Research 26610013. The second author was supported by Grant-in-Aid for Young Scientists (B) 24740046, 16K17585 and Grant-in-Aid for Challenging Exploratory Research 26610016. The third author was supported by Grant-in-Aid for JSPS Research Fellow 15J06855 and the Program for Leading Graduate Schools, MEXT, Japan.

Received 11 May 2016

Accepted 24 June 2016

Published 6 March 2018