Asian Journal of Mathematics

Volume 21 (2017)

Number 6

Fano–Ricci limit spaces and spectral convergence

Pages: 1015 – 1062

DOI: http://dx.doi.org/10.4310/AJM.2017.v21.n6.a2

Authors

Akito Futaki (Graduate School of Mathematical Sciences, University of Tokyo, Japan)

Shouhei Honda (Mathematical Institute, Tohoku University, Sendai, Japan)

Shunsuke Saito (Graduate School of Mathematical Sciences, University of Tokyo, Japan)

Abstract

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

Full Text (PDF format)

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