Fano-Ricci limit spaces and spectral convergence

Akito Futaki, Shouhei Honda, Shunsuke Saito

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1015-1062
Number of pages48
JournalAsian Journal of Mathematics
Volume21
Issue number6
DOIs
Publication statusPublished - 2017 Jan 1

Keywords

  • Fano manifold
  • Gromov-hausdorff convergence
  • Kähler-Ricci soliton

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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