### 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 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 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 language | English |
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Pages (from-to) | 1015-1062 |

Number of pages | 48 |

Journal | Asian Journal of Mathematics |

Volume | 21 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2017 Jan 1 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Asian Journal of Mathematics*,

*21*(6), 1015-1062. https://doi.org/10.4310/AJM.2017.v21.n6.a2