Compactness of Kähler–Ricci solitons on Fano manifolds

Bin Guo; Duong H. Phong; Jian Song; Jacob Sturm

doi:10.4310/PAMQ.2022.v18.n1.a9

Compactness of Kähler–Ricci solitons on Fano manifolds

Bin Guo, Duong H. Phong, Jian Song, Jacob Sturm

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

In this short paper, we improve the result of Phong– Song–Sturm on degeneration of Fano Kähler–Ricci solitons by re-moving the assumption on the uniform bound of the Futaki in-variant. Let KR(n) be the space of Kähler–Ricci solitons on n-dimensional Fano manifolds. We show that after passing to a subse-quence, any sequence in KR(n) converge in the Gromov–Hausdorff topology to a Kähler–Ricci soliton on an n-dimensional Q-Fano va-riety with log terminal singularities.

Original language	English (US)
Pages (from-to)	305-316
Number of pages	12
Journal	Pure and Applied Mathematics Quarterly
Volume	18
Issue number	1
DOIs	https://doi.org/10.4310/PAMQ.2022.v18.n1.a9
State	Published - 2022
Externally published	Yes

All Science Journal Classification (ASJC) codes

Mathematics(all)

Keywords

Fano manifolds
Kähler-Ricci solitons

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10.4310/PAMQ.2022.v18.n1.a9

Cite this

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title = "Compactness of K{\"a}hler–Ricci solitons on Fano manifolds",

abstract = "In this short paper, we improve the result of Phong– Song–Sturm on degeneration of Fano K{\"a}hler–Ricci solitons by re-moving the assumption on the uniform bound of the Futaki in-variant. Let KR(n) be the space of K{\"a}hler–Ricci solitons on n-dimensional Fano manifolds. We show that after passing to a subse-quence, any sequence in KR(n) converge in the Gromov–Hausdorff topology to a K{\"a}hler–Ricci soliton on an n-dimensional Q-Fano va-riety with log terminal singularities.",

keywords = "Fano manifolds, K{\"a}hler-Ricci solitons",

author = "Bin Guo and Phong, {Duong H.} and Jian Song and Jacob Sturm",

note = "Funding Information: Received February 27, 2020. ∗Work supported in part by National Science Foundation grants DMS-1711439, DMS-12-66033 and DMS-1710500. Publisher Copyright: {\textcopyright} 2022, International Press, Inc.. All rights reserved.",

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doi = "10.4310/PAMQ.2022.v18.n1.a9",

language = "English (US)",

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pages = "305--316",

journal = "Pure and Applied Mathematics Quarterly",

issn = "1558-8599",

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T1 - Compactness of Kähler–Ricci solitons on Fano manifolds

AU - Guo, Bin

AU - Phong, Duong H.

AU - Song, Jian

AU - Sturm, Jacob

N1 - Funding Information: Received February 27, 2020. ∗Work supported in part by National Science Foundation grants DMS-1711439, DMS-12-66033 and DMS-1710500. Publisher Copyright: © 2022, International Press, Inc.. All rights reserved.

PY - 2022

Y1 - 2022

N2 - In this short paper, we improve the result of Phong– Song–Sturm on degeneration of Fano Kähler–Ricci solitons by re-moving the assumption on the uniform bound of the Futaki in-variant. Let KR(n) be the space of Kähler–Ricci solitons on n-dimensional Fano manifolds. We show that after passing to a subse-quence, any sequence in KR(n) converge in the Gromov–Hausdorff topology to a Kähler–Ricci soliton on an n-dimensional Q-Fano va-riety with log terminal singularities.

AB - In this short paper, we improve the result of Phong– Song–Sturm on degeneration of Fano Kähler–Ricci solitons by re-moving the assumption on the uniform bound of the Futaki in-variant. Let KR(n) be the space of Kähler–Ricci solitons on n-dimensional Fano manifolds. We show that after passing to a subse-quence, any sequence in KR(n) converge in the Gromov–Hausdorff topology to a Kähler–Ricci soliton on an n-dimensional Q-Fano va-riety with log terminal singularities.

KW - Fano manifolds

KW - Kähler-Ricci solitons

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DO - 10.4310/PAMQ.2022.v18.n1.a9

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SN - 1558-8599

VL - 18

SP - 305

EP - 316

JO - Pure and Applied Mathematics Quarterly

JF - Pure and Applied Mathematics Quarterly

IS - 1

ER -

Compactness of Kähler–Ricci solitons on Fano manifolds

Abstract

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