Compactness of Kähler–Ricci solitons on Fano manifolds

Bin Guo (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.; and Department of Mathematics & Computer Science, Rutgers University, Newark, New Jersey, U.S.A.)

Duong H. Phong (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Jian Song (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.)

Jacob Sturm (Department of Mathematics & Computer Science, Rutgers University, Newark, New Jersey, U.S.A.)

Abstract

In this short paper, we improve the result of Phong–Song–Sturm on degeneration of Fano Kähler–Ricci solitons by removing the assumption on the uniform bound of the Futaki invariant. Let $\mathcal{KR}(n)$ be the space of Kähler–Ricci solitons on $n$‑dimensional Fano manifolds.We show that after passing to a subsequence, any sequence in $\mathcal{KR}(n)$ converge in the Gromov–Hausdorff topology to a Kähler–Ricci soliton on an $n$‑dimensional $\mathbb{Q}$‑Fano variety with $\operatorname{log}$ terminal singularities.

Keywords

Kähler–Ricci solitons, Fano manifolds

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The authors’ work was supported in part by National Science Foundation grants DMS-1711439, DMS-12-66033 and DMS-1710500.

Received 27 February 2020

Accepted 23 August 2020

Published 10 February 2022