Compactness of Kahler-Ricci solitons on Fano manifolds

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

Paper · May 8, 2018

Compactness of Kahler-Ricci solitons on Fano manifolds

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

math.DGarXiv:1805.03084

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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 $K R (n)$ be the space of Kähler-Ricci solitons on $n$ -dimensional Fano manifolds. We show that after passing to a subsequence, any sequence in $K R (n)$ converge in the Gromov-Hausdorff topology to a Kähler-Ricci soliton on an $n$ -dimensional $Q$ -Fano variety with log terminal singularities.

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Submitted: May 8, 2018
Subjects: math.DG

Comments: 10 pages

Ricci solitons Kähler–Ricci solitons Singularities Kähler geometry Limit spaces & RCD Kähler–Einstein & K-stability Analytic minimal model program

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

6C.11
Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods
Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.
9B.4
Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds
Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.
11B.3
Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities
Hamilton’s compactness theorem and the missing injectivity radius bound.

[1] 6C.11
Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods
Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

[2] 9B.4
Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds
Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

[3] 11B.3
Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities
Hamilton’s compactness theorem and the missing injectivity radius bound.

Compactness of Kahler-Ricci solitons on Fano manifolds

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