[Paper Review] Compactness of Kahler-Ricci solitons on Fano manifolds

Bin Guo, D. H. Phong+2 more|arXiv (Cornell University)|May 8, 2018

Geometry and complex manifolds15 references3 citations

TL;DR

This paper establishes the Gromov-Hausdorff compactness of sequences of Kähler-Ricci solitons on Fano manifolds of fixed complex dimension $n \geq 2$, without requiring a uniform bound on the Futaki invariant. By leveraging Birkar's birational geometry results and Perelman's $μ$-functional, the authors prove that any such sequence converges to a $\mathbb{Q}$-Fano variety with log terminal singularities, equipped with a Kähler-Ricci soliton metric and extended holomorphic vector field.

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 log terminal singularities.

Motivation & Objective

• To establish compactness of the space $\mathcal{KR}(n)$ of Kähler-Ricci solitons on $n$-dimensional Fano manifolds in the Gromov-Hausdorff topology.
• To remove the prior assumption of a uniform bound on the Futaki invariant in earlier compactness results for Kähler-Ricci solitons.
• To show that the Gromov-Hausdorff limit of any sequence in $\mathcal{KR}(n)$ is a $\mathbb{Q}$-Fano variety with log terminal singularities and a well-defined Kähler-Ricci soliton structure.
• To derive uniform bounds on scalar curvature, diameter, and Futaki invariant for all Kähler-Ricci solitons in $\mathcal{KR}(n)$, independent of the specific manifold.

Proposed method

• Use Birkar's recent result in birational geometry to establish a uniform lower bound $\epsilon(n) > 0$ for the Ricci curvature of any $n$-dimensional Fano manifold.
• Apply the uniform lower bound on Ricci curvature to derive a uniform lower bound on Perelman's $\mu$-functional for all Kähler-Ricci solitons in $\mathcal{KR}(n)$.
• Leverage the uniform $\mu$-functional bound to obtain a uniform $C^0$-estimate, enabling Gromov-Hausdorff convergence of sequences in $\mathcal{KR}(n)$.
• Use the Cheeger-Colding theory for Bakry-Émery Ricci curvature and Perelman's pseudo-locality theorem to establish local smooth convergence on the regular part of the limit space.
• Apply the main theorem of [23] and non-collapsing estimates to ensure the limit metric space is compact or complete with controlled singularities.
• Show that the Kähler-Ricci soliton metric and holomorphic vector field extend continuously to the entire limit space, including the singular locus.

Experimental results

Research questions

• RQ1 Can the Gromov-Hausdorff compactness of Kähler-Ricci solitons on Fano manifolds be established without assuming a uniform bound on the Futaki invariant?
• RQ2 Does the limit of a sequence of Kähler-Ricci solitons on $n$-dimensional Fano manifolds necessarily carry the structure of a Kähler-Ricci soliton on a $\mathbb{Q}$-Fano variety with log terminal singularities?
• RQ3 Can uniform bounds on scalar curvature, diameter, and the Futaki invariant be derived for all Kähler-Ricci solitons in $\mathcal{KR}(n)$?
• RQ4 Is Perelman's $\mu$-functional uniformly bounded below for all Kähler-Ricci solitons on $n$-dimensional Fano manifolds?
• RQ5 Does the singular set of the Gromov-Hausdorff limit of such sequences have Hausdorff dimension at most $2n-4$?

Key findings

• Any sequence in $\mathcal{KR}(n)$ converges in the Gromov-Hausdorff topology to a compact metric length space $(X_\infty, d_\infty)$ with a singular set $\Sigma_\infty$ of Hausdorff dimension at most $2n-4$.
• The limit space $(X_\infty, d_\infty)$ is isometric to the metric completion of the regular part $(X_\infty \setminus \Sigma_\infty, g_\infty)$, and carries a Kähler-Ricci soliton structure with $Ric(g_\infty) = g_\infty + L_{\mathcal{V}_\infty}g_\infty$.
• The Kähler metric $g_\infty$ extends to a Kähler current on the entire $\mathbb{Q}$-Fano variety $X_\infty$ with bounded local potential, and the vector field $\mathcal{V}_\infty$ extends to a global holomorphic vector field.
• There exist constants $F(n), D(n), K(n) > 0$ such that for all $(X,g,u) \in \mathcal{KR}(n)$, the Futaki invariant satisfies $\mathcal{F}_X \leq F(n)$, the diameter is bounded by $D(n)$, and the scalar curvature satisfies $0 < R \leq K(n)$.
• The $\mu$-functional for any Kähler-Ricci soliton in $\mathcal{KR}(n)$ is uniformly bounded below by a constant depending only on $n$, due to the uniform lower Ricci curvature bound from Birkar's result.
• The compactness result extends to complete or closed gradient shrinking Ricci solitons under a uniform lower bound on the $\mu$-functional, with similar convergence and singular set dimension bounds.