[Paper Review] The Kahler-Ricci flow and K-stability

Gábor Székelyhidi|arXiv (Cornell University)|Mar 11, 2008

Geometry and complex manifolds17 references3 citations

TL;DR

This paper establishes that under uniform curvature bounds and bounded Mabuchi energy, K-polystability of a Fano manifold implies the existence of a Kähler-Einstein metric via the Kähler-Ricci flow. The key contribution is a perturbation result showing that small deformations of cscK manifolds remain cscK if K-polystable, which is applied to prove convergence of the flow under these conditions.

ABSTRACT

We consider the Kähler-Ricci flow on a Fano manifold. We show that if the curvature remains uniformly bounded along the flow, the Mabuchi energy is bounded below, and the manifold is K-polystable, then the manifold admits a Kähler-Einstein metric. The main ingredient is a result that says that a sufficiently small perturbation of a cscK manifold admits a cscK metric if it is K-polystable.

Motivation & Objective

• To establish conditions under which the Kähler-Ricci flow on a Fano manifold converges to a Kähler-Einstein metric.
• To bridge the gap between K-polystability and the existence of Kähler-Einstein metrics by analyzing the flow under curvature and energy constraints.
• To prove that sufficiently small deformations of cscK manifolds admit cscK metrics if K-polystable, providing a key technical tool.
• To show that K-polystability obstructs the existence of non-trivial test-configurations with smooth central fibers, linking geometric and algebraic stability.

Proposed method

• Utilizes the Kähler-Ricci flow on Fano manifolds, leveraging Perelman-type estimates to control curvature and complex structure convergence.
• Applies a perturbation result: if a cscK manifold is deformed slightly and remains K-polystable, it admits a cscK metric.
• Employs moment map theory and the Hilbert-Mumford criterion to analyze non-polystable points in the deformation space.
• Constructs a test-configuration with smooth central fiber from a non-polystable deformation, using holomorphic $ S^1 $-equivariant families over a disk.
• Uses the $ ho $-action of a one-parameter subgroup to define a holomorphic family of complex structures and lifts it to a $ oldsymbol{C}^* $-equivariant family over $ oldsymbol{C} $.
• Relies on the Mabuchi functional's boundedness and curvature uniformity to ensure convergence of the complex structure to a Kähler-Ricci soliton.

Experimental results

Research questions

• RQ1 Under what geometric conditions does the Kähler-Ricci flow on a Fano manifold converge to a Kähler-Einstein metric?
• RQ2 Can K-polystability be used to guarantee the existence of a cscK metric in a small deformation of a cscK manifold?
• RQ3 What is the relationship between the failure of polystability in a deformation and the existence of non-product test-configurations?
• RQ4 How do curvature bounds and Mabuchi energy boundedness constrain the limit complex structure of the Kähler-Ricci flow?
• RQ5 To what extent does the existence of a non-trivial test-configuration with smooth central fiber obstruct K-polystability?

Key findings

• If the Riemann curvature tensor is uniformly bounded along the Kähler-Ricci flow on a Fano manifold and the Mabuchi functional is bounded below, then the flow converges to a Kähler-Ricci soliton.
• A small deformation $ (M', L') $ of a cscK manifold $ (M, L) $ admits a cscK metric if it is K-polystable, even under a weak form of K-polystability involving only smooth central fibers.
• If a deformation $ (M', L') $ is not K-polystable, then there exists a non-product test-configuration with smooth central fiber $ (M_0, L_0) $ that admits a cscK metric and is itself a small deformation of $ (M, L) $.
• The existence of such a test-configuration implies that $ (M', L') $ fails K-polystability, establishing a converse to the existence result.
• The limit complex structure $ J_0 $ obtained from the flow is Kähler-Einstein if the Mabuchi functional is bounded below and curvature is uniformly bounded.
• The proof shows that K-polystability is both necessary and sufficient for cscK existence in small deformations, under the given geometric constraints.