[Paper Review] Kähler-Ricci solitons on toric Fano orbifolds

Yalong Shi, Xiaohua Zhu|arXiv (Cornell University)|Feb 14, 2011

Geometry and complex manifolds13 references4 citations

TL;DR

This paper establishes the existence of T-invariant Kähler-Ricci solitons on toric Fano orbifolds by reducing the soliton equation to a real Monge-Ampère equation via the torus action and proving uniform a priori estimates, generalizing Wang and Zhu's result from smooth manifolds to the orbifold setting. The soliton metric is Kähler-Einstein if and only if the Futaki invariant vanishes.

ABSTRACT

We prove the existence of Kähler-Ricci solitons on toric Fano orbifolds, hence extend the the theorem of Wang and Zhu [WZ] to the orbifold case.

Motivation & Objective

• To extend the existence result of Kähler-Ricci solitons from toric Fano manifolds to the broader class of toric Fano orbifolds.
• To establish the existence of T-invariant Kähler-Ricci soliton metrics on toric Fano orbifolds using geometric analysis techniques.
• To confirm a generalized version of Nakagawa's conjecture on the existence of Kähler-Einstein metrics on toric Gorenstein Fano orbifolds via the vanishing of the Futaki invariant.

Proposed method

• Utilize the torus action on toric Fano orbifolds to rewrite the Kähler-Ricci soliton equation as a real Monge-Ampère equation in the Legendre transform setting.
• Apply a continuity method by considering a one-parameter family of equations parameterized by t ∈ [ε₀, 1], starting from a known solution at t = 1.
• Establish uniform a priori estimates for the solution φ_t using the Legendre transform and properties of the dual polytope P.
• Prove boundedness of sup φ_t and inf φ_t via maximum principle and Sobolev embedding theorems on the dual polytope P.
• Use the Harnack-type estimate and Sobolev embedding to control the oscillation of the Legendre transform u, ensuring uniform C⁰ bounds.
• Leverage the fact that the singular set of a normal orbifold has codimension ≥ 2 to extend holomorphic vector fields from the regular part to the whole orbifold.

Experimental results

Research questions

• RQ1 Does the existence of Kähler-Ricci solitons on toric Fano manifolds extend to the orbifold case?
• RQ2 Can the continuity method be adapted to prove existence of solitons on toric Fano orbifolds via Monge-Ampère equations?
• RQ3 Is the vanishing of the Futaki invariant a sufficient condition for the existence of Kähler-Einstein metrics on toric Fano orbifolds, as conjectured by Nakagawa?
• RQ4 How do a priori estimates for the Kähler potential behave uniformly in the continuity path for orbifold singularities?
• RQ5 What is the role of the dual polytope P in controlling the solution of the soliton equation on toric Fano orbifolds?

Key findings

• The existence of T-invariant Kähler-Ricci solitons is proven for all toric Fano orbifolds, generalizing Wang and Zhu's result from smooth manifolds to orbifolds.
• The soliton metric is Kähler-Einstein if and only if the Futaki invariant of the orbifold vanishes, confirming a generalized version of Nakagawa's conjecture.
• Uniform a priori estimates for the solution φ_t of the continuity path are established, with sup φ_t ≤ C and inf φ_t ≥ -C for constants independent of t.
• The proof relies on the Legendre transform of the Kähler potential and Sobolev embedding on the dual polytope P to control oscillation and ensure convergence.
• Examples are provided: a Fano orbifold with non-vanishing Futaki invariant admits a soliton but not a Kähler-Einstein metric, while one with vanishing Futaki invariant admits a Kähler-Einstein metric.
• The method confirms that the uniqueness theorem of Tian and Zhu extends to Fano orbifolds, and the Kähler-Einstein condition is equivalent to the vanishing of the Futaki invariant.