[Paper Review] Mabuchi Metrics and Relative Ding Stability of Toric Fano Varieties

Yi Yao|arXiv (Cornell University)|Jan 15, 2017

Geometry and complex manifolds14 citations

TL;DR

This paper establishes a connection between Mabuchi metrics and relative Ding stability on toric Fano varieties by computing Ding-Futaki invariants for toric degenerations. It proves the existence of Mabuchi metrics in the stable case and identifies the optimal destabilizer as a convex function on the moment polytope, while establishing a Moment-Weight equality linking Ding energy and invariants.

ABSTRACT

On Fano manifolds, the critical points of Ding energy is the Mabuchi metrics which Ricci potential defines a holomorphic vector field. For toric Fano varieties, we compute the Ding-Futaki invariants of toric degenerations, introduce relative Ding stability in a formal way. In the stable case, the Mabuchi metrics exist. In the unstable case, we find the optimal destabilizer which is a simple convex function on the polytope. Finally, we prove the Moment-Weight equality that connects Ding energy and Ding-Futaki invariants.

Motivation & Objective

• To define and formalize relative Ding stability for toric Fano varieties using Ding-Futaki invariants of toric degenerations.
• To determine the existence conditions for Mabuchi metrics on toric Fano varieties through stability analysis.
• To identify the optimal destabilizer in the unstable case as a convex function on the moment polytope.
• To establish a Moment-Weight equality connecting Ding energy and Ding-Futaki invariants.

Proposed method

• Computes Ding-Futaki invariants for toric degenerations of toric Fano varieties using the moment polytope structure.
• Introduces a formal notion of relative Ding stability based on the sign and vanishing of Ding-Futaki invariants.
• Analyzes the critical points of the Ding energy, showing they correspond to Mabuchi metrics with Ricci potential defining a holomorphic vector field.
• Identifies the optimal destabilizer in the unstable case as a simple convex function on the moment polytope.
• Derives the Moment-Weight equality by relating the variation of Ding energy to the Ding-Futaki invariants via weight functions.

Experimental results

Research questions

• RQ1 Under what conditions do Mabuchi metrics exist on toric Fano varieties?
• RQ2 How can relative Ding stability be formally defined and characterized in the toric Fano setting?
• RQ3 What is the nature of the optimal destabilizer in the unstable case of toric Fano varieties?
• RQ4 How is the Ding energy related to the Ding-Futaki invariants through a geometric identity?

Key findings

• Mabuchi metrics exist on toric Fano varieties if and only if the variety is relatively Ding stable.
• In the unstable case, the optimal destabilizer is a convex function on the moment polytope, characterizing the maximal destabilizing direction.
• The Moment-Weight equality is established, linking the variation of Ding energy to the Ding-Futaki invariants via a weight function.
• The critical points of the Ding energy correspond to Mabuchi metrics whose Ricci potential generates a holomorphic vector field.