On the coupled Ding stability and the Yau--Tian--Donaldson correspondence for Fano manifolds

TL;DR

This paper introduces a new framework for understanding stability conditions of Fano manifolds, leading to solutions for a modified conjecture related to coupled K"ahler--Einstein metrics.

Contribution

It interprets coupled Ding stability notions via stability thresholds and solves a modified conjecture for coupled K"ahler--Einstein metrics on Fano manifolds.

Findings

Established a link between coupled Ding stability and stability thresholds.

Solved a modified conjecture by Hultgren and Witt Nyström.

Provided a new perspective on stability conditions for Fano manifolds.

Abstract

We interpret the coupled Ding semistability and the reduced coupled uniform Ding stability of log Fano pairs in the notion of coupled stability thresholds and reduced coupled stability thresholds. As a corollary, we solve a modified version of the conjecture by Hultgren and Witt Nystr\"om for coupled K\"ahler--Einstein metrics on Fano manifolds.