K-stability of Fano weighted hypersurfaces via plt flags and convex geometry

TL;DR

This paper introduces a new framework combining birational and convex geometry techniques to analyze the K-stability of weighted Fano hypersurfaces, providing criteria for stability and examples of instability.

Contribution

It develops a novel approach using plt flags and convex geometry to determine K-stability of weighted Fano hypersurfaces, extending previous methods.

Findings

All quasi-smooth weighted Fano hypersurfaces of index 1 with up to two weights > 1 are K-stable.

Constructed examples of K-unstable hypersurfaces of low indices.

Established lower bounds for stability thresholds using Abban-Zhuang method.

Abstract

We develop a framework to study the K-stability of weighted Fano hypersurfaces based on a combination of birational and convex-geometric techniques. As an application, we prove that all quasi-smooth weighted Fano hypersurfaces of index 1 with at most two weights greater than 1 are K-stable. We also construct several examples of K-unstable quasi-smooth weighted Fano hypersurfaces of low indices. To prove these results, we establish lower bounds for stability thresholds using the method of Abban-Zhuang, which reduces the problem to lower-dimensional cases. A key feature of our approach is the use of plt flags that are not necessarily admissible.