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arXiv · 2601.02974

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

Livia Campo·Kento Fujita·Taro Sano·Luca Tasin
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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.

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Livia Campo, Kento Fujita, Taro Sano, Luca Tasin. 2026-01-06. K-stability of Fano weighted hypersurfaces via plt flags and convex geometry. https://arxiv.org/abs/2601.02974

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