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

Livia Campo, Kento Fujita, Taro Sano, Luca Tasin

TL;DR

The work develops a robust framework for K-stability of quasi-smooth weighted Fano hypersurfaces by blending birational methods with convex-geometric tools. Central to the approach is the Abban–Zhuang method, which reduces local stability questions to lower-dimensional data via plt flags and weighted blowups, together with Okounkov-body barycenters to bound stability thresholds. The authors prove a general K-stability result for index-1 weighted hypersurfaces with at most two weights greater than one, showing δ ≥ (n+1)/n > 1, and construct explicit low-index unstable examples to illustrate limitations. They also extend the analysis to the two-weight setting, obtaining δ ≥ (n+1)/(n+1/a) > 1, and provide corollaries that connect to moduli and openness phenomena in K-stability. Overall, the paper integrates convex-geometric techniques with birational constructions to yield concrete stability criteria and new examples in the weighted Fano setting.

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.

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

TL;DR

The work develops a robust framework for K-stability of quasi-smooth weighted Fano hypersurfaces by blending birational methods with convex-geometric tools. Central to the approach is the Abban–Zhuang method, which reduces local stability questions to lower-dimensional data via plt flags and weighted blowups, together with Okounkov-body barycenters to bound stability thresholds. The authors prove a general K-stability result for index-1 weighted hypersurfaces with at most two weights greater than one, showing δ ≥ (n+1)/n > 1, and construct explicit low-index unstable examples to illustrate limitations. They also extend the analysis to the two-weight setting, obtaining δ ≥ (n+1)/(n+1/a) > 1, and provide corollaries that connect to moduli and openness phenomena in K-stability. Overall, the paper integrates convex-geometric techniques with birational constructions to yield concrete stability criteria and new examples in the weighted Fano setting.

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.
Paper Structure (18 sections, 58 theorems, 313 equations, 1 figure)

This paper contains 18 sections, 58 theorems, 313 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Known results for standard hypersurfaces
  4. Known results for weighted hypersurfaces
  5. Preliminaries
  6. Blowing-up weighted projective spaces and their hypersurfaces
  7. Weighted projective spaces
  8. Standard weighted blowups
  9. On quasi-smoothness
  10. Cutting process
  11. Covering process
  12. Barycenters of convex sets and stability thresholds on surfaces
  13. Bounds on local stability thresholds of singular points
  14. A non-Eckardt case
  15. An Eckardt case
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $a >1$ and $n \ge 3$ be integers. Let $X=X_d \subset \mathbb{P}(1^{n+1},a)$ be a quasi-smooth Fano weighted hypersurface of degree $d$, dimension $n$ and index $1$. Then In particular, $X$ is K-stable.

Figures (1)

  • Figure 1: A convex subset $\mathbf{O} \subset \mathbb{R}^2$. The coordinates of the barycenter of $\mathbf{O}$ are maximized when the figure of $\mathbf{O}$ is equal to the dashed area.

Theorems & Definitions (138)

  • Theorem 1.1
  • Example 1.2: See Corollary \ref{['corollary_ninja']}
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1: MR3896135MR4067358AZ22
  • Definition 2.2
  • Definition 2.3: Fujita2023Fujita:2024aa
  • Remark 2.4
  • Theorem 2.5: AZ22 (see also Fujita:2024aa)
  • Definition 3.1
  • ...and 128 more