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Delta invariants of weighted hypersurfaces

Taro Sano, Luca Tasin

TL;DR

The paper addresses the problem of establishing K-stability for large classes of Fano weighted hypersurfaces by obtaining sharp lower bounds for delta-invariants. It combines the Abban--Zhuang stability framework with a detailed analysis of linear systems on flags of weighted hypersurfaces, extending these techniques to singular (quasi-smooth) settings. The main contribution is a general bound $\delta(\mathcal{O}_X(1)) \ge \frac{(n+1)a_r}{d}$ under divisibility conditions, which implies $\delta(-K_X) \ge 1$ and, for $n \ge 3$, K-stability when $X$ is Fano with appropriate index constraints. The results yield K-stability for wide families of quasi-smooth and smooth weighted hypersurfaces (e.g., index 1 cases with $a_{n+1}|d$) and provide a framework that covers many low-dimensional classifications, thereby advancing the understanding of Kähler–Einstein metrics on these singular Fano varieties.

Abstract

We give a lower bound for the delta invariant of the fundamental divisor of a quasi-smooth weighted hypersurface. As a consequence, we prove K-stability of a large class of quasi-smooth Fano hypersurfaces of index 1 and of all smooth Fano weighted hypersurfaces of index 1 and 2. The proofs are based on the Abban--Zhuang method and on the study of linear systems on flags of weighted hypersurfaces.

Delta invariants of weighted hypersurfaces

TL;DR

The paper addresses the problem of establishing K-stability for large classes of Fano weighted hypersurfaces by obtaining sharp lower bounds for delta-invariants. It combines the Abban--Zhuang stability framework with a detailed analysis of linear systems on flags of weighted hypersurfaces, extending these techniques to singular (quasi-smooth) settings. The main contribution is a general bound under divisibility conditions, which implies and, for , K-stability when is Fano with appropriate index constraints. The results yield K-stability for wide families of quasi-smooth and smooth weighted hypersurfaces (e.g., index 1 cases with ) and provide a framework that covers many low-dimensional classifications, thereby advancing the understanding of Kähler–Einstein metrics on these singular Fano varieties.

Abstract

We give a lower bound for the delta invariant of the fundamental divisor of a quasi-smooth weighted hypersurface. As a consequence, we prove K-stability of a large class of quasi-smooth Fano hypersurfaces of index 1 and of all smooth Fano weighted hypersurfaces of index 1 and 2. The proofs are based on the Abban--Zhuang method and on the study of linear systems on flags of weighted hypersurfaces.
Paper Structure (10 sections, 19 theorems, 82 equations)

This paper contains 10 sections, 19 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Low dimensional cases
  3. Structure of the paper and the ingredients
  4. Preliminaries
  5. Weighted complete intersections
  6. Numerical observations
  7. Linear systems on weighted hypersurfaces
  8. Stability thresholds and Abban--Zhuang method
  9. Abban--Zhuang method
  10. Main results

Key Result

Theorem 1.1

[=Theorem thm:main] Let $X=X_d \subset \mathbb{P}(a_0, \ldots, a_{n+1})$ be a well-formed quasi-smooth weighted hypersurface of degree $d$ which is not a linear cone. Assume that there is $r$ such that $a_r >1$ and $a_r | d$. Then Moreover, if $X$ is Fano of index $\iota_X: = \sum_{i=0}^{n+1} a_i -d$ and $\frac{(n+1)a_{r}}{d} \ge \iota_X$, then $\delta(-K_X) \ge 1$ and if $n \ge 3$, then $X$ is K

Theorems & Definitions (59)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • proof
  • Remark 1.5
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • ...and 49 more