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Asymptotic Chow stability of symmetric reflexive toric varieties

King Leung Lee

TL;DR

The paper investigates asymptotic Chow stability for symmetric reflexive toric varieties, introducing a Futaki-Ono invariant framework and a regular-boundary/special-polytope concept that yields a practical stability criterion. It proves that special polytopes are asymptotically Chow polystable and develops a triangulation-based sufficient condition that extends stability to non-special symmetric polytopes, with explicit examples like $D(X_8)$ and $D(X_9)$. It provides a thorough catalog of stability across dimensions, including all 2D cases, several 3D families, and higher-dimensional counterexamples (e.g., $D([-1,1]^n)$ for $n\, ext{ge}\,6$), and discusses how duality can affect stability. Together, these results offer concrete tools and classifications for assessing Chow polystability of toric varieties, with implications for singular Kähler geometry and moduli problems.

Abstract

In this note, we study the asymptotic Chow stability of toric varieties. We provide examples of symmetric reflexive toric varieties that are not asymptotic Chow semistable. On the other hand, we also show that any weakly symmetric reflexive toric varieties which have regular triangulation (special) are asymptotic Chow polystable. After that, we provide another criteria that can show a symmetric reflexive toric variety is asymptotic Chow polystable. In particular, we give two examples that are asymptotic Chow polystable, but not special. We also provide some examples of special polytopes, mainly in 2 or 3 dimensions, and some in higher dimensions.

Asymptotic Chow stability of symmetric reflexive toric varieties

TL;DR

The paper investigates asymptotic Chow stability for symmetric reflexive toric varieties, introducing a Futaki-Ono invariant framework and a regular-boundary/special-polytope concept that yields a practical stability criterion. It proves that special polytopes are asymptotically Chow polystable and develops a triangulation-based sufficient condition that extends stability to non-special symmetric polytopes, with explicit examples like and . It provides a thorough catalog of stability across dimensions, including all 2D cases, several 3D families, and higher-dimensional counterexamples (e.g., for ), and discusses how duality can affect stability. Together, these results offer concrete tools and classifications for assessing Chow polystability of toric varieties, with implications for singular Kähler geometry and moduli problems.

Abstract

In this note, we study the asymptotic Chow stability of toric varieties. We provide examples of symmetric reflexive toric varieties that are not asymptotic Chow semistable. On the other hand, we also show that any weakly symmetric reflexive toric varieties which have regular triangulation (special) are asymptotic Chow polystable. After that, we provide another criteria that can show a symmetric reflexive toric variety is asymptotic Chow polystable. In particular, we give two examples that are asymptotic Chow polystable, but not special. We also provide some examples of special polytopes, mainly in 2 or 3 dimensions, and some in higher dimensions.
Paper Structure (19 sections, 23 theorems, 121 equations, 6 figures)

This paper contains 19 sections, 23 theorems, 121 equations, 6 figures.

Table of Contents

  1. introduction
  2. Chow stability of toric variety and criteria
  3. Some special class of toric varieties
  4. Product class
  5. Symmetric Double cone type
  6. special polytopes
  7. Chow stability of special polytopes
  8. Chow stabilities of special polytopes
  9. regular triangulation of $n$ simplex
  10. A sufficient condition of Chow stabilities on toric varieties
  11. $D(X_8)$ and $D(X_9)$
  12. Examples of stability of symmetric reflexive polytopes
  13. 1 and 2 dimensional
  14. 3 dimensional
  15. higher dimensional
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $P$ be a integral convex polytope of a toric variety $X_P$, and let $G<SL(n,\mathbb{Z})$ be the biggest finite group acting on $P$ by multiplication. A $n$ dimensional toric variety $X_P$ is asymptotic Chow semistable iff for any $k\in \mathbb{N}$, and for any convex $G$ invariant function $f$ o

Figures (6)

  • Figure 1: triangulation of 2 simplex and rectangle.
  • Figure 2: $X_i$ for $i=3,4,6,8,9$.
  • Figure 3: $\triangle_0\subset X_3$, $\triangle_0\subset X_4$ and $\triangle_0\subset X_6$.
  • Figure 4: $X_i \times [-1,1]$
  • Figure 5: $D(X_i)$
  • ...and 1 more figures

Theorems & Definitions (72)

  • Theorem 1.1: Ono13
  • Definition 1.2: Definition \ref{['regular triangulation']}
  • Definition 1.3: Definition \ref{['special polytope']}
  • Theorem 1.4: Theorem \ref{['special polytope is stable']}
  • Theorem 1.5: Theorem \ref{['general condition of Chow stable']}
  • Corollary 1.6: Corollary \ref{['D(X_i) are Chow stable']}
  • Definition 2.1
  • Lemma 2.2: corollary 4.7, Ono13; also Fut04
  • Definition 2.3
  • Definition 2.4
  • ...and 62 more