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Numerical semistability of projective toric varieties

Naoto Yotsutani

Abstract

Let $X \to \mathbb P^N$ be a smooth linearly normal projective variety. It was proved by Paul that the $K$-energy of $(X, {ω_{FS}}|_{X})$ restricted to the Bergman metrics is bounded from below if and only if the pair of (rescaled) Chow/Hurwitz forms of $X$ is numerically semistable. In this paper, we provide a necessary and sufficient condition for a given smooth toric variety $X_P$ to be numerically semistable with respect to $\mathcal O_{X_P}(i)$ for a positive integer $i$. Applying this result to a smooth polarized toric variety $(X_P, L_P)$, we prove that $(X_P, L_P)$ is asymptotically numerically semistable if and only if it is K-semistable for toric degenerations.

Numerical semistability of projective toric varieties

Abstract

Let be a smooth linearly normal projective variety. It was proved by Paul that the -energy of restricted to the Bergman metrics is bounded from below if and only if the pair of (rescaled) Chow/Hurwitz forms of is numerically semistable. In this paper, we provide a necessary and sufficient condition for a given smooth toric variety to be numerically semistable with respect to for a positive integer . Applying this result to a smooth polarized toric variety , we prove that is asymptotically numerically semistable if and only if it is K-semistable for toric degenerations.
Paper Structure (16 sections, 22 theorems, 136 equations)

This paper contains 16 sections, 22 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hilbert-Mumford's semistability
  4. Semistability of pairs
  5. The Chow forms and the discriminant forms
  6. Higher associated hypersurfaces and the Hurwitz forms
  7. $A$-Resultants and $A$-Discriminants
  8. A construction of the secondary polytopes
  9. Massive GKZ vectors and the discriminant polytopes
  10. Hurwitz vectors and Hurwitz polytopes
  11. Paul's criterion for numerical semistability
  12. The key propositions
  13. Proof of the main theorem
  14. The relationship among other notions of GIT-stability
  15. Asymptotic numerical semistability and K-semistability
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $X\subset \mathbb{P}^N$ be a linearly normal smooth projective variety of degree $d\geqslant 2$, where $G=\mathrm{SL}(N+1,\mathbb C)$ acts to $\mathbb{P}^N$ linearly. Let $R_X$ (resp. $\mathrm{Hu}_X$) be the Chow form (resp. the Hurwitz form) of $X$. For any $\sigma \in G$, there are norms $\lef

Theorems & Definitions (43)

  • Theorem 1.1: Theorem A in Sean12
  • Theorem 1.2: Theorem C in Sean12
  • Theorem 1.3
  • Corollary 1.4: See, Theorem \ref{['thm:PandK']}
  • Proposition 2.1: Hilbert-Mumford criterion
  • Proposition 2.2: The numerical criterion
  • Proposition 2.3: Corollary $2.3$ in LLSW19
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 33 more