Numerical semistability of projective toric varieties

2024/01/04 by Naoto Yotsutani, Yotsutani, Naoto
Mathematics · #14M25 #51M20 #53C55 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2401.02010

openalex publication_date 2024/01/04 · openalex created_date 2024/01/13 · openalex updated_date 2026/07/28

Abstract

Let X → \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_XP(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.

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