Numerical semistability of projective toric varieties

Abstract

Let $X \to \mathbb P^N$ be a smooth linearly normal projective variety. It was proved by Paul that the $K$-energy of $(X, {\omega_{FS}}|_{X})$ 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.