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.

arXiv commentThe title has been slightly changed. We adapt the following terminology for semistability of a pair. Semistability is defined to be disjoint orbit closures. Numerical semistability is defined as the inclusion of weight polytopes