Semistability of pairs for projective toric varieties

Naoto Yotsutani

Published 2024-01-04Version 1

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 semistable. In this paper, we provide a necessary and sufficient condition for a given smooth toric variety $X_P$ to be semistable of pairs 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 semistable of pairs if and only if it is K-semistable for toric degenerations.