Algebro-geometric semistability of polarized toric manifolds

Abstract

Let $\Delta \subset {\mathbb{R}}^n$ be an $n$-dimensional integral Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_{\Delta }$ and the very ample $({\mathbb{C}}^{\times }{)}^n$-equivariant line bundle $L_{\Delta }$ on $X_{\Delta }$ associated with $\Delta$. In the present paper, we give a necessary and sufficient condition for Chow semistability of $(X_{\Delta },L_{\Delta }^i)$ for a maximal torus action. We then see that asymptotic (relative) Chow semistability implies (relative) K-semistability for toric degenerations, which is proved by Ross and Thomas, without any knowledge of Riemann-Roch theorem and test configurations.

7 pages

References in corpus (1)

• A necessary condition for Chow semistability of polarized toric manifolds

Cited by in corpus (1)

• Asymptotic Chow polystability in Kähler geometry