Asian Journal of Mathematics

Algebro-geometric semistability of polarized toric manifolds

Hajime Ono

Full-text: Open access


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 a very ample $(\mathbb{C}×)^n$-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 $( X_{\Delta}, {L^i}_{\Delta})$ 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.

Article information

Asian J. Math., Volume 17, Number 4 (2013), 609-616.

First available in Project Euclid: 22 August 2014

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14L24: Geometric invariant theory [See also 13A50] 14M25: Toric varieties, Newton polyhedra [See also 52B20] 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]

Chow stability K-stability polarized toric manifold


Ono, Hajime. Algebro-geometric semistability of polarized toric manifolds. Asian J. Math. 17 (2013), no. 4, 609--616.

Export citation