Algebro-geometric semistability of polarized toric manifolds

2010/09/01 by Hajime Ono, Ono, Hajime
Mathematics · #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #math.AG #math.CO #math.DG

paper · pdf · doi:10.48550/arxiv.1009.0087

7 pages

arxiv created 2010/09/01 · arxiv updated 2010/09/02

Abstract

Let Δ⊂ ℝⁿ be an n-dimensional integral Delzant polytope. It is well-known that there exist the n-dimensional compact toric manifold X_Δ and the very ample (ℂ^×)ⁿ-equivariant line bundle L_Δ on X_Δ associated with Δ. In the present paper, we give a necessary and sufficient condition for Chow semistability of (X_Δ,L_Δⁱ) 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.

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