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arXiv · 2610.05179

K-unstable Toric Varieties and Secondary Polytopes

Chi Li·Li Sheng·Yi Yao
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Abstract

For K-unstable toric varieties, Székelyhidi's optimal test-function $Θ_{P}$ is a mysterious concave function over the moment polytope $P$, which gives the maximal destabilizer for K-stability, and encodes the limiting behavior of the divergent Calabi flow. As balanced norms quantize cscK metrics, we show that $Θ_{P}$ can be quantized by the maximal destabilizers for Chow-stability, which are given by the shortest GKZ vectors, i.e., the least norm point on the secondary polytope for the set $P\cap k^{-1}\mathbb{Z}^{n}$. Properties and algorithm for general sGKZ vectors are given. Our result may provide a new method to detect the K-unstability of toric varieties.

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Chi Li, Li Sheng, Yi Yao. 2026-10-04. K-unstable Toric Varieties and Secondary Polytopes. https://arxiv.org/abs/2610.05179

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