K-unstable Toric Varieties and Secondary Polytopes

Abstract

For K-unstable toric varieties, Székelyhidi's optimal test-function ${\Theta }_P$ 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 ${\Theta }_P$ 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.

46 pages, 8 figures