The alpha spectrum of K-polystable toric $\mathbb{Q}$-Fano varieties

Abstract

We explicitly determine the numerical spectrum of the ordinary, non-equivariant global invariant for $n$-dimensional K-polystable toric $\mathbb{Q}$-Fano varieties. Specifically, we prove that $$\left\{ \alpha(X)\mid X\text{ is an $n$-dimensional K-polystable toric $\mathbb{Q}$-Fano variety}\right \} = \mathbb{Q}\cap \left[\frac{1}{n+1},\frac{1}{2}\right].$$ This result establishes a complete toric realization theorem, which strengthens a question raised by Liu and Zhuang and refines the recent construction results of Liu and Zhu. The initial construction of the examples in this work was suggested by GPT-5.6 Sol, and was subsequently refined and rigorously developed by the author.