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

Abstract: We explicitly determine the numerical spectrum of the ordinary, non-equivariant $α$-invariant for $n$-dimensional K-polystable toric $\mathbb{Q}$-Fano varieties. Specifically, we prove that $\left{α(X)\mid X\text{ is an }n\text{-dimensional K-polystable toric }\mathbb{Q}\text{-Fano variety}\right}=\mathbb{Q}\cap\left[\frac{1}{n+1},\frac{1}{2}\right].$ 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 paper proves that the alpha spectrum of n-dimensional K-polystable toric Q-Fano varieties is exactly the set of rational numbers in [1/(n+1), 1/2].
• It constructs explicit cyclic-polytopal examples using coprime integers A and B, with the target value r=B/(A+B), and verifies K-polystability through a vanishing barycenter.
• The result includes values below 2/(2n+1), reaches arbitrarily close to the Fujita–Odaka lower bound, and shows that toric examples cannot have alpha invariant greater than 1/2 because of their infinite torus automorphism groups.

Overview and main result

This paper determines, in closed form, the set of values attained by the ordinary, non-equivariant global alpha invariant across all $n$-dimensional K-polystable toric $\mathbb{Q}$-Fano varieties. Specifically, the main theorem states that for every $n \geq 1$ and every rational number

$$r \in \mathbb{Q} \cap \left[\frac{1}{n+1}, \frac{1}{2}\right],$$

there exists an $n$-dimensional K-polystable (hence K-semistable) toric $\mathbb{Q}$-Fano variety $X$ with $\alpha(X) = r$. The result thus gives a complete realization theorem: the alpha spectrum of K-polystable toric $\mathbb{Q}$-Fanos is exactly $\mathbb{Q} \cap [1/(n+1), 1/2]$.

The context is the Fujita–Odaka lower bound $\mathbb{Q}$0 for K-semistable $\mathbb{Q}$1-Fanos, attained by $\mathbb{Q}$2, and Jiang's gap conjecture asserting that a K-semistable Fano manifold with $\mathbb{Q}$3 must be $\mathbb{Q}$4. Liu and Zhuang asked whether, in dimension $\mathbb{Q}$5, there exist K-semistable $\mathbb{Q}$6-Fanos with $\mathbb{Q}$7. Liu and Zhu answered this affirmatively with K-polystable toric examples achieving $\mathbb{Q}$8, and asked whether values below $\mathbb{Q}$9 could be realized. The present paper answers this fully in the toric setting: every rational value in the admissible interval is realized, including the entire sub-interval $n \geq 1$0 that was previously open. A notable corollary of the upper-bound argument is that no toric example can satisfy the second part of Liu–Zhuang's question, since toric $n \geq 1$1-Fanos always have infinite torus automorphisms and hence $n \geq 1$2.

Toric preliminaries and the necessary bounds

For a toric $n \geq 1$3-Fano variety $n \geq 1$4 with fan ray generators $n \geq 1$5, the paper works with the polar convention $n \geq 1$6, where $n \geq 1$7, so that $n \geq 1$8 is the anti-canonical polytope. Two standard facts anchor the argument:

• K-polystability criterion: $n \geq 1$9 is K-semistable, equivalently K-polystable, if and only if the volume barycenter of $$r \in \mathbb{Q} \cap \left[\frac{1}{n+1}, \frac{1}{2}\right],$$0 vanishes, using results of Blum–Jonsson and Berndtsson.
• Alpha invariant formula: with $$r \in \mathbb{Q} \cap \left[\frac{1}{n+1}, \frac{1}{2}\right],$$1, one has $$r \in \mathbb{Q} \cap \left[\frac{1}{n+1}, \frac{1}{2}\right],$$2.

Since $$r \in \mathbb{Q} \cap \left[\frac{1}{n+1}, \frac{1}{2}\right],$$3 is rational, $$r \in \mathbb{Q} \cap \left[\frac{1}{n+1}, \frac{1}{2}\right],$$4 is rational. The upper bound $$r \in \mathbb{Q} \cap \left[\frac{1}{n+1}, \frac{1}{2}\right],$$5 follows immediately from the infinitude of the torus $$r \in \mathbb{Q} \cap \left[\frac{1}{n+1}, \frac{1}{2}\right],$$6 together with Liu–Zhuang's lemma that $$r \in \mathbb{Q} \cap \left[\frac{1}{n+1}, \frac{1}{2}\right],$$7 forces finite automorphism group. Combined with the Fujita–Odaka bound, this yields the necessary containment $$r \in \mathbb{Q} \cap \left[\frac{1}{n+1}, \frac{1}{2}\right],$$8 for K-semistable toric $$r \in \mathbb{Q} \cap \left[\frac{1}{n+1}, \frac{1}{2}\right],$$9-Fanos; the main theorem shows this containment is sharp.

The construction

Fix a target value $n$0 in lowest terms, and set $n$1. Then $n$2 and the condition on $n$3 translates exactly to the inequality $n$4. The lattice is realized as $n$5 with $n$6, and the polytope $n$7 is defined as the convex hull of the $n$8 points

$n$9

with indices cyclic modulo $\mathbb{Q}$0. These are variable-width analogues of the cyclic polytopes used by Liu and Zhu, with the uniform step size replaced by the pair $\mathbb{Q}$1 encoding the target alpha value.

The verification proceeds through several elementary but careful checks. The polar polytope $\mathbb{Q}$2 admits a clean description via the cyclic difference map: identifying $\mathbb{Q}$3 with $\mathbb{Q}$4 by $\mathbb{Q}$5, one obtains

$\mathbb{Q}$6

i.e., the intersection of the traceless hyperplane with the box $\mathbb{Q}$7. From this description, every vertex of $\mathbb{Q}$8 is shown to be primitive: a vertex has $\mathbb{Q}$9 coordinates equal to $X$0, $X$1 equal to $X$2, and one free coordinate $X$3, and primitivity reduces to $X$4. Primitivity of the vertices guarantees that $X$5 is $X$6-Fano: $X$7 is $X$8-Cartier, $X$9 is klt, and the support function of $\alpha(X) = r$0 is strictly convex, so $\alpha(X) = r$1 is ample.

K-polystability and the alpha computation

Two short arguments complete the proof. First, $\alpha(X) = r$2 is invariant under the cyclic permutation $\alpha(X) = r$3 of coordinates, which fixes only the origin in $\alpha(X) = r$4 and preserves Lebesgue measure; hence $\alpha(X) = r$5 and $\alpha(X) = r$6 is K-polystable. Second, since the pairing is maximized at the generators,

$\alpha(X) = r$7

using $\alpha(X) = r$8, and therefore

$\alpha(X) = r$9

The construction is consistent with prior work: for $\mathbb{Q}$0 one recovers $\mathbb{Q}$1, precisely the Liu–Zhu example. The new content is that the full interval is realized, including values arbitrarily close to the Fujita–Odaka lower bound $\mathbb{Q}$2 (approached as $\mathbb{Q}$3). This shows that in the toric $\mathbb{Q}$4-Fano category there is no gap above the lower bound, in contrast with the smooth setting governed by Jiang's conjecture.

Further properties of the examples

The paper records, without proof, several structural features of $\mathbb{Q}$5 that delineate the boundary between the $\mathbb{Q}$6-Fano and smooth/terminal categories:

• Gorensteinness and index: $\mathbb{Q}$7 is Gorenstein exactly when $\mathbb{Q}$8; otherwise its index is $\mathbb{Q}$9.
• $\mathbb{Q} \cap [1/(n+1), 1/2]$0-factoriality: this holds only when $\mathbb{Q} \cap [1/(n+1), 1/2]$1, or $\mathbb{Q} \cap [1/(n+1), 1/2]$2 or $\mathbb{Q} \cap [1/(n+1), 1/2]$3.
• Singularities: the unique smooth (and terminal) example is $\mathbb{Q} \cap [1/(n+1), 1/2]$4; $\mathbb{Q} \cap [1/(n+1), 1/2]$5 is canonical if and only if $\mathbb{Q} \cap [1/(n+1), 1/2]$6.
• Class group: with $\mathbb{Q} \cap [1/(n+1), 1/2]$7 and $\mathbb{Q} \cap [1/(n+1), 1/2]$8, one has $\mathbb{Q} \cap [1/(n+1), 1/2]$9.
• Volume: $\mathbb{Q}$00, where $\mathbb{Q}$01 with $\mathbb{Q}$02.

These data confirm that the family is genuinely singular in general: outside the exceptional cases listed above, the varieties are neither $\mathbb{Q}$03-factorial nor terminal, which is consistent with the expectation that Jiang-type gap phenomena in the smooth category cannot be probed with toric methods.

Limitations and open questions

The realization theorem is intrinsically toric and $\mathbb{Q}$04-Fano. The paper's own upper-bound argument shows that toric varieties can never produce K-semistable $\mathbb{Q}$05-Fanos with $\mathbb{Q}$06, so the second part of Liu–Zhuang's original question requires non-toric methods. Likewise, all examples beyond $\mathbb{Q}$07 are singular, so the construction sheds no direct light on Jiang's gap conjecture for Fano manifolds. The paper also leaves the proofs of the structural properties in its final section as "standard exercises," and does not address whether the realized spectrum is stable under deformations or whether non-toric K-polystable $\mathbb{Q}$08-Fanos realize the same interval. A natural question the author records via Yuchen Liu is what further invariants of the constructed varieties can be controlled within the family.

Conclusion

The paper settles the alpha spectrum problem for K-polystable toric $\mathbb{Q}$09-Fanos completely: the spectrum is exactly $\mathbb{Q}$10, with both endpoints forced by general theory and every intermediate rational value realized by an explicit two-parameter family of cyclic polytopes $\mathbb{Q}$11. The proof combines the Blum–Jonsson alpha formula with a barycenter computation and a primitivity check, and it refines the earlier constructions of Liu and Zhu while confirming that the toric category cannot address the remaining smooth-category questions.