K-polystable toric Fano varieties with small alpha invariants

Abstract: For every $n\geq 2$, we exhibit an $n$-dimensional K-polystable toric $\mathbb{Q}$-Fano variety $X_n$, defined by the face fan of an explicit lattice polytope, and whose alpha invariant is exactly $\tfrac{2}{2n+1}$. This answers a question of Liu and Zhuang whether there exists an $n$-dimensional K-semistable $\mathbb{Q}$-Fano variety whose alpha invariant is between $\tfrac{1}{n+1}$ and $\tfrac1n$. The main result of this paper was obtained by Chatgpt 5.5 pro, and the Danus system based on the Rethlas system.

• The paper constructs, for every dimension n≥2, an explicit K-polystable toric Q-Fano variety with alpha invariant exactly 2/(2n+1), strictly between 1/(n+1) and 1/n.
• The authors use a combinatorial dual-polytope construction, proving the required vertex structure, lattice primitivity, origin-centered barycenter, and extremal pairing that determines the invariant.
• The result confirms that Tian’s criterion cannot be uniformly strengthened for singular Q-Fano varieties, while leaving open whether smaller values occur and whether Jiang’s conjecture holds for smooth Fano manifolds.

The paper constructs, for every dimension $n \geq 2$, an explicit $n$-dimensional toric $\mathbb{Q}$-Fano variety $X_n$ that is K-polystable and whose alpha invariant equals exactly $\tfrac{2}{2n+1}$ (2607.04005). This value lies strictly between the sharp lower bound $\tfrac{1}{n+1}$ for K-semistable Fano varieties and $\tfrac{1}{n}$, thereby answering affirmatively a question of Liu and Zhuang on the sharpness of Tian's criterion.

Background and motivation

For a $\mathbb{Q}$-Fano variety $X$, the alpha invariant (global log canonical threshold) is defined as

$$\alpha(X) = \inf\{\operatorname{lct}(X; D) \mid D \sim_{\mathbb{Q}} -K_X,\ D \geq 0\}.$$

Tian's criterion states that an $n$0-dimensional Fano variety with $n$1 (respectively $n$2) is K-stable (respectively K-semistable). Fujita and Odaka established the lower bound $n$3 for K-semistable $n$4-Fano varieties of dimension $n$5, and this is sharp: $n$6 attains exactly $n$7. Jiang showed moreover that $n$8 is the only K-semistable Fano manifold achieving this minimum, and proposed the conjecture that a K-semistable Fano manifold with $n$9 must be $\mathbb{Q}$0. Liu and Zhuang reformulated this as a question: does there exist an $\mathbb{Q}$1-dimensional K-semistable $\mathbb{Q}$2-Fano variety with

$\mathbb{Q}$3

An affirmative answer shows that no strengthening of Tian's criterion with a uniform threshold above $\mathbb{Q}$4 can hold for singular varieties, since the constructed examples are K-*poly*stable yet have alpha invariant $\mathbb{Q}$5 for all $\mathbb{Q}$6.

The dual polytope construction

The construction is purely combinatorial. Working in the dual lattices

$\mathbb{Q}$7

the paper defines, with $\mathbb{Q}$8, the points

$\mathbb{Q}$9

and sets $X_n$0, with $X_n$1 the polar polytope. The key technical device is the cyclic difference coordinate map $X_n$2 with $X_n$3, which identifies $X_n$4 with the zero-sum lattice and converts $X_n$5 into the transparent hyperplane slice

$X_n$6

The combinatorial core is then established by elementary but careful arguments:

• Vertices: $X_n$7 has exactly $X_n$8 vertices, namely the points $X_n$9 (indexed by ordered pairs $\tfrac{2}{2n+1}$0) with difference coordinates $\tfrac{2}{2n+1}$1, $\tfrac{2}{2n+1}$2, and $\tfrac{2}{2n+1}$3 otherwise. The proof hinges on the number-theoretic fact that $\tfrac{2}{2n+1}$4, which rules out all coordinates being endpoints of the box.
• Primitivity: each $\tfrac{2}{2n+1}$5 is a primitive lattice point, so these are genuine ray generators of a smooth-in-codimension-one sense fan.
• Barycenter: the cyclic coordinate permutation preserves $\tfrac{2}{2n+1}$6 and its only fixed point is $\tfrac{2}{2n+1}$7, forcing the barycenter of $\tfrac{2}{2n+1}$8 to be the origin.
• Extremal pairing: $\tfrac{2}{2n+1}$9, attained at $\tfrac{1}{n+1}$0 against $\tfrac{1}{n+1}$1.

The toric $\tfrac{1}{n+1}$2-Fano variety

Taking $\tfrac{1}{n+1}$3 as the face fan of $\tfrac{1}{n+1}$4 and $\tfrac{1}{n+1}$5, the ray generators are the vertices $\tfrac{1}{n+1}$6, the anticanonical divisor is $\tfrac{1}{n+1}$7, and the anticanonical polytope is exactly $\tfrac{1}{n+1}$8. The paper verifies that $\tfrac{1}{n+1}$9 is $\tfrac{1}{n}$0-Cartier via the facet supporting functions, ample since $\tfrac{1}{n}$1 is bounded and full-dimensional, and that $\tfrac{1}{n}$2 is klt since every nonzero $\tfrac{1}{n}$3 in a maximal cone has positive log discrepancy. Thus $\tfrac{1}{n}$4 is an $\tfrac{1}{n}$5-dimensional toric $\tfrac{1}{n}$6-Fano variety.

K-polystability and alpha invariant

Two standard results then finish the argument. By Berman's toric criterion, K-polystability of a toric $\tfrac{1}{n}$7-Fano variety is equivalent to the barycenter of the anticanonical polytope being the origin; since $\tfrac{1}{n}$8, $\tfrac{1}{n}$9 is K-polystable, hence K-semistable. By the Blum–Jonsson toric formula,

$\mathbb{Q}$0

and the extremal pairing $\mathbb{Q}$1 yields

$\mathbb{Q}$2

Since $\mathbb{Q}$3 for all $\mathbb{Q}$4 (indeed $\mathbb{Q}$5 and $\mathbb{Q}$6), the main theorem follows: for every $\mathbb{Q}$7 there exists a K-polystable toric $\mathbb{Q}$8-Fano variety of dimension $\mathbb{Q}$9 with alpha invariant exactly $X$0.

Limitations and open questions

The examples are necessarily singular, so Jiang's conjecture for K-semistable Fano manifolds remains untouched; the paper explicitly notes that the conjecture may still hold. The paper also makes no claim regarding part (2) of the Liu–Zhuang question. The natural refinement posed is whether the value $X$1 is itself optimal: does there exist a K-semistable $X$2-Fano variety of dimension $X$3 with

$X$4

The paper notes, as a limitation of the automated systems involved, that some related references in the literature may have been missed. It is also worth recording, as the authors do, that the proof sketch was obtained by ChatGPT 5.5 Pro and subsequently verified and written up with the assistance of the Danus system built on the Rethlas framework, with human verification and polishing.

Conclusion

The paper settles a question on the sharpness of Tian's criterion by giving an explicit, fully combinatorial family of K-polystable toric $X$5-Fano varieties $X$6 with $X$7 for every $X$8. The construction demonstrates that the gap between the Fujita–Odaka lower bound $X$9 and Tian's threshold $$\alpha(X) = \inf\{\operatorname{lct}(X; D) \mid D \sim_{\mathbb{Q}} -K_X,\ D \geq 0\}.$$0 is populated by K-polystable examples, while leaving open both the manifold version of the phenomenon and the optimality of the value $$\alpha(X) = \inf\{\operatorname{lct}(X; D) \mid D \sim_{\mathbb{Q}} -K_X,\ D \geq 0\}.$$1 within the interval.