Jiang’s gap conjecture for K-semistable Fano manifolds

Prove that an n-dimensional K-semistable Fano manifold X satisfies \(\alpha(X)<1/n\) if and only if \(X\cong\mathbb{P}^n\), thereby establishing the conjectured gap between the general lower bound \(\alpha(X)\geq 1/(n+1)\) and the value \(1/n\) in the smooth setting.

Background

The paper recalls the Fujita–Odaka lower bound α(X)≥1/(n+1)\alpha(X)\geq 1/(n+1) for n-dimensional K-semistable QQ-Fano varieties, with equality attained by Pn\mathbb{P}^n. In the smooth setting, Jiang proposed a stronger gap statement asserting that values below $1/n$ occur only for projective space.

The paper’s main theorem determines the alpha-invariant spectrum for K-polystable toric QQ-Fano varieties, but it does not resolve Jiang’s conjecture for general K-semistable Fano manifolds or even for the full smooth setting described in the conjecture.

References

In the smooth setting, Jiang proved that equality in eq:FO-lower characterizes $Pn$ and proposed the following gap conjecture.

\begin{conj}[{Conjecture~1.6}] Let $X$ be an $n$-dimensional K-semistable Fano manifold. Then $$ \alpha(X)<\frac1n\quad\Longleftrightarrow\quad X\congPn. $$ \end{conj}

eq:FO-lower:

α(X)≥1n+1;\alpha(X)\geq\frac1{n+1};

— The alpha spectrum of K-polystable toric $\mathbb{Q}$-Fano varieties  (2608.14115 - Wu, 14 Aug 2026) in Section 1, Introduction, Conjecture 1.6