Tian’s alpha-invariant conjecture for projectively normal polarized varieties

Prove that for a polarized algebraic variety with a very ample line bundle L whose associated embedding is projectively normal, the global log canonical threshold satisfies glct(X,L)=α₁(X,L).

Background

The paper introduces the global log canonical threshold glct(X,L) and Tian’s m-th alpha-invariant α_m(X,L). It notes the general inequality glct(X,L)≤α_m(X,L) whenever the relevant linear system is nonempty.

The authors then state a conjecture attributed to Tian asserting equality for a very ample line bundle that defines a projectively normal embedding, meaning that the associated graded section algebra is generated by its degree-one sections. The paper uses this conjectural framework to interpret equalities established for the specific families of surfaces studied, but does not prove the conjecture in general.

References

Tian subsequently proposed the following conjecture. Suppose that $L$ is very ample and defines a projectively normal embedding under its associated morphism, that is, the graded algebra $$\bigoplus\limits_{i \ge 0 }H{0}\left(X, \mathcal{O}{X}\left(iL \right) \right)$$ is generated by elements in $H{0}\left(X, \mathcal{O}{X}\left(L \right) \right)$. Then $\glct \left(X,L \right) = \alpha_1(X, L)$.

— Global log canonical thresholds for Burniat surfaces and some infinite families of surfaces of general type as $\mathbb{Z}_2^n$-covers  (2609.35678 - Bin et al., 28 Sep 2026) in Section 2, subsection “Some facts on global log canonical thresholds,” immediately preceding Conjecture 5.4