A Frobenius version of Tian’s alpha-invariant

Abstract For a pair ( X , L ) (X,L) consisting of a projective variety 𝑋 over a perfect field of characteristic p > 0 p>0 and an ample line bundle 𝐿 on 𝑋, we introduce and study a positive characteristic analog of Tian’s 𝛼-invariant, which we call the α F \alpha_{F} -invariant . We utilize the theory of 𝐹-singularities in positive characteristics, and our approach is based on replacing klt singularities with the closely related notion of global 𝐹-regularity. We show that the α F \alpha_{F} -invariant of a pair ( X , L ) (X,L) can be understood in terms of the global Frobenius splittings of the linear systems | m ⁢ L | m ≥ 0 \lvert mL\rvert_{m\geq 0} . We establish inequalities relating the α F \alpha_{F} -invariant with the 𝐹-signature, and use that to prove the positivity of the α F \alpha_{F} -invariant for all globally 𝐹-regular projective varieties (with respect to any ample 𝐿 on 𝑋). When 𝑋 is a Fano variety and L = − K X L=-K_{X} , we prove that the α F \alpha_{F} -invariant of 𝑋 is always bounded above by 1 2 \frac{1}{2} and establish tighter comparisons with the 𝐹-signature. We also show that, for toric Fano varieties, the α F