The Alpha invariant of a complex Fano manifold was introduced by Tian to detect its K-stability, an algebraic condition that implies the existence of a Kähler–Einstein metric. Demailly later reinterpreted the Alpha invariant algebraically in terms of a singularity invariant called the log canonical threshold. In this talk, we will present an analog of the Alpha invariant for Fano varieties in positive characteristics, called the Frobenius-Alpha invariant. This analog is obtained by replacing “log canonical threshold” with “F-pure threshold”, a singularity invariant defined using the Frobenius map. We will review the definition of these invariants and the relations between them. The main theorem proves some interesting properties of the Frobenius-Alpha invariant; namely, we will show that its value is always at most 1/2 and make connections to a version of local volume called the F-signature.

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Last updated: 01 Oct 2023