A Frobenius version of Tian's alpha invariant, and the F-signature of Fano varieties

Abstract: 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. Time permitting, we will also discuss the semicontinuity properties of the Frobenius-Alpha invariant.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html