On the uniform positivity of 𝐹-signature under reduction modulo 𝑝

Abstract

Carvajal-Rojas, Schwede and Tucker asked whether the mod 𝑝 reductions of a complex klt type singularity have uniformly positive 𝐹-signature for all but finitely many primes 𝑝. In this paper, we give an affirmative answer to this conjecture for pure subrings of regular local rings, including reductive quotient singularities. We also show that the conjecture can be reduced to the Gorenstein case. Finally, we discuss the connection with 𝐹-alpha invariants for log Fano pairs, introduced by Pande as characteristic 𝑝 analogs of Tian’s alpha invariants.