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

Carvajal-Rojas, Schwede and Tucker asked whether the mod $p$ reductions of a complex klt type singularity have uniformly positive $F$-signature for almost all primes $p$. In this paper, we give an affirmative answer to this conjecture in the case of pure subrings of regular local rings–for example, 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–a characteristic $p$ analog of Tian’s alpha invariants introduced by Pande–for log Fano pairs.