Adjoint ideals and a correspondence between log canonicity and $F$-purity

Abstract

This paper presents three results on $F$-singularities. First, we give a new proof of Eisenstein’s restriction theorem for adjoint ideal sheaves using the theory of $F$-singularities. Second, we show that a conjecture of Mustaţă and Srinivas implies a conjectural correspondence of $F$-purity and log canonicity. Finally, we prove this correspondence when the defining equations of the variety are very general.