Extendability of differential forms via Cartier operators

Abstract

Let (X, B) be a pair of a normal variety over a perfect field of positive characteristic and a reduced divisor. We prove that if the Cartier isomorphism on the log smooth locus of (X, B) extends to the whole X, then (X, B) satisfies the logarithmic extension theorem. As applications, we show that the logarithmic (resp. regular) extension theorem holds for a quotient singularity by a linearly reductive group scheme (resp. a finite group scheme order prime to the characteristic). We also prove that the logarithmic extension theorem for one-forms for singularities of higher codimension holds under assumptions about Serre's condition.