Ordinary varieties and the comparison between multiplier ideals and test ideals

Abstract

We consider the following conjecture: if $X$ is a smooth and irreducible $n$-dimensional projective variety over a field $k$ of characteristic zero, then there is a dense set of reductions $X_s$ to positive characteristic such that the action of the Frobenius morphism on $H^n(X_s,{\mathcal{O}}_{X_s})$ is bijective. There is another conjecture relating certain invariants of singularities in characteristic zero (the multiplier ideals) with invariants in positive characteristic (the test ideals). We prove that the former conjecture implies the latter one in the case of ambient nonsingular varieties.