Ordinary varieties and the comparison between multiplier ideals and test ideals

TL;DR

This paper explores the relationship between multiplier ideals and test ideals, proposing conjectures about their connection across characteristics and proving implications for nonsingular varieties.

Contribution

It establishes that a conjecture about Frobenius action in positive characteristic implies a conjecture linking multiplier and test ideals in characteristic zero for nonsingular varieties.

Findings

Proves the implication from Frobenius action conjecture to the multiplier-test ideal conjecture.

Provides evidence for the connection between invariants of singularities in different characteristics.

Abstract

We consider the following conjecture: if X is a smooth projective variety over a field of characteristic zero, then there is a dense set of reductions X_s of X to positive characteristic such that the action of the Frobenius morphism on the top Zariski cohomology of the structure sheaf of X_s is bijective. We also consider a 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.