Ordinary varieties and the comparison between multiplier ideals and test ideals II

TL;DR

This paper explores the relationship between multiplier ideals and test ideals across different characteristics, proving that a conjecture about test ideals implies a conjecture about Frobenius actions on cohomology.

Contribution

It demonstrates that the conjecture linking test ideals to multiplier ideals implies a conjecture about Frobenius actions on cohomology, establishing a key implication in characteristic p and zero.

Findings

Proved that the test ideal conjecture implies the Frobenius action conjecture.

Established the connection between multiplier ideals and test ideals in different characteristics.

Extended previous results by showing the implication in the reverse direction.

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 to positive characteristic such that the action of the Frobenius morphism on the top Zariski cohomology of the structure sheaf on X_s is bijective. We also consider the conjecture relating the multiplier ideals of an ideal J on a nonsingular variety in characteristic zero, and the test ideals of the reductions of J to positive characteristic. We prove that the latter conjecture implies the former one. The converse was proved in a joint paper of the author with V. Srinivas.