  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.
