Weakening the closed-fiber hypotheses in the unramified-cohomology vanishing theorem

Determine to what extent the assumption that the irreducible components of the reduced closed fiber satisfy the Tate conjecture can be weakened in the vanishing theorem for the Tate module of third unramified cohomology of a smooth projective surface over a local field.

Background

The paper proves vanishing of the Tate module of third unramified cohomology under hypotheses including potentially good reduction of the Albanese variety and the Tate conjecture for the irreducible components of a semistable closed fiber. The authors note that the potentially good reduction hypothesis cannot generally be removed, citing counterexamples of Parimala–Suresh.

The unresolved issue is whether the condition imposed on the components of the closed fiber can be replaced by a weaker geometric or cohomological condition while retaining the same vanishing conclusion.