Monodromy property under tensoring with an inertia representation

Determine whether the monodromy property for a perverse sheaf passes to its tensor product with a representation of the inertia group, without invoking the theory of weights.

Background

The monodromy property is defined by requiring that the filtration induced by the monodromy spectral sequence agree with the monodromy filtration on the perverse cohomology of a proper pushforward. In the treatment of nearby cycles, the relevant perverse sheaf is tensored with an indecomposable unipotent representation of pro-l-adic inertia.

The paper explains that the theory of weights is used to establish the needed result in the arithmetic setting. Without that theory, the authors do not know how to prove that the monodromy property is preserved under tensoring a perverse sheaf with an inertia representation.

References

The reason we allow for a perverse sheaf tensored by a unipotent representation of inertia is the monodromy property: without the theory of weights, we do not know how to show that the monodromy property passes from a perverse sheaf to a perverse sheaf tensored with a representation of the inertia.

— Spreading out perverse sheaves  (2610.06707 - Zurbuchen, 5 Oct 2026) in Section 5, subsection “Geometrically irreducible perverse sheaves”