Motivic incarnation of the microlocalization–vanishing cycles isomorphism

Develop a motivic-level construction that realizes the isomorphism Ø_g(T'(M)[-r]) ≅ u_z(M) from Theorem 1.1 in an appropriate motivic framework (for example, in a category of motives or a Grothendieck-type ring), thereby producing a “motivic incarnation” that reflects this microlocalization–vanishing cycles relation.

Background

Theorem 1.1 establishes a precise identification between vanishing cycles along g and microlocalization for mixed Hodge modules at the level of bi-filtered D-modules. A motivic formulation would lift this relation to a more universal setting, potentially connecting it to invariants in algebraic and arithmetic geometry.

The authors indicate interest in such a motivic counterpart but presently lack a method to construct it.

References

It would be very nice to find a "motivic" incarnation of this isomorphism, though we cannot see how that might be done at the moment.

— Fourier transform and Radon transform for mixed Hodge modules  (2405.19127 - Dirks, 2024) in Section 1. Introduction

However, it is unclear that whether one can define $DN(-)$ for diagrams of varieties so we propose a different solution to this.

— Motivic nearby functors on perverse Nori motives  (2608.17412 - Pham, 18 Aug 2026) in Section 6.1, proof of the proposition on commutation with Braden transformations