Stability of weak holonomicity under projective direct images

Prove that weak holonomicity is preserved by projective direct images of coadmissible complexes in the rigid analytic differential-operator formalism.

Background

The paper proves holonomicity of certain complexes by using the known stability of holonomic complexes under projective pushforward. In contrast, the corresponding stability statement for weak holonomicity is not available.

Establishing this stability would help upgrade the holonomic complex result for Gauss–Manin connections to holonomicity and weak holonomicity statements for the individual higher direct-image cohomology modules beyond the strict normal-crossing case.

References

While it is natural to expect that the individual higher pushforwards $Ri j_*\mathcal Eq$ are also holonomic and weakly holonomic beyond the snc case, such a statement currently rests on resolving some fundamental open questions on (weak) holonomicity. For instance, it is known that pushforward along a projective morphism preserves holonomic $\mathcal{C}$-complexes by Theorem 5.10.(iii), which is how we can go from Theorem~\ref{introthm:snc} to Theorem~\ref{introthm:mainhol}, but stability of weak holonomicity has so far not been established.

— Gauss--Manin connections extend to holonomic coadmissible D-cap-modules  (2609.34312 - Bode et al., 28 Sep 2026) in Section 1, Introduction, paragraph beginning “While it is natural to expect”