Determine the invariants computed by twisted logarithmic de Rham complexes of multiarrangements

Determine what algebraic, geometric, or topological invariants are computed by the cohomology of untwisted and twisted logarithmic de Rham complexes associated with hyperplane multiarrangements.

Background

The paper introduces logarithmic de Rham complexes for multiarrangements as extensions of the corresponding complexes for ordinary hyperplane arrangements. Although these complexes are stable under the de Rham and twisted differentials and exhibit top-degree cohomology concentration in several important families, their intrinsic meaning is not established.

Understanding the invariants represented by these cohomology groups would provide a geometric or topological interpretation of the algebraic constructions and clarify their relationship with complements of arrangements and local-system cohomology.

References

However, while the notion of a logarithmic differential form on a hyperplane arrangement is available, and the vector space of logarithmic de Rham forms $\Omega\bullet(A,m)$ is stable under the de Rham differential, it is not at all clear what information the cohomology of the corresponding complex carries.

— Twisted logarithmic de Rham complexes and the rational hit conjecture  (2610.26028 - Dotsenko, 19 Oct 2026) in Section 1.1, Introduction

Two natural questions for subsequent work concern possible generalizations to exceptional complex reflection groups and possible approaches to questions of for $\ell>1$ in the same way, making use of suitable multiarrangements of subspaces of codimension $\ell$, as well as the associated logarithmic vector fields and the arising Lie--Rinehart homology. It is not yet clear to us whether these questions, especially the second one, have promise, but they are certainly worth examining.

— Twisted logarithmic de Rham complexes and the rational hit conjecture  (2610.26028 - Dotsenko, 19 Oct 2026) in Section 1.3, paragraph beginning “Two natural questions for subsequent work”