Middle convolution for affine hyperplane arrangements and Aomoto realizability

Abstract

We construct middle convolution for local systems on complements of affine hyperplane arrangements. Our aim is to study Aomoto realizability: whether a local system can be realized by a flat logarithmic connection on a trivial bundle with constant residue matrices. For a chosen direction, we minimally enlarge the arrangement so that projection becomes a bundle of punctured affine lines. On the associated relative stratification, we extend Katz's middle convolution to constructible perverse sheaves and prove compatibility with its action on the fibers. It induces middle convolution on local systems and satisfies composition and inversion laws under fiberwise hypotheses. Using logarithmic de Rham cohomology, we compare this construction with Haraoka's algebraic middle convolution of logarithmic Pfaffian systems under explicit spectral conditions. The comparison requires no irreducibility assumption and allows integral differences between nonintegral residue eigenvalues, extending the scope of the classical comparison theorems. We also prove flatness of the algebraic convolution for every parameter. Consequently, middle convolution preserves Aomoto realizability on the enlarged arrangement. For inverse middle convolution, we give sufficient conditions for removing the additional poles and recovering an Aomoto realization of the original local system on the initial arrangement.

arXiv commentSubstantially revised and reorganized. The construction now applies to arbitrary affine hyperplane arrangements via Y-closure. The relative perverse sheaf construction and the cohomological comparison proof have been refined, and an independent proof of flatness for all parameters has been added