关于具有好直线的超平面排列的Riemann-Hilbert问题
Middle convolution for affine hyperplane arrangements and Aomoto realizability
AI总结：
研究超平面排列补集上的Riemann-Hilbert问题变体，通过推广Katz的中部卷积为函子，证明其保持该问题的可解性。
摘要：
我们研究了超平面排列补集上的Riemann-Hilbert问题的一个变体。该问题询问补集上的给定局部系统是否可以实现为具有常系数的对数Pfaffian系统的解层。在本文中，我们将Katz的中部卷积推广为超平面补集上局部系统的函子，并证明它保持该问题的可解性。
英文摘要：
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.
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