On the Riemann-Hilbert problem for hyperplane arrangements with a good line

Technical Summary: On the Riemann-Hilbert Problem for Hyperplane Arrangements with a Good Line

Problem Statement
This paper addresses a variant of the Riemann-Hilbert problem on the complement $M(\mathcal{A})$ of an affine hyperplane arrangement $\mathcal{A}$ in $\mathbb{C}^\ell$. The central question is whether a given local system $\mathcal{L}$ on $M(\mathcal{A})$ can be realized as the solution sheaf of a logarithmic Pfaffian system with constant coefficients. Unlike the classical case where the divisor is normally crossing, a general hyperplane arrangement defines a divisor that is not normally crossing, rendering the problem nontrivial. The authors aim to characterize the essential image of the de Rham functor restricted to the category of such logarithmic Pfaffian systems, denoted $\text{Pf}(\log \mathcal{A})$.

Methodology
The authors employ a functorial approach, generalizing the concept of middle convolution to the setting of hyperplane arrangements. The methodology proceeds through the following steps:

1. Geometric Setup (Good Lines): The authors introduce the notion of a "good line" $Y$ for an affine arrangement $\mathcal{A}$. A line $Y$ through the origin is "good" if for every rank-2 element $X$ in the intersection poset $L(\mathcal{A})$, the sum $X + Y$ remains in $L(\mathcal{A})$. This condition ensures that the projection $\pi_Y: M(\mathcal{A}) \to M(\mathcal{A}_Y)$ (where $\mathcal{A}_Y$ consists of hyperplanes parallel to $Y$) admits a fiber bundle structure with fibers isomorphic to $\mathbb{C}$ minus a finite set of points. This generalizes Terao's results for central arrangements to the affine case.

2. Functor Construction for Local Systems: Using the fiber bundle structure induced by a good line, the authors define a middle convolution functor $MC_\chi$ on the category of local systems $\text{Loc}(M(\mathcal{A}), \mathbb{C})$. This is constructed as a generalization of Katz's middle convolution on $\mathbb{C} \setminus \{n \text{ points}\}$. The functor is defined via the image of a morphism between derived direct images ($R^1 pr_{2!}$ and $R^1 pr_{2*}$) of an external tensor product involving a rank-1 local system associated with a character $\chi$.

3. Functor Construction for Connections: Parallel to the local system construction, the authors utilize Haraoka's definition of a middle convolution functor $mc_\lambda$ acting on the category of logarithmic Pfaffian systems with constant coefficients, $\text{Pf}(\log \mathcal{A})$. This functor operates by modifying the coefficient matrices of the connection forms along the direction of the good line $Y$.

4. Compatibility Analysis: The core technical effort involves proving the compatibility between these two functors under the de Rham correspondence. The authors establish an isomorphism between the middle convolution of a local system and the de Rham functor applied to the middle convolution of the corresponding Pfaffian system. This requires careful analysis of period maps, multivalued sections, and the cohomology of the fibers, specifically under Assumption 4.3, which imposes non-resonance conditions on the eigenvalues of the coefficient matrices.

Key Contributions and Results

• Generalization of Goodness: The paper extends Terao's characterization of fiber bundle structures to affine hyperplane arrangements by defining "good lines" and proving that the existence of such a line implies a locally trivial fibration structure on the complement (Theorem 1.8).
• Definition of $MC_\chi$: The authors define the middle convolution functor $MC_\chi$ for local systems on hyperplane complements with a good line (Definition 2.3) and prove it satisfies a composition law analogous to the classical case (Theorem 2.12).
• Compatibility Theorem: The primary result is Theorem 4.8, which establishes an isomorphism of local systems:
  $$MC_\chi \circ DR_{Pf}(\nabla_\mathcal{A}) \cong DR_{Pf} \circ mc_\lambda(\nabla_\mathcal{A})$$
  where $\chi(1) = \exp(2\pi i \lambda)$ and $\nabla_\mathcal{A}$ satisfies the non-resonance Assumption 4.3. This generalizes previous results by Haraoka (for braid arrangements) and Dettweiler–Reiter (for $\mathbb{C} \setminus \{n \text{ points}\}$).
• Preservation of Solvability: As a direct corollary, Theorem 5.2 demonstrates that the middle convolution functor preserves the solvability of the Riemann-Hilbert problem for logarithmic Pfaffian systems. Specifically, if a local system $\mathcal{L}$ admits a solution $\nabla_\mathcal{A}$ satisfying the necessary assumptions, then $MC_\chi(\mathcal{L})$ also admits a solution. Conversely, if the convoluted system admits a solution, the original system does as well (provided the system satisfies property $\wp$).

Significance
The paper claims to provide a structural tool for investigating the Riemann-Hilbert problem in the context of hyperplane arrangements. By showing that the middle convolution functor preserves the property of being realizable as a solution to a logarithmic Pfaffian system with constant coefficients, the authors offer a method to generate new solvable local systems from known ones. This extends the applicability of middle convolution techniques, previously limited to central arrangements or specific configurations, to a broader class of affine arrangements possessing a "good line." The results unify the geometric perspective of fiber bundles on arrangement complements with the algebraic manipulation of connection coefficients.