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

Shunya Adachi, Kazuki Hiroe

TL;DR

This work analyzes a Riemann-Hilbert-type question on the arrangement complement $M( ext{A})$, asking when a local system is realizable as the solution sheaf of a logarithmic Pfaffian system with constant coefficients, with flat connections defined by $ abla_A=d-oldsymbol{ abla}_A$ and $oldsymbol{ abla}_A= extstyleigl(igoplus_H A_Higr) rac{df_H}{f_H}$. It then develops a generalized middle convolution $ ext{MC}_{oldsymbol{ mi}}$ along a good line $Y$, proves its compatibility with Katz’s classical construction via fibers, and establishes a period-map framework that ties de Rham realizations to local systems. A parallel theory for logarithmic Pfaffian systems with constant coefficients is built, defining a Pfaffian middle convolution $ ext{mc}_{oldsymbol{ mi}}$ and proving a compatibility result $ ext{MC}_{oldsymbol{ mi}} ext{DR}_{ ext{Pf}}igl( abla_Aigr) o ext{DR}_{ ext{Pf}}igl( ext{mc}_{oldsymbol{ mi}}( abla_A)igr)$ under a genericity assumption. The paper also extends Terao’s fiber-bundle perspective to affine arrangements, proves a composition law for middle convolution in the Pfaffian setting, and shows that middle convolution preserves the solvability of the Riemann-Hilbert problem for $ ext{Loc}(M( ext{A}))$, thereby generating new solvable instances and deepening connections between local systems, de Rham theory, and integrable structures.

Abstract

We study a variant of the Riemann-Hilbert problem on the complements of hyperplane arrangements. This problem asks whether a given local system on the complement can be realized as the solution sheaf of a logarithmic Pfaffian system with constant coefficients. In this paper, we generalize Katz's middle convolution as a functor for local systems on hyperplane complements and show that it preserves the solvability of this problem.

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

TL;DR

This work analyzes a Riemann-Hilbert-type question on the arrangement complement , asking when a local system is realizable as the solution sheaf of a logarithmic Pfaffian system with constant coefficients, with flat connections defined by and . It then develops a generalized middle convolution along a good line , proves its compatibility with Katz’s classical construction via fibers, and establishes a period-map framework that ties de Rham realizations to local systems. A parallel theory for logarithmic Pfaffian systems with constant coefficients is built, defining a Pfaffian middle convolution and proving a compatibility result under a genericity assumption. The paper also extends Terao’s fiber-bundle perspective to affine arrangements, proves a composition law for middle convolution in the Pfaffian setting, and shows that middle convolution preserves the solvability of the Riemann-Hilbert problem for , thereby generating new solvable instances and deepening connections between local systems, de Rham theory, and integrable structures.

Abstract

We study a variant of the Riemann-Hilbert problem on the complements of hyperplane arrangements. This problem asks whether a given local system on the complement can be realized as the solution sheaf of a logarithmic Pfaffian system with constant coefficients. In this paper, we generalize Katz's middle convolution as a functor for local systems on hyperplane complements and show that it preserves the solvability of this problem.
Paper Structure (22 sections, 24 theorems, 176 equations)

This paper contains 22 sections, 24 theorems, 176 equations.

Table of Contents

  1. Riemann-Hilbert problem for logarithmic Pfaffian systems
  2. Main results
  3. Fiber bundle structure on complement of hyperplane arrangement
  4. Hyperplane arrangement with modular coatom
  5. Fibration on affine hyperplane arrangement
  6. Middle convolution on complements of hyperplane arrangements with a good line
  7. Complement of simple Weierstrass polynomial
  8. Middle convolution on complements with a good line
  9. Comparison with Katz middle convolution
  10. Composition low for middle convolution
  11. Middle convolution as coimage
  12. Middle convolution for logarithmic Pfaffian systems with constant coefficients
  13. Category of logarithmic Pfaffian systems with constant coefficients
  14. Convolution functor along a line
  15. Middle convolution functor along a line
  16. ...and 7 more sections

Key Result

Theorem 3

Let $\nabla_{A}\in \mathrm{Pf}(\log\mathcal{A})$ be a logarithmic Pfaffian system satisfying Assumption as:generic0 for some $\lambda\in \mathbb{C}\setminus\mathbb{Z}$. Let $\chi\colon \mathbb{Z}\to \mathbb{C}^{\times}$ be the character defined by $\chi(1)=\exp(2\pi i\lambda)$. Then there exists an

Theorems & Definitions (54)

  • Theorem 3: Theorem \ref{['thm:MCdeRham']}
  • Theorem 4: Theorem \ref{['thm:RHMC']}
  • Definition 1.1
  • Theorem 1.2: Terao, Theorem 2.9 in Tera1
  • Proposition 1.3
  • proof : Proof of Proposition \ref{['prop:vclosed']}
  • Definition 1.4: cone of affine arrangement
  • Definition 1.5: decone of central arrangement
  • Definition 1.6
  • Remark 1.7
  • ...and 44 more