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Periods of hyperplane arrangements and multiple polylogarithms

Riccardo Tosi

Abstract

We compute the periods associated with a special class of hyperplane arrangements. In particular, we exhibit a combinatorial condition on the intersection lattice of a hyperplane arrangement that ensures that its associated periods are linear combinations of special values of multiple polylogarithms. Our method generalizes Brown's approach to the periods of moduli spaces of curves of genus zero. We apply this result to the reflection arrangement of the full monomial group, whose periods are shown to be linear combinations of values of multiple polylogarithms at roots of unity.

Periods of hyperplane arrangements and multiple polylogarithms

Abstract

We compute the periods associated with a special class of hyperplane arrangements. In particular, we exhibit a combinatorial condition on the intersection lattice of a hyperplane arrangement that ensures that its associated periods are linear combinations of special values of multiple polylogarithms. Our method generalizes Brown's approach to the periods of moduli spaces of curves of genus zero. We apply this result to the reflection arrangement of the full monomial group, whose periods are shown to be linear combinations of values of multiple polylogarithms at roots of unity.
Paper Structure (27 sections, 58 theorems, 219 equations)

This paper contains 27 sections, 58 theorems, 219 equations.

Table of Contents

  1. Hyperplane arrangements and modular elements
  2. Compactifications of complements of hyperplane arrangements
  3. Hyperplane arrangements
  4. De Concini-Procesi compactifications
  5. Examples
  6. Supersolvable arrangements
  7. Arrangements with enough modular elements
  8. Retractions to boundary divisors
  9. Modular elements and building sets
  10. Construction of retractions
  11. Periods of hyperplane arrangements
  12. The reduced bar complex
  13. Definition and first properties
  14. The reduced bar complex of fiber type arrangements
  15. Analytic manifolds with corners
  16. ...and 12 more sections

Key Result

Theorem A

Assume that $\mathcal{A}$ has enough modular elements. Let $A,B\subseteq \overline Y_\mathcal{A}\setminus Y_\mathcal{A}$ be two divisors without common irreducible components. Consider an admissible relative homology class $\Delta\in H_l(\overline Y_\mathcal{A}\setminus A, B\setminus (A\cap B))$ and lies in the $k$-subalgebra of $\mathbb{C}$ generated by Moreover, this $k$-algebra can be endowed

Theorems & Definitions (145)

  • Theorem A: see Theorem \ref{['theo: periods of hyperplane arrangements with enough modular elements']}
  • Theorem B: see Theorem \ref{['theo: full monomial group']}
  • Definition 1.1.1
  • Remark 1.1.2
  • Definition 1.1.3
  • Definition 1.1.4
  • Definition 1.1.5
  • Definition 1.1.6
  • Definition 1.1.7
  • Remark 1.1.8
  • ...and 135 more