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Motivic principal value integrals for hyperplane arrangements

Nero Budur, Quan Shi, Huaiqing Zuo

Abstract

A conjecture of Denef-Jacobs-Veys relates motivic principal value integrals of multivalued rational top-forms with cohomology support loci of rank one local systems. We give a stronger positive answer to this conjecture for hyperplane arrangements.

Motivic principal value integrals for hyperplane arrangements

Abstract

A conjecture of Denef-Jacobs-Veys relates motivic principal value integrals of multivalued rational top-forms with cohomology support loci of rank one local systems. We give a stronger positive answer to this conjecture for hyperplane arrangements.

Paper Structure

This paper contains 10 sections, 24 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multivalued rational $n$-forms and local systems
  4. Motivic principal value integrals.
  5. Hyperplane arrangements
  6. Generic hyperplane arrangements
  7. General hyperplane arrangements
  8. The constant term.
  9. Generic forms
  10. Non-essential and decomposable cases.

Key Result

Theorem 1.2

Let $X$ be the canonical log resolution of a hyperplane arrangement $\mathcal{A}\subset\mathbb P^n$, and let $\omega^{1/q}$ be a multivalued rational $n$-form on $X$ without logarithmic poles and such that the support of $\mathrm{div}(\omega^{1/q})$ contains the strict transforms of all dense edges.

Theorems & Definitions (57)

  • Conjecture 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Conjecture 1.7
  • Theorem 1.8
  • Remark 2.1
  • Remark 2.2
  • ...and 47 more