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A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups

Takuro Abe, Gerhard Röhrle, Christian Stump, Masahiko Yoshinaga

Abstract

Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration of that module. Our approach is based on a detailed analysis of a flat connection applied to the primitive vector field. This generalizes and unifies analogous results for real reflection groups.

A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups

Abstract

Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration of that module. Our approach is based on a detailed analysis of a flat connection applied to the primitive vector field. This generalizes and unifies analogous results for real reflection groups.

Paper Structure

This paper contains 11 sections, 26 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multi-arrangements of hyperplanes and their derivations
  4. Unitary reflection groups and their reflection arrangements
  5. Flat connections for well-generated unitary reflection groups
  6. A general version of Saito's criterion and universal vector fields
  7. A general version of Saito's criterion
  8. Properties of universal vector fields on multi-arrangements
  9. A Hodge filtration and universality for reflection arrangements
  10. Construction of the bases
  11. Universal vector fields for reflection arrangements

Key Result

Lemma 2.3

We have $\mu > \nu$ if and only if $\mathfrak D({\mathscr A},\mu) \subsetneqq \mathfrak D({\mathscr A},\nu)$. In particular, we have $\mathfrak D({\mathscr A},\mu) \subseteq {{\operatorname{Der}}_S}$ if and only if $\mu \geq 0$.

Theorems & Definitions (51)

  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.5
  • Proposition 2.17
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.4
  • proof
  • ...and 41 more