Arrangements, multiderivations, and adjoint quotient map of type ADE

TL;DR

This paper surveys the algebraic geometry of sheaves of logarithmic vector fields in hyperplane arrangements and establishes a natural isomorphism between the relative de Rham cohomology of ADE-type adjoint quotient maps and modules of multiderivations.

Contribution

It provides a new isomorphism linking de Rham cohomology to multiderivations for ADE-type adjoint quotient maps, combining geometric and algebraic insights.

Findings

Isomorphism between de Rham cohomology and multiderivations for ADE types

Gauss-Manin derivative of Kostant-Kirillov form used in proof

Survey of sheaves of logarithmic vector fields in hyperplane arrangements

Abstract

The first part of this paper is a survey on algebro-geometric aspects of sheaves of logarithmic vector fields of hyperplane arrangements. In the second part we prove that the relative de Rham cohomology (of degree two) of ADE-type adjoint quotient map is naturally isomorphic to the module of certain multiderivations. The isomorphism is obtained by the Gauss-Manin derivative of the Kostant-Kirillov form.