Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors

TL;DR

This paper improves a theorem on computing hyperplane complement cohomology with local system coefficients, showing fewer conditions need checking, and links resonance conditions to Kac-Kazhdan reducibility criteria in Lie algebra representations.

Contribution

It demonstrates that Aomoto non-resonance conditions can be verified on a smaller set of edges and connects these conditions to Kac-Kazhdan criteria for certain local systems.

Findings

Resonance conditions coincide with Kac-Kazhdan reducibility conditions.

Fewer edges need to be checked for non-resonance conditions.

Applicable to local systems related to Knizhnik-Zamolodchikov equations.

Abstract

In this note we strenghten a theorem by Esnault-Schechtman-Viehweg which states that one can compute the cohomology of a complement of hyperplanes in a complex affine space with coefficients in a local system using only logarithmic global differential forms, provided certain "Aomoto non-resonance conditions" for monodromies are fulfilled at some "edges" (intersections of hyperplanes). We prove that it is enough to check these conditions on a smaller subset of edges. We show that for certain known one dimensional local systems over configuration spaces of points in a projective line defined by a root system and a finite set of affine weights (these local systems arise in the geometric study of Knizhnik-Zamolodchikov differential equations), the Aomoto resonance conditions at non-diagonal edges coincide with Kac-Kazhdan conditions of reducibility of Verma modules over affine Lie algebras.