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arXiv · hep-th/9411083

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

V. Schechtman·H. Terao·A. Varchenko
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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.

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V. Schechtman, H. Terao, A. Varchenko. 1994-12-22. Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors. https://arxiv.org/abs/hep-th/9411083

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