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Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors
High Energy Physics - Theory · Submitted 11 Nov 1994
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that it is enough to check these conditions on a smaller subset of edges.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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). | background 0.94 | 0.08 | 0.13 | asserted 0.98 |
| s1 | We prove that it is enough to check these conditions on a smaller subset of edges. | result 0.84 | 0.95 | 0.49 | asserted 1.00 |
| s2 | 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. | result 0.92 | 0.92 | 0.34 | asserted 1.00 |
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The paper
Original 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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References
How the literature uses it
3 arXiv papers cite this in 5 sentences
Papers citing it · 3 sentences without a confident label
Cites it as background 2 papers
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Cohomology groups of this type are important in the study of combinatorial and algebraic properties of hyperplane arrangements and braid groups [ref] .
Aspects of Scattering Amplitudes and Moduli Space Localization · Scattering Equations · 0.91
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If the monodromy of $Φ$ is non-trivial, that is, if at least one of the numbers $m_1/κ, …, m_n/κ$ is not an integer, then [equation] see for example [ref] .
Twisted de Rham Complex on Line and Singular Vectors in $\hat{{\mathfrak{sl}_2}}$ Verma Modules · Twisted de Rham complex · 0.83
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