Arrangements of hyperplanes I: Rational functions and Jeffrey-Kirwan residue

1999/03/30 by Michel Brion, Brion, Michel, Michèle Vergne +2 · 2 citations
Mathematics · #44A10 #52B30 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:44A10 #msc:52B30

paper · pdf · doi:10.48550/arxiv.math/9903178

33 pages, LaTEX2e, to appear in the Annales Scientifiques de l'Ecole Normale Superieure

arxiv created 1999/03/30 · openalex publication_date 1999/03/30 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the space R_Δ of rational functions of several variables with poles on a fixed arrangement Δ of hyperplanes. We obtain a decomposition of R_Δ as a module over the ring of differential operators with constant coefficients. We generalize to the space R_Δ the notions of principal part and of residue, and we describe its relations to Laplace transforms of locally polynomial functions. This explains algebraic aspects of work by L. Jeffreys and F. Kirwan about integrals of equivariant cohomology classes on Hamiltonian manifolds. As another application, we will construct multidimensional versions of Eisenstein series in a subsequent article, and we will obtain another proof of a residue formula of A. Szenes for Witten zeta functions.

Cited by

• A Vafa-Intriligator formula for semi-positive quotients of linear spaces
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Related

• The residue formula and the Tolman-Weitsman theorem
• Arrangements of hyperplanes II: Szenes formula and Eisenstein series
• Hyperplane arrangements and K-theory
• The Monodromy Conjecture for hyperplane arrangements
• Monodromy of hypergeometric functions arising from arrangements of hyperplanes