The monodromy conjecture for hyperplane arrangements
type: Item
instance of: scholarly article
title: The monodromy conjecture for hyperplane arrangements
zbMATH Open document ID: 1227.32035
author: Nero Budur
author: Mircea Mustaţă
author: Zach Teitler
published in: Geometriae Dedicata
publication date: 2011-08-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/0906.1991
review text: Let \(f\in\mathbb{C}[x_1,\dots,x_n]\) be a non-constant polynomial with \(f(0)=0\). The topological monodromy conjecture stated in [\textit{J. Denef} and \textit{F. Loeser}, Astérisque No. 201--203, Exp. No. 741, 359--386 (1991; Zbl 0777.32017)] relates the poles of the local topological zeta-function of \(f\) to eigenvalues of the monodromy on the cohomology of the Milnor fibre and to roots of its Bernstein-Sato polynomial. If \(f\) defines a hyperplane arrangement in \(\mathbb{C}^n\), the authors prove the first relation and reduce the proof of the second one to a weaker conjecture. They also prove the weaker conjecture for some special cases of hyperplane arrangements, in particular for those which are of moderate type. Furthermore, they recall the \(p\)-adic version of the monodromy conjecture given in [\textit{J.-I. Igusa}, An introduction to the theory of local zeta functions. AMS/IP Studies in Advanced Mathematics. 14. Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0959.11047)] and prove a similar result in the case of \(\mathbb{Q}\)-hyperplane arrangements.
reviewed by: Michael Bogner
Mathematics Subject Classification ID: 32S40
Mathematics Subject Classification ID: 32S22
zbMATH DE Number: 5939476
zbMATH Keywords: monodromy conjecture
zbMATH Keywords: hyperplane arrangements
MaRDI profile type: Publication
OpenAlex ID: W2049221857
arXiv ID: 0906.1991
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DOI: 10.1007/S10711-010-9560-1
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description: scientific article