On monodromy for a class of surfaces
type: Item
instance of: scholarly article
title: On monodromy for a class of surfaces
zbMATH Open document ID: 1140.14013
author: Ann Lemahieu
author: Willem Veys
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2008-01-03T00:00:00Z
review text: The authors study the eigenvalues of monodromy for a class of polynomials \(f\in \mathbb{C}[x,y,z]\) such that \(f\) admits an embedded resolution of singularities of a certain combinatorial type (blow-up of a toric constellation). The main result is the following: If \(E_j\) is an exceptional component with numerical data \((N_j,\nu_j)\) such that \(\chi_{top}(E_j^o)>0\), then \(\exp (-2\pi i \nu_j/N_j)\) is a monodromy eigenvalue of \(f\). The authors use this result to give a partial proof of the monodromy conjecture for the topological zeta function \(Z_{top,f}(s)\) associated to \(f\) at the origin [\textit{J. Denef} and \textit{F. Loeser}, J. Am. Math. Soc. 5, No. 4, 705--720 (1992; Zbl 0777.32017)]. They show that, if \(s\) is a pole of \(Z_{top,f}(s)\) of expected order one, then \(\exp (2\pi is)\) is a monodromy eigenvalue of \(f\).
reviewed by: Johannes Nicaise
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 58K10
zbMATH DE Number: 5223051
zbMATH Keywords: resolution of singularities
zbMATH Keywords: toric constellations
zbMATH Keywords: monodromy conjecture
MaRDI profile type: Publication
full work available at URL: j.crma.2007.10.031
OpenAlex ID: W1991987943
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DOI: 10.1016/J.CRMA.2007.10.031
description: scientific article