Relative differential forms and complex polynomials
type: Item
instance of: scholarly article
title: Relative differential forms and complex polynomials
zbMATH Open document ID: 0980.32007
DOI: 10.1016/S0007-4497(00)01055-1
author: P. Bonnet
author: Alexandru Dimca
published in: Bulletin des Sciences Mathématiques
publication date: 2001-03-19T00:00:00Z
review text: The authors study the algebraic de Rham complex for polynomials \(f\colon {\mathbb C}^n \to {\mathbb C}\). For \(f\) with isolated singularities on \({\mathbb C}^n\) and with generic fibre cohomologically \((n-2)\)-connected their main result states that \(H^0(\Omega_f^*,d)={\mathbb C}[f]\) and \(H^k(\Omega_f^*,d)=0\) for \(0<k<n-1\). If moreover \(f\) is cohomologically tame or \(M\)-tame the \({\mathbb C}[t]\)-module \(H^{n-1}(\Omega_f^*,d)\) is free and of finite rank equal to the total Milnor number of \(f\). In case \(f\) is semi-weighted homogeneous an explicit basis is produced. The final two sections describe in detail the torsion in \(H^{n-1}(\Omega_f^*,d)\) in the case \(n=2\), when the polynomial \(f\) has non-isolated singularities or non-connected fibres.
Mathematics Subject Classification ID: 32S05
Mathematics Subject Classification ID: 32S55
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 32S40
zbMATH DE Number: 1547230
zbMATH Keywords: de Rham complex
zbMATH Keywords: tame polynomial
reviewed by: Jan Stevens
MaRDI profile type: Publication
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full work available at URL: s0007-4497(00)01055-1
OpenAlex ID: W1976452289
Wikidata QID: Q128118321
description: scientific article; zbMATH DE number 1547230