Logarithmic cohomology of the complement of a plane curve
type: Item
instance of: scholarly article
title: Logarithmic cohomology of the complement of a plane curve
zbMATH Open document ID: 1010.32016
published in: Commentarii Mathematici Helvetici
publication date: 2002-06-17T00:00:00Z
review text: The main object of study in this paper is the de Rham complex of holomorphic differential forms with logarithmic poles along a divisor in a complex manifold. It had been showed earlier by the last three authors that if the divisor is a locally quasi-homogeneous free divisor then the above mentioned complex calculates the cohomology of the complement of the divisor (``the logarithmic comparison theorem holds''). In the present paper the authors study a partial converse to this statement, i.e. they study plane curve germs (which are always free divisors) and find that the mentioned complex calculates the cohomology of the complement only if the curve germ is quasi-homogeneous. The main argument of the proof shows that if the condition holds then there is a vector field germ \(\chi\) such that \(\chi\cdot h=h\), where \(h\) is the local equation of the curve. Then they can use Saito's result proving that \(h\) is quasi-homogeneous in certain coordinates. In higher dimensions Saito's result can not be applied, so the authors only conjecture the existence of the the vector field \(\chi\).
reviewed by: Richárd Rimányi
Mathematics Subject Classification ID: 32S20
Mathematics Subject Classification ID: 32S40
Mathematics Subject Classification ID: 14F40
zbMATH DE Number: 1756817
zbMATH Keywords: free divisor
zbMATH Keywords: logarithmic de Rham complex
zbMATH Keywords: plane curve
zbMATH Keywords: local quasi-homogeneity
author: Francisco Javier Calderón Moreno
author: David Mond
author: Luis Narváez-Macarro
author: Francisco J. Castro-Jiménez
MaRDI profile type: Publication
full work available at URL: s00014-002-8330-6
OpenAlex ID: W1982722340
DOI: 10.1007/S00014-002-8330-6
description: scientific article; zbMATH DE number 1756817