Twisted de Rham cohomology groups of logarithmic forms
type: Item
instance of: scholarly article
title: Twisted de Rham cohomology groups of logarithmic forms
zbMATH Open document ID: 0905.14010
author: Kazuhiko Aomoto
author: Michitake Kita
author: Peter Orlik
author: Hiroaki Terao
published in: Advances in Mathematics
publication date: 1999-01-24T00:00:00Z
review text: For an \(l\)-dimensional complex affine space \(V\) with coordinate ring \(S\), let \(\Omega^{p}[V]\) be the \(S\)-module of global regular \(p\)-forms on \(V\), \(\Omega^{p}(V) \) the space of global rational forms. The module \(\Omega^{p}(\mathcal G)\) of logarithmic \(p\)-forms with poles along \(\mathcal G\) is defined as \[ \Omega^{p}(\mathcal G)=\{\omega \in \Omega^{p}(V) \mid Q\omega \in \Omega^{p} [V],\;Q(d\omega) \in \Omega^{p+1}[V] \} \] where \(\mathcal G\) is an affine \(l\)-arrangement with \(n\) hyperplanes \(H_{i}\) in general position, \(Q=\prod{\alpha_{i}}\), \(\alpha_{i}\) is a linear polynomial and \(H_{i}= \ker(\alpha_{i})\). Taking a polynomial \(f\) of degree \( q+1>0 \), the operator \[ \nabla_{f}=d+ \sum{\lambda_{i} \frac{d \alpha_{j}}{\alpha_{j}} \wedge} + df \wedge \] gives rise to a cochain complex \((\Omega^{\bullet}(\mathcal{G}),\nabla_{f})\). The main result of this paper is the computation of the cohomology groups of the twisted de Rham complex of the operator \(\nabla_{f}\), including vanishing, dimension, basis and so on.
reviewed by: Na Jisheng
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 32C35
Mathematics Subject Classification ID: 32L10
zbMATH DE Number: 1044218
zbMATH Keywords: logarithmic \(p\)-form
zbMATH Keywords: twisted de Rham cohomology group
zbMATH Keywords: complex affine space
MaRDI profile type: Publication
full work available at URL: aima.1997.1631
OpenAlex ID: W2007188098
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DOI: 10.1006/AIMA.1997.1631
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description: scientific article; zbMATH DE number 1044218