On a twisted de Rham complex
type: Item
instance of: scholarly article
title: On a twisted de Rham complex
zbMATH Open document ID: 0947.14007
author: C. Sabbah
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2000-11-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9805087
review text: Let \(\widetilde f:\widetilde X\to\mathbb{P}^1\) be a morphism from a smooth projective variety \(\widetilde X\) over \(\mathbb{C}\) to the projective line and let \(f:X\to\mathbb{A}^1\) be its restriction over the affine line \(\mathbb{A}^1\). A result of \textit{M. Kontsevitch} and \textit{S. Barannikov} says that the hypercohomology spaces on \(X\) of the complexes \((\Omega_X^\bullet, d-df\wedge)\) and \((\Omega_X^\bullet, df\wedge)\) have the same (finite) dimension. Motivated by this result, the author of the present paper proves the following generalization: Given a mixed Hodge module \((M,F)\) on \(X\), the hypercohomology spaces on \(X\) of the complexes \((\Omega_X^\bullet \otimes_{{\mathcal O}_X}M,\;\nabla-df\wedge)\) and \((\Omega^\bullet_X\otimes_{{\mathcal O}_X} \text{gr}^FM,\;\text{gr}^F\nabla-df \wedge)\) have the same (finite) dimension. An analytic version of this result concerning the vanishing cycle functor is also established.
reviewed by: Fumio Hazama
Mathematics Subject Classification ID: 14D07
Mathematics Subject Classification ID: 32S40
Mathematics Subject Classification ID: 14D05
Mathematics Subject Classification ID: 32G20
Mathematics Subject Classification ID: 14F10
zbMATH DE Number: 1321877
zbMATH Keywords: de Rham complex
zbMATH Keywords: differential complex
zbMATH Keywords: hypercohomology spaces
zbMATH Keywords: mixed Hodge module
zbMATH Keywords: vanishing cycle
MaRDI profile type: Publication
OpenAlex ID: W2963057136
arXiv ID: math/9805087
cites work: Q3731741
cites work: Q3762606
cites work: Equations différentielles à points singuliers réguliers
cites work: Relèvements modulo \(p^ 2\) et décomposition du complexe de de Rham. (Lifting modulo \(p^ 2\) and decomposition of the de Rham complex)
cites work: Algebraic study of systems of partial differential equations
cites work: Q3999005
cites work: Q3344309
cites work: Sur les images directes de \({\mathcal D}\)-modules
cites work: Differential equations with polynomial coefficients
cites work: Hypergeometric periods for a tame polynomial
cites work: On a generalization of de Rham lemma
cites work: Modules de Hodge polarisables. (Polarisable Hodge modules)
cites work: Mixed Hodge modules
cites work: Decomposition theorem for proper Kähler morphisms
cites work: Q3216881
DOI: 10.2748/TMJ/1178224856
description: scientific article