Coniveau of classes of flat bundles trivialized on a finite smooth covering of a complex manifold
type: Item
instance of: scholarly article
title: Coniveau of classes of flat bundles trivialized on a finite smooth covering of a complex manifold
zbMATH Open document ID: 0841.57036
DOI: 10.1007/BF00961402
author: Hélène Esnault
published in: \(K\)-Theory
publication date: 1995-06-06T00:00:00Z
review text: Let \((E,\bigtriangledown)\) be an algebraic vector bundle with a flat connection on a smooth complex variety, \(X\). There are (analytic) Chern classes \(c_i (E,\bigtriangledown) \in H^{2i - 1} (X_{an}, \mathbb{C}/ \mathbb{Z} (i))\), mapping to the Deligne-Beilinson Chern class, \(c_i^{\mathfrak D} (E) \in H^{2i - 1} \bigl( X_{an}; \mathbb{C}/ \mathbb{Z} (i) \bigr)/F^i \subseteq H^{2i}_{\mathfrak D} \bigl( X; \mathbb{Z} (i) \bigr)\). The author shows that \(c_i (E,\bigtriangledown)\) lies in the bottom stage of the coniveau filtration, \(L^{i - 1} H^{2i - 1} (X_{an}; \mathbb{Q} (i)/ \mathbb{Z} (i))\). From this she deduces that the corresponding classes in étale cohomology also lie in \[ L^i H^{2i - 1}_{\text{ét}} (X; \mathbb{Q}/ \mathbb{Z}), \] which affirmatively answers a question of [\textit{B. Kahn}, \(K\)-theory 5, No. 6, 555-566 (1992; Zbl 0770.12006)] in the smooth case.
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 14C35
zbMATH DE Number: 720849
zbMATH Keywords: algebraic vector bundle
zbMATH Keywords: flat connection
zbMATH Keywords: smooth complex variety
zbMATH Keywords: Chern classes
zbMATH Keywords: Deligne-Beilinson Chern class
reviewed by: Victor P. Snaith
MaRDI profile type: Publication
cites work: Gersten's conjecture and the homology of schemes
cites work: Théorie de Hodge. III
cites work: Characteristic classes of flat bundles
cites work: Characteristic classes of flat bundles. II
cites work: Chern classes of vector bundles with holomorphic connections on a complete smooth complex variety
cites work: Chern classes of complex Galois representations
cites work: Q3924325
cites work: Q3871006
cites work: Rationality of secondary classes
cites work: Q3495455
full work available at URL: https://doi.org/10.1007/bf00961402
OpenAlex ID: W2062200645
Recommended article: On somes characteristic classes of flat bundles in complex geometry
Recommended article: Flat bundles over some compact complex manifolds
Recommended article: Étale triviality of finite vector bundles over compact complex manifolds
Recommended article: Q4416599
Recommended article: Q4549300
Recommended article: Flat bundles on affine manifolds
Recommended article: Bounded characteristic classes and flat bundles
Recommended article: Q5385332
Recommended article: Q4463315
Recommended article: Characteristic classes for flat bundles and their formulas
description: scientific article; zbMATH DE number 720849