Higher order operations in Deligne cohomology
type: Item
instance of: scholarly article
title: Higher order operations in Deligne cohomology
zbMATH Open document ID: 0847.55014
DOI: 10.1007/BF01241130
author: Christopher Deninger
published in: Inventiones Mathematicae
publication date: 1995-12-05T00:00:00Z
full work available at URL: https://eudml.org/doc/144277
review text: Massey products were originally introduced in algebraic topology to discover geometric phenomena which are not reflected by the ordinary cohomology ring of a topological space. The easiest of them are products \(\langle \alpha, \beta, \gamma \rangle\) of three cohomology classes; they are defined if \(\alpha \cup \beta = 0 = \beta \cup \gamma\) and are cosets of a certain subgroup of a cohomology group. In the present paper, the author introduces such Massey products in the Deligne cohomology of smooth spaces. They are defined using Čech resolutions; there are, however, also formulas in terms of representing differential forms. These formulas are used to calculate explicitly Massey products on an elliptic curve \(E\) over \(\mathbb{Q}\). Together with the Beilinson conjectures this gives a conjectural expression for \(L(E,3)\) in terms of certain nonclassical double Eisenstein-Kronecker series. In recent work, Goncharov has found such expressions for modular elliptic curves, using a completely different approach. This shows that the author's Massey products are not always zero. The two approaches together give another nice corroboration of Beilinson's conjectures.
Mathematics Subject Classification ID: 55S30
Mathematics Subject Classification ID: 14F99
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 784003
zbMATH Keywords: Massey products
zbMATH Keywords: Deligne cohomology
zbMATH Keywords: Čech resolutions
zbMATH Keywords: differential forms
zbMATH Keywords: elliptic curve
zbMATH Keywords: Beilinson conjectures
reviewed by: Wilhelm Singhof
MaRDI profile type: Publication
Wikidata QID: Q56813090
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description: scientific article; zbMATH DE number 784003