Higher order operations in Deligne cohomology
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.
scientific article; zbMATH DE number 784003
scientific article