geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
The Beilinson-Deligne cup product of two holomorphic functions $f,g \in H^0(X,\mathbb{G}_m)$ is a holomorphic line bundle with connection, hence a degree-2 cocycle in Deligne cohomology
This kind of line bundle is often referred to as the Deligne line bundle, following (Deligne 91).
They were originally motivated and used in a geometric construction of the Beilinson regulartor? $c_{2,2}$, see at Beilinson regulator – Geometic construction.
The universal version of this is the Deligne bundle on $\mathbb{G}_m \times \mathbb{G}_m$ with $f$ and $g$ the projection onto the first and onto the second factor, respectively (e.g. Brylinski 00, p. 15).
Review in the general context of Deligne cohomology is in
Review in the context of holomorphic line 2-bundles and Chern-Simons line 3-bundles is in
Last revised on September 4, 2014 at 06:50:47. See the history of this page for a list of all contributions to it.