nLab
Deligne line bundle
Contents
Context
Differential cohomology
differential cohomology

Ingredients
Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Fiber integration
Application to gauge theory
Complex geometry
Contents
Idea
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

$f \cup g \in H^2(X, \mathbb{Z} \to \Omega^0 \to \Omega^1)
\,.$

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 regulator $c_{2,2}$ , 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 ).

References
Pierre Deligne , Le symbole modéré . Publ. Math. IHES 73, (1991), 147-181.
Review in the general context of Deligne cohomology is in

Hélène Esnault , Eckart Viehweg , p.5 of Deligne-Beilinson cohomology in Rapoport, Schappacher, Schneider (eds.) Beilinson’s Conjectures on Special Values of L-Functions . Perspectives in Math. 4, Academic Press (1988) 43 - 91 (pdf )
Review in the context of holomorphic line 2-bundles and Chern-Simons line 3-bundles is in

Last revised on April 10, 2020 at 20:43:15.
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