nLab Deligne line bundle

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

• Jean-Luc Brylinski, p. 15 of Gerbes on complex reductive Lie groups (arXiv:math/0002158)