nLab holomorphic line 2-bundle

Idea

Over a site of complex analytic spaces, where the multiplicative group $\mathbb{G}_m$ classifies non-vanishing holomorphic functions and $\mathbf{B}\mathbb{G}_m$ classifies holomorphic line bundles, then a holomorphic line 2-bundle is a $\mathbb{G}_m$-principal 2-bundle, modulated by maps to $\mathbf{B}^2 \mathbb{G}_m$.

This means that the moduli stack of holomorphic line 2-bundles on a complex analytic space or more generally on a complex analytic ∞-groupoid $X$ is the Brauer stack $\mathbf{Br}(X) \coloneqq [X,\mathbf{B}^2 \mathbb{G}_m]$ (the line 2-bundle itself is the associated ∞-bundle to the $\mathbf{B}\mathbb{G}_m$-principal ∞-bundle which is the homotopy fiber of a given map $X \to \mathbf{B}^2 \mathbb{G}_m$). In particular equivalence classes of holomorphic line 2-bundles form the elements of the bigger Brauer group of $X$ (the Brauer group proper if they are torsion).

Discussion in terms of bundle gerbes includes (Chatterjee 98,Brylinski 00 Mathai-Stevenson 02, section 7).

Properties

Dixmier-Douady class

The Dixmier-Douady class of holomorphic line 2-bundles, hence the higher analog of the first Chern class, is given by the connecting homomorphism on degee 2 of the long exact sequence in cohomology which is induced by the exponential exact sequence in complex analytic geometry:

$$DD\;\colon\; H^2(-,\mathbb{G}_m) \longrightarrow H^3(-,\mathbb{Z}) \,.$$

Relation to higher twistor transforms

Holomorphic line 2-bundles appear in the higher degree analogs of twistor transforms. See (Chatterjee 98) and see twistor – References – Application to self-dual 2-forms

Related concepts

• higher complex analytic geometry

• holomorphic line n-bundle

• algebraic line 2-bundle

References

General

Discussion in relation to Beilinson regulators is in

• Jean-Luc Brylinski, Holomorphic gerbes and the Beilinson regulator, Astérisque 226 (1994): 145-174 (pdf)

Early discussion in terms of bundle gerbes includes

• David Chatterjee, section 5 of On Gerbs, 1998 (pdf)

Discussion with an eye towards of holomorphic twisted K-theory is in

• Varghese Mathai, Danny Stevenson, Chern character in twisted K-theory: equivariant and holomorphic cases, Commun.Math.Phys. 236 (2003) 161-186 (arXiv:hep-th/0201010v5)

An equivariant example arising from more algebro-geometric origin is in

• Giovanni Felder, Andre Henriques, Carlo A. Rossi, Chenchang Zhu, A gerbe for the elliptic gamma function, Duke Math. J. 141(2008), 1-74 arXiv:math/0601337 (doi)

Discussion connecting explicitly to the holomorphic Brauer group includes

• Oren Ben-Bassat, Gerbes and the Holomorphic Brauer Group of Complex Tori, Journal of Noncommutative Geometry, Volume 6, Issue 3 (2012) 407-455 (arXiv:0811.2746)

• Edoardo Ballico, Oren Ben-Bassat, Meromorphic Line Bundles and Holomorphic Gerbes, Math. Res. Lett. 18 (2011), 6, 1-14 (arXiv:1101.2216)

See also

• Oren Ben-Bassat, Equivariant Gerbes on Complex Tori (arXiv:1102.2312)

Basic line 2-bundles on reductive groups

The existence of the “basic” 2-line bundle (see at Chern-Simons line 3-bundle) on a complex reductive group (such as $SL(n,\mathbb{C})$) is mentioned in

• Jean-Luc Brylinski, around theorem 5.4.10 (p. 226-227) of Loop spaces and characteristic classes, Birkhäuser

The actual construction appears in

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