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holomorphic line 2-bundle

Context

Bundles

Complex geometry

Contents

Idea

Over a site of complex analytic spaces, where the multiplicative group 𝔾 m\mathbb{G}_m classifies non-vanishing holomorphic functions and B𝔾 m\mathbf{B}\mathbb{G}_m classifies holomorphic line bundles, then a holomorphic line 2-bundle is a 𝔾 m\mathbb{G}_m-principal 2-bundle, modulated by maps to B 2𝔾 m\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 XX is the Brauer stack Br(X)[X,B 2𝔾 m]\mathbf{Br}(X) \coloneqq [X,\mathbf{B}^2 \mathbb{G}_m] (the line 2-bundle itself is the associated ∞-bundle to the B𝔾 m\mathbf{B}\mathbb{G}_m-principal ∞-bundle which is the homotopy fiber of a given map XB 2𝔾 mX \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 XX (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:H 2(,𝔾 m)H 3(,). 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

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

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

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

Discussion connecting explicitly to the holomorphic Brauer group includes

See also

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,)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

Last revised on December 8, 2016 at 10:35:38. See the history of this page for a list of all contributions to it.