# nLab holomorphic line 2-bundle

Contents

### Context

#### Bundles

bundles

fiber bundles in physics

complex geometry

# Contents

## 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

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

## 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

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