(see also Chern-Weil theory, parameterized homotopy theory)
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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).
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:
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
Discussion in relation to Beilinson regulators is in
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
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
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
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.