nLab complex analytic stack

Idea

A stack over the site of complex manifolds, hence a 1-truncated complex analytic ∞-stack.

Beware that standard terminology such as for algebraic stack might suggest that a complex analytic stack is a geometric stack over the complex analytic site, but usage as in (Behrend-Noohi 05) differs from that would-be convention.

Examples

For $G$ a complex Lie group then its delooping in complex analytic stack is the moduli stack $\mathbf{B}G$ for holomorphic $G$-principal bundles. For instance for holomorphic vector bundles if $G$ is the general linear group $GL(n,\mathbb{C})$.

Related concepts

• complex orbifold

• algebraic stack, smooth stack, topological stack,

• higher complex analytic geometry, complex analytic ∞-stack

• holomorphic line 2-bundle

References

• Kai Behrend, Behrang Noohi, section 3.2 of Uniformization of Deligne-Mumford curves (arXiv:0504309)

• Jacob Lurie, sections 11 and 12 of Closed Immersions