holomorphic line n-bundle in nLab

Contents

Contents

Idea

The concept of line n-bundle in complex analytic geometry, classified by maps into the nn-fold delooping B n𝔾 m\mathbf{B}^n \mathbb{G}_m of the multiplicative group.

• for n=0n = 0 these are holomorphic functions classified by the group of units;

• for n=1n= 1 these are holomorphic line bundles, classified by Picard group, modulated by Picard stack;

• for n=2n =2 these are holomorphic line 2-bundles, classified by Brauer group, modulated by Brauer stack

Properties

(…)

• algebraic line n-bundle

• complex analytic ∞-stack

References

• Alexander Grothendieck, Le groupe de Brauer : II. Théories cohomologiques. Séminaire Bourbaki, 9 (1964-1966), Exp. No. 297, 21 p. (Numdam)