line bundle

In algebraic geometry, the term line bundle refers to a locally free coherent sheaf of rank 1, also called an invertible sheaf. In manifold theory, it refers to a real or complex one dimensional vector bundle. These notions are equivalent on a non-singular complex algebraic variety $X$ : given a one dimensional vector bundle, its sheaf of holomorphic sections is locally free and of rank 1. Similarly, given a locally free sheaf $\mathcal{F}$ of rank one, the space $$\mathcal{L}=\cup_{x\in X}\mathcal{F}_x/\mathfrak{m}_x\mathcal{F}_x,$$ given the coarsest topology for which sections of $\mathcal{F}$ define continuous functions in a vector bundle of complex dimension 1 over $X$ , with the obvious map taking the stalk over a point to that point.