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 $ℱ$ of rank one, the space

$$ℒ={\cup }_{x\in X}{ℱ}_x/{𝔪}_x{ℱ}_x,$$

given the coarsest topology for which sections of $ℱ$ 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.