line bundle

In algebraic geometryMathworldPlanetmathPlanetmath, the term line bundleMathworldPlanetmath 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 bundleMathworldPlanetmath. These notions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on a non-singularPlanetmathPlanetmath complex algebraic variety X: given a one dimensional vector bundle, its sheaf of holomorphic sectionsPlanetmathPlanetmathPlanetmath is locally free and of rank 1. Similarly, given a locally free sheaf of rank one, the space


given the coarsest topologyMathworldPlanetmath for which sections of define continuous functionsMathworldPlanetmathPlanetmath in a vector bundle of complex dimension 1 over X, with the obvious map taking the stalk over a point to that point.

Title line bundle
Canonical name LineBundle
Date of creation 2013-03-22 13:31:04
Last modified on 2013-03-22 13:31:04
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 5
Author bwebste (988)
Entry type Definition
Classification msc 14-00