# 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

$$\mathcal{L}={\cup}_{x\in X}{\mathcal{F}}_{x}/{\U0001d52a}_{x}{\mathcal{F}}_{x},$$ |

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.

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 |