nLab
canonical bundle

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

For XX a space with a notion of dimension dimXdim X \in \mathbb{N} and a notion of (Kähler) differential forms on it, the canonical bundle or canonical sheaf over XX is the line bundle (or its sheaf of sections) of nn-forms on XX, the dim(X)dim(X)-fold exterior product

L canΩ X n L_{can} \coloneqq \Omega^n_X

of the bundle Ω X 1\Omega^1_X of 1-forms.

The first Chern class of this bundle is also called the canonical characteristic class or just the canonical class of XX.

Often this bundle is regarded via its sheaf of sections.

A square root of the canonical class, hence another characteristic class Θ\Theta such that the cup product 2Θ=ΘΘ2 \Theta = \Theta \cup \Theta equals the canonical class is called a Theta characteristic (see also metalinear structure).

Examples

Notice that if XX is for instance a complex manifold regarded over the complex numbers, then Kähler differential forms are holomorphic forms.

The following table lists classes of examples of square roots of line bundles

line bundlesquare rootchoice corresponds to
canonical bundleTheta characteristicover Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundlehalf-density bundle
canonical bundle of Lagrangian submanifoldmetalinear structuremetaplectic correction
determinant line bundlePfaffian line bundle
quadratic secondary intersection pairingpartition function of self-dual higher gauge theoryintegral Wu structure

References

In the context of algebraic geometry:

  • Vladimir Lazić Lecture 7. Canonical bundle, I and II (2011) (pdf I, pdf II)

See also

Revised on January 7, 2014 10:23:35 by Urs Schreiber (89.204.135.107)