#
nLab

canonical bundle

### Context

#### Differential geometry

#### Cohomology

**cohomology**

### Special and general types

### Special notions

### Variants

### Operations

### Theorems

# Contents

## Idea

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

$L_{can} \coloneqq \Omega^n_X$

of the bundle $\Omega^1_X$ of 1-forms.

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

Often this bundle is regarded via its sheaf of sections.

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

## Examples

### In complex analytic geometry

For $X$ complex manifold regarded over the complex numbers, then Kähler differential forms are *holomorphic* forms. Hence the canonical bundle for $dim_{\mathbb{C}}(X) = n$ is $\Omega^{n,0}$ (see also at *Dolbeault complex*), a complex line bundle.

For $X$ a Riemann surface of genus $g$, the degree of the canonical bundle is $2 g - 2$. This means it is divisible by 2 and hence there are “Theta characteristic” square roots.

In particular the first Chern class of the canonical bundle on the 2-sphere is *twice* that of the basic line bundle on the 2-sphere, the generator in $H^2(S^2, \mathbb{Z}) \simeq \mathbb{Z}$. See also at *geometric quantization of the 2-sphere*.

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

## 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 February 9, 2017 05:16:27
by

Urs Schreiber
(83.208.22.80)