# nLab canonical bundle

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

cohomology

# Contents

## Idea

For $X$ a space with a notion of dimension $\mathrm{dim}X\in ℕ$ and a notion of (Kähler) differential forms on it, the canonical bundle over $X$ is the line bundle of $n$-forms on $X$, the $\mathrm{dim}\left(X\right)$-fold exterior product

${L}_{\mathrm{can}}≔{\Omega }_{X}^{n}$L_{can} \coloneqq \Omega^n_X

of the bundle ${\Omega }_{X}^{1}$ 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

Notice that if $X$ 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

## References

In the context of algebraic geometry:

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