nLab
canonical bundle

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 over $X$ is the line bundle of $n$-forms on $X$, the $\dim (X)$-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.

Related concepts

• density bundle

• volume form

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

line bundle	square root	choice corresponds to
canonical bundle	Theta characteristic	over Riemann surface: spin structure
density bundle	half-density bundle
canonical bundle of Lagrangian submanifold	metalinear structure	metaplectic correction
determinant line bundle	Pfaffian line bundle
quadratic secondary intersection pairing	partition function of self-dual higher gauge theory	integral 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

• Wikipedia, Canonical bundle