nLab canonical bundle

Idea

For $X$ a space with a notion of dimension $n = dim(X) \in \mathbb{N}$ 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 $X$.

The inverse of the canonical line bundle (i.e. that with minus its first Chern class) is called the anticanonical line bundle.

Over an algebraic variety, the divisor corresponding to the canonical line bundle is called the canonical divisor.

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.

Related concepts

• density bundle

• volume form

• Kodaira vanishing theorem

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 and Hermitian manifold (e.g.Kähler manifold): 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