# nLab canonical bundle

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

bundles

cohomology

# Contents

## 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.

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)