nLab
canonical bundle

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

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Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\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

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

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

L canΩ X n L_{can} \coloneqq \Omega^n_X

of the bundle Ω X 1\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 XX.

Often this bundle is regarded via its sheaf of sections.

A square root of the canonical class, hence another characteristic class Θ\Theta such that the cup product 2Θ=ΘΘ2 \Theta = \Theta \cup \Theta equals the canonical class is called a Theta characteristic (see also metalinear structure).

Examples

In complex analytic geometry

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

For XX a Riemann surface of genus gg, the degree of the canonical bundle is 2g22 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,)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

line bundlesquare rootchoice corresponds to
canonical bundleTheta characteristicover Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundlehalf-density bundle
canonical bundle of Lagrangian submanifoldmetalinear structuremetaplectic correction
determinant line bundlePfaffian line bundle
quadratic secondary intersection pairingpartition function of self-dual higher gauge theoryintegral 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

Revised on May 26, 2017 05:51:04 by Urs Schreiber (92.218.150.85)