nLab canonical bundle



Differential geometry

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infinitesimal cohesion

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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& \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 }


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Special notions


Extra structure





For XX a space with a notion of dimension n=dim(X)n = dim(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.

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 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).


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


In the context of algebraic geometry:

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

See also

Last revised on August 30, 2023 at 16:15:56. See the history of this page for a list of all contributions to it.