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In mathematics, the tautological line bundle is a particular natural line bundle on a projective space. It is a special case of the tautological bundle on a Grassmannians. The tautological line bundle is important in the study of characteristic classes.

The older term canonical line bundle is now strongly deprecated, as it conflicts with universally accepted terminology in the theory of complex manifolds, where the canonical line bundle of a complex m-manifold is defined to be the bundle of (m, 0)-forms (top forms). For this reason, the term tautological is now widely considered to be preferable to avoid confusion.

Definition

Form the cartesian product Pⁿ(R) × Rⁿ⁺¹, with the first factor denoting real projective n-space. We consider the subset

$E({\gamma }_n):=\{(x,v)\in {\mathbf{P}}^n(\mathbf{R})\times {\mathbf{R}}^{n+1}:v\in x\}.$

We have an obvious projection map π : E(γ_n) → Pⁿ(R), with (x, v) ↦ x. Each fibre of π is then the line x inside Euclidean (n+1)-space. Giving each fibre the induced vector space structure we obtain the bundle

${\gamma }_n:=(E({\gamma }_n)\to {\mathbf{P}}^n(\mathbf{R})),$

the tautological line bundle over Pⁿ(R).

Complex and quaternionic cases

The above definition continues to makes sense if we replace the field R by either the complex numbers C or the quaternions. Thus we obtain the complex line bundle

${\gamma }_{n,\mathbf{C}}:=(E({\gamma }_{n,\mathbf{C}})\to {\mathbf{P}}^n(\mathbf{C})),$

whose fibres are isomorphic to C ≅ R², and the quaternionic line bundle

${\gamma }_{n,\mathbf{H}}:=(E({\gamma }_{n,\mathbf{H}})\to {\mathbf{P}}^n(\mathbf{H})),$

whose fibres are isomorphic to H ≅ R⁴.

Tautological line bundle in algebraic geometry

In algebraic geometry, this notion exists over any commutative unital ring.

Over a field, its dual line bundle is the line bundle associated to the hyperplane divisor H, whose global sections are the linear forms. Its Chern class is −H. This is an example of an anti-ample line bundle. Over C, this is equivalent to saying that it is a negative line bundle, meaning that minus its Chern class is the de Rham class of a Kähler form.

Facts

• The tautological line bundle γ_n is locally trivial but not trivial, for n ≥ 1. This remains true over other fields.

In fact, it is straightforward to show that, for n = 1, the real tautological line bundle is none other than the well-known bundle whose total space is the Möbius strip. For a full proof of the above fact, see.

See also

• Stiefel-Whitney class.

References

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• [M+S] J. Milnor & J. Stasheff, Characteristic Classes, Princeton, 1974.
• Griffiths and Harris, Principles of Algebraic Geometry, Academic Press