Tautological line bundle

The canonical or tautological line bundle on a projective space appears frequently in mathematics, often in the study of characteristic classes. Note that there is possible confusion with the theory of the canonical class in algebraic geometry; for which reason the name tautological is preferred in some contexts.

For generalizations to Grassmannians, see also tautological bundle.

1 Definition
- 1.1 Complex and quaternionic cases
2 Tautological line bundle in algebraic geometry
3 Facts
4 See also
5 References

Definition

Form the cartesian product $\mathbf{R} P^n\times\mathbf{R}^{n%2B1}$ , with the first factor denoting real projective n-space. We consider the subset

$E(\gamma^n):=\big\{(x,v)\in\mathbf{R}P^n\times\mathbf{R}^{n%2B1}:v\in x\big\}.$

We have an obvious projection map $\pi:E(\gamma^n)\to\mathbf{R}P^n$ , with $(x,v)\mapsto x$ . Each fibre of $\pi$ is then the line $x$ inside Euclidean (n+1)-space. Giving each fibre the induced vector space structure we obtain the bundle

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

the canonical line bundle over $\mathbf{R}P^n$ .

Complex and quaternionic cases

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

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

whose fibres are isomorphic to $\mathbf{C} \cong \mathbf{R}^2$ , and the quaternionic line bundle

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

whose fibres are isomorphic to $\mathbf{H} \cong \mathbf{R}^4$ .