Canonical 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.

Contents

Definition

Form the cartesian product $\mathbb RP^n\times\mathbb R^{n+1}$ , with the first factor denoting real projective NaodW29-math30784404864acfc00000002-space. We consider the subset

$E(\gamma^n):=\big\{(\{\pm\;x\},v)\in\mathbb RP^n\times\mathbb R^{n+1}:v=\lambda x,\;\lambda\in\mathbb R\big\}.$

We have an obvious projection map $\pi:E(\gamma^n)\to\mathbb RP^n$ , with $(\{\pm\;x\},v)\mapsto\{\pm\;x\}$ . Each fibre of $π$ is then the line through $x$ and $- x$ inside Euclidean NaodW29-math30784404864acfc00000009-space. Giving each fibre the induced vector space structure we obtain the bundle

$\gamma^n:=(E(\gamma^n)\to\mathbb RP^n),$

the canonical line bundle over $\mathbb RP^n$ .

Facts

$γ n$ is locally trivial but not trivial, for $n\geq 1$ .

In fact, it is straightforward to show that, for $n = 1$ , the canonical line bundle is none other than the well-known bundle whose total space is the Möbius band. For a full proof of the above fact, see [M+S].

References

[M+S] J. Milnor & J. Stasheff, Characteristic Classes, Princeton, 1974.

Categories: Algebraic topology