# Projective bundle

In mathematics, a **projective bundle** is a fiber bundle whose fibers are projective spaces.

By definition, a scheme *X* over a Noetherian scheme *S* is a **P**^{n}-bundle if it is locally a projective *n*-space; i.e., and transition automorphisms are linear. Over a regular scheme *S* such as a smooth variety, every projective bundle is of the form for some vector bundle (locally free sheaf) *E*.[1]

## The projective bundle of a vector bundle

Every vector bundle over a variety *X* gives a projective bundle by taking the projective spaces of the fibers, but not all projective bundles arise in this way: there is an obstruction in the cohomology group *H*^{2}(*X*,O*). In particular, if *X* is a compact Riemann surface, the obstruction vanishes i.e. *H*^{2}(*X*,O*)=0.

The projective bundle of a vector bundle *E* is the same thing as the Grassmann bundle of 1-planes in *E*.

The projective bundle **P**(*E*) of a vector bundle *E* is characterized by the universal property that says:[2]

- Given a morphism
*f*:*T*→*X*, to factorize*f*through the projection map*p*:**P**(*E*) →*X*is to specify a line subbundle of*f*^{*}*E*.

For example, taking *f* to be *p*, one gets the line subbundle *O*(-1) of *p*^{*}*E*, called the tautological line bundle on **P**(*E*). Moreover, this *O*(-1) is a universal bundle in the sense that when a line bundle *L* gives a factorization *f* = *p* ∘ *g*, *L* is the pullback of *O*(-1) along *g*. See also Cone#*O*(1) for a more explicit construction of *O*(-1).

On **P**(*E*), there is a natural exact sequence (called the tautological exact sequence):

where *Q* is called the tautological quotient-bundle.

Let *E* ⊂ *F* be vector bundles (locally free sheaves of finite rank) on *X* and *G* = *F*/*E*. Let *q*: **P**(*F*) → *X* be the projection. Then the natural map *O*(-1) → *q*^{*}*F* → *q*^{*}*G* is a global section of the sheaf hom Hom(*O*(-1), *q*^{*}G) = *q*^{*} *G* ⊗ *O*(1). Moreover, this natural map vanishes at a point exactly when the point is a line in *E*; in other words, the zero-locus of this section is **P**(*E*).

A particularly useful instance of this construction is when *F* is the direct sum *E* ⊕ 1 of *E* and the trivial line bundle (i.e., the structure sheaf). Then **P**(*E*) is a hyperplane in **P**(*E* ⊕ 1), called the hyperplane at infinity, and the complement of **P**(*E*) can be identified with *E*. In this way, **P**(*E* ⊕ 1) is referred to as the projective completion (or "compactification") of *E*.

The projective bundle **P**(*E*) is stable under twisting *E* by a line bundle; precisely, given a line bundle *L*, there is the natural isomorphism:

such that [3] (In fact, one gets *g* by the universal property applied to the line bundle on the right.)

## Cohomology ring and Chow group

Let *X* be a complex smooth projective variety and *E* a complex vector bundle of rank *r* on it. Let *p*: **P**(*E*) → *X* be the projective bundle of *E*. Then the cohomology ring H^{*}(**P**(*E*)) is an algebra over H^{*}(*X*) through the pullback *p*^{*}. Then the first Chern class ζ = *c*_{1}(*O*(1)) generates H^{*}(**P**(*E*)) with the relation

where *c*_{i}(*E*) is the *i*-th Chern class of *E*. One interesting feature of this description is that one can *define* Chern classes as the coefficients in the relation; this is the approach taken by Grothendieck.

Over fields other than the complex field, the same description remains true with Chow ring in place of cohomology ring (still assuming *X* is smooth). In particular, for Chow groups, there is the direct sum decomposition

As it turned out, this decomposition remains valid even if *X* is not smooth nor projective.[4] In contrast, *A*_{k}(*E*) = *A*_{k-r}(*X*), via the Gysin homomorphism, morally because that the fibers of *E*, the vector spaces, are contractible.

## See also

- Proj construction
- cone (algebraic geometry)
- ruled surface (an example of a projective bundle)
- Severi–Brauer variety

## References

- Hartshorne, Ch. II, Exercise 7.10. (c).
- Hartshorne, Ch. II, Proposition 7.12.
- Hartshorne, Ch. II, Lemma 7.9.
- Fulton, Theorem 3.3.

- Elencwajg, G.; Narasimhan, M. S. (1983), "Projective bundles on a complex torus",
*Journal für die reine und angewandte Mathematik*,**340**(340): 1–5, doi:10.1515/crll.1983.340.1, ISSN 0075-4102, MR 0691957 - William Fulton. (1998),
*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.,**2**(2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323 - Hartshorne, Robin (1977),
*Algebraic Geometry*, Graduate Texts in Mathematics,**52**, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157