Projective bundle

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

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 H²(X,O*).

The projective bundle of a vector bundle E is the same thing as the Grassmann bundle $G_{1}(E)$ of 1-planes in E.

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

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).

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:

$g:\mathbf {P} (E){\overset {\sim }{\to }}\mathbf {P} (E\otimes L)$

such that $g^{*}({\mathcal {O}}(-1))\simeq {\mathcal {O}}(-1)\otimes p^{*}L.$ (In fact, one gets g by the universal property applied to the line bundle on the right.)

Cohomology ring

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 = c₁(O(1)) generates H^*(P(E)) with the relation

$\zeta ^{r}+c_{1}(E)\zeta ^{r-1}+\cdots +c_{r}(E)=0$

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 (stil assuming X is smooth). In particular, for Chow groups, there is the direct sum decomposition

$A_{k}(\mathbf {P} (E))=\bigoplus _{i=0}^{r-1}\zeta ^{i}A_{k-r+1+i}(X).$

As it turned out, this decomposition remains valid even if X is not smooth nor projective.

See also

• Proj construction
• cone (algebraic geometry)

References

• Elencwajg, G.; Narasimhan, M. S. (1983), "Projective bundles on a complex torus", Journal für die reine und angewandte Mathematik, 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., vol. 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, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157