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:^[1]

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).

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 a fiber of 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:

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

such that $g^{*}({\mathcal {O}}(-1))\simeq {\mathcal {O}}(-1)\otimes p^{*}L.$ ^[2] (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 = 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.^[3] 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.

References

↑ 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: 1–5, doi:10.1515/crll.1983.340.1, ISSN 0075-4102, MR 691957
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

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