Projective bundle

Template:Short description 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 Pⁿ-bundle if it is locally a projective n-space; i.e., $X{\times }_{S}U\simeq {\mathbb{P}}_U^n$ and transition automorphisms are linear. Over a regular scheme S such as a smooth variety, every projective bundle is of the form $\mathbb{P}(E)$ for some vector bundle (locally free sheaf) E.

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*). To see why, recall that a projective bundle comes equipped with transition functions on double intersections of a suitable open cover. On triple overlaps, any lift of these transition functions satisfies the cocycle condition up to an invertible function. The collection of these functions forms a 2-cocycle which vanishes in H²(X,O*) only if the projective bundle is the projectivization of a vector bundle. In particular, if X is a compact Riemann surface then H²(X,O*)=0, and so this obstruction vanishes.

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 Template:Nowrap 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):

$0\to {𝒪}_{𝐏(E)}(-1)\to p^{*}E\to Q\to 0$

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 Template:Nowrap is a global section of the sheaf hom Template:Nowrap. 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:

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

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

Examples

Many non-trivial examples of projective bundles can be found using fibrations over

${\mathbb{P}}^1$

such as Lefschetz fibrations. For example, an elliptic K3 surface

$X$

is a K3 surface with a fibration

$\pi :X\to {\mathbb{P}}^1$

such that the fibers

$E_p$

for

$p\in {\mathbb{P}}^1$

are generically elliptic curves. Because every elliptic curve is a genus 1 curve with a distinguished point, there exists a global section of the fibration. Because of this global section, there exists a model of

$X$

giving a morphism to the projective bundle

$X\to \mathbb{P}({𝒪}_{{\mathbb{P}}^1}(4)\oplus {𝒪}_{{\mathbb{P}}^1}(6)\oplus {𝒪}_{{\mathbb{P}}^1})$

defined by the Weierstrass equation

$y^{2}z+a_{1}xyz+a_{3}y{z}^2=x^3+a_2{x}^{2}z+a_{4}x{z}^2+a_6{z}^3$

where

$x,y,z$

represent the local coordinates of

${𝒪}_{{\mathbb{P}}^1}(4),{𝒪}_{{\mathbb{P}}^1}(6),{𝒪}_{{\mathbb{P}}^1}$

, respectively, and the coefficients

$a_i\in H^0({\mathbb{P}}^1,{𝒪}_{{\mathbb{P}}^1}(2i))$

are sections of sheaves on

${\mathbb{P}}^1$

. Note this equation is well-defined because each term in the Weierstrass equation has total degree

$12$

(meaning the degree of the coefficient plus the degree of the monomial. For example,

$\text{deg}(a_{1}xyz)=2+(4+6+0)=12$

).

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

$A_k(𝐏(E))=\underset{i=0}{\overset{r-1}{⨁}}{\zeta }^i{A}_{k-r+1+i}(X).$

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

References

• Template:Citation
• Template:Citation
• Template:Hartshorne AG