Projective bundle

Short description: Fiber bundle whose fibers are projective spaces

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., [math]\displaystyle{ X \times_S U \simeq \mathbb{P}^n_U }[/math] and transition automorphisms are linear. Over a regular scheme S such as a smooth variety, every projective bundle is of the form [math]\displaystyle{ \mathbb{P}(E) }[/math] 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 [math]\displaystyle{ G_1(E) }[/math] 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).

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

[math]\displaystyle{ 0 \to \mathcal{O}_{\mathbf{P}(E)}(-1) \to p^* E \to Q \to 0 }[/math]

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:

[math]\displaystyle{ g: \mathbf{P}(E) \overset{\sim}\to \mathbf{P}(E \otimes L) }[/math]

such that [math]\displaystyle{ g^*(\mathcal{O}(-1)) \simeq \mathcal{O}(-1) \otimes p^* L. }[/math] (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 [math]\displaystyle{ \mathbb{P}^1 }[/math] such as Lefschetz fibrations. For example, an elliptic K3 surface [math]\displaystyle{ X }[/math] is a K3 surface with a fibration

[math]\displaystyle{ \pi:X \to \mathbb{P}^1 }[/math]

such that the fibers [math]\displaystyle{ E_p }[/math] for [math]\displaystyle{ p \in \mathbb{P}^1 }[/math] 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 [math]\displaystyle{ X }[/math] giving a morphism to the projective bundle

[math]\displaystyle{ X \to \mathbb{P}(\mathcal{O}_{\mathbb{P}^1}(4)\oplus\mathcal{O}_{\mathbb{P}^1}(6)\oplus\mathcal{O}_{\mathbb{P}^1}) }[/math]

defined by the Weierstrass equation

[math]\displaystyle{ y^2z + a_1xyz + a_3yz^2 = x^3 + a_2x^2z + a_4xz^2 + a_6z^3 }[/math]

where [math]\displaystyle{ x,y,z }[/math] represent the local coordinates of [math]\displaystyle{ \mathcal{O}_{\mathbb{P}^1}(4), \mathcal{O}_{\mathbb{P}^1}(6), \mathcal{O}_{\mathbb{P}^1} }[/math], respectively, and the coefficients

[math]\displaystyle{ a_i \in H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(2i)) }[/math]

are sections of sheaves on [math]\displaystyle{ \mathbb{P}^1 }[/math]. Note this equation is well-defined because each term in the Weierstrass equation has total degree [math]\displaystyle{ 12 }[/math] (meaning the degree of the coefficient plus the degree of the monomial. For example, [math]\displaystyle{ \text{deg}(a_1xyz) = 2 + (4 + 6 + 0) = 12 }[/math]).

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

[math]\displaystyle{ \zeta^r + c_1(E) \zeta^{r-1} + \cdots + c_r(E) = 0 }[/math]

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

[math]\displaystyle{ A_k(\mathbf{P}(E)) = \bigoplus_{i=0}^{r-1} \zeta^i A_{k-r+1+i}(X). }[/math]

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

• Elencwajg, G.; Narasimhan, M. S. (1983), "Projective bundles on a complex torus", Journal für die reine und angewandte Mathematik 1983 (340): 1–5, doi:10.1515/crll.1983.340.1, ISSN 0075-4102 
• Fulton, William (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 
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9