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.,
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] 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.
of 1-planes in E. The projective bundle P(E) of a vector bundle E is characterized by the universal property that says:[2] 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.
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.)
Many non-trivial examples of projective bundles can be found using fibrations over
For example, an elliptic K3 surface
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
giving a morphism to the projective bundle[4]
defined by the Weierstrass equation
Note this equation is well-defined because each term in the Weierstrass equation has total degree
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 where ci(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.
[5] In contrast, Ak(E) = Ak-r(X), via the Gysin homomorphism, morally because that the fibers of E, the vector spaces, are contractible.