Projective bundle

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 P"n"-bundle if it is locally a projective "n"-space; i.e., formula_1 and transition automorphisms are linear. Over a regular scheme "S" such as a smooth variety, every projective bundle is of the form formula_2 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 "H"2("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 "H"2("X",O*) only if the projective bundle is the projectivization of a vector bundle. In particular, if "X" is a compact Riemann surface then "H"2("X",O*)=0, and so this obstruction vanishes.
The projective bundle of a vector bundle "E" is the same thing as the Grassmann bundle formula_3 of 1-planes in "E".
The projective bundle P("E") of a vector bundle "E" is characterized by the universal property that says:
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". Let "q": P("F") → "X" be the projection. Then the natural map is a global section of the sheaf hom . 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 formula_6 (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 formula_7 such as Lefschetz fibrations. For example, an elliptic K3 surface formula_8 is a K3 surface with a fibrationformula_9such that the fibers formula_10 for formula_11 are generically elliptic curves. Because every elliptic curves 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 formula_8 giving a morphism to the projective bundleformula_13defined by the Weierstrass equationformula_14where formula_15 represent the local coordinates of formula_16, respectively, and the coefficientsformula_17are sections of sheaves on formula_7. Note this equation is well-defined because each term in the Weierstrauss equation has total degree formula_19 (meaning the degree of the coefficient plus the degree of the monomial. For example, formula_20).
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 = "c"1("O"(1)) generates H*(P("E")) with the relation
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
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.