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 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.^[1]

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:^[2]

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):

0\to {\mathcal {O}}_{\mathbf {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 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:

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

such that $g^{*}({\mathcal {O}}(-1))\simeq {\mathcal {O}}(-1)\otimes p^{*}L.$ ^[3] (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^[4]

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

defined by the Weierstrass equation

$y^{2}z+a_{1}xyz+a_{3}yz^{2}=x^{3}+a_{2}x^{2}z+a_{4}xz^{2}+a_{6}z^{3}$

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

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

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}(\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.^[5] 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 1977, Ch. II, Exercise 7.10. (c).
^ Hartshorne 1977, Ch. II, Proposition 7.12.
^ Hartshorne 1977, Ch. II, Lemma 7.9.
^ Propp, Oron Y. (2019-05-22). "Constructing explicit K3 spectra". arXiv:1810.08953 [math.AT].
^ Fulton 1998, Theorem 3.3.

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, MR 0691957, S2CID 122557310
Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 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, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

[1] Hartshorne 1977, Ch. II, Exercise 7.10. (c).

[2] Hartshorne 1977, Ch. II, Proposition 7.12.

[3] Hartshorne 1977, Ch. II, Lemma 7.9.

[4] Propp, Oron Y. (2019-05-22). "Constructing explicit K3 spectra". arXiv:1810.08953 [math.AT].

[5] Fulton 1998, Theorem 3.3.

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