- Source: 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
×
S
U
≃
P
U
n
{\displaystyle 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
P
(
E
)
{\displaystyle \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
)
{\displaystyle 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 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
→
O
P
(
E
)
(
−
1
)
→
p
∗
E
→
Q
→
0
{\displaystyle 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
:
P
(
E
)
→
∼
P
(
E
⊗
L
)
{\displaystyle g:\mathbf {P} (E){\overset {\sim }{\to }}\mathbf {P} (E\otimes L)}
such that
g
∗
(
O
(
−
1
)
)
≃
O
(
−
1
)
⊗
p
∗
L
.
{\displaystyle g^{*}({\mathcal {O}}(-1))\simeq {\mathcal {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
P
1
{\displaystyle \mathbb {P} ^{1}}
such as Lefschetz fibrations. For example, an elliptic K3 surface
X
{\displaystyle X}
is a K3 surface with a fibration
π
:
X
→
P
1
{\displaystyle \pi :X\to \mathbb {P} ^{1}}
such that the fibers
E
p
{\displaystyle E_{p}}
for
p
∈
P
1
{\displaystyle 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
{\displaystyle X}
giving a morphism to the projective bundle
X
→
P
(
O
P
1
(
4
)
⊕
O
P
1
(
6
)
⊕
O
P
1
)
{\displaystyle 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
x
y
z
+
a
3
y
z
2
=
x
3
+
a
2
x
2
z
+
a
4
x
z
2
+
a
6
z
3
{\displaystyle 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
{\displaystyle x,y,z}
represent the local coordinates of
O
P
1
(
4
)
,
O
P
1
(
6
)
,
O
P
1
{\displaystyle {\mathcal {O}}_{\mathbb {P} ^{1}}(4),{\mathcal {O}}_{\mathbb {P} ^{1}}(6),{\mathcal {O}}_{\mathbb {P} ^{1}}}
, respectively, and the coefficients
a
i
∈
H
0
(
P
1
,
O
P
1
(
2
i
)
)
{\displaystyle a_{i}\in H^{0}(\mathbb {P} ^{1},{\mathcal {O}}_{\mathbb {P} ^{1}}(2i))}
are sections of sheaves on
P
1
{\displaystyle \mathbb {P} ^{1}}
. Note this equation is well-defined because each term in the Weierstrass equation has total degree
12
{\displaystyle 12}
(meaning the degree of the coefficient plus the degree of the monomial. For example,
deg
(
a
1
x
y
z
)
=
2
+
(
4
+
6
+
0
)
=
12
{\displaystyle {\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
ζ
r
+
c
1
(
E
)
ζ
r
−
1
+
⋯
+
c
r
(
E
)
=
0
{\displaystyle \zeta ^{r}+c_{1}(E)\zeta ^{r-1}+\cdots +c_{r}(E)=0}
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
A
k
(
P
(
E
)
)
=
⨁
i
=
0
r
−
1
ζ
i
A
k
−
r
+
1
+
i
(
X
)
.
{\displaystyle 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. In contrast, Ak(E) = Ak-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, 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