Projective bundle

The projective bundle of a real or complex vector bundle $E\to X$ has fiber the space of one-dimensional subspaces of $E_x$. It carries a tautological bundle $\lambda \subset p^{\ast }E$; in the real case this is the projectivization of a real vector bundle.

The same construction applies to an algebraic vector bundle over an algebraic variety: local trivializations glue products with projective space. In the lines convention its universal tautological bundle is a line bundle; the alternative projectivization by quotients convention parametrizes one-dimensional quotient spaces. Both conventions give dimension $\dim X+\mathrm{rank}(E)-1$ and a proper morphism to the base.

This convention for a projective bundle parametrizes one-dimensional quotients of the fibres of a vector bundle $E$. Its universal quotient is ${\pi }^{\ast }E\twoheadrightarrow {\mathcal{O}}_{\mathbb{P}(E)}(1)$, and ${\pi }_{\ast }\mathcal{O}(1)=E$. Tensoring $E$ by a line bundle on the base leaves the projective bundle unchanged but tensors its universal quotient by the pullback of that line bundle. The alternative convention of holomorphic projectivization by lines replaces $E$ by its dual and reverses the tautological sign.

On the projectivization by quotients, the fibre over a one-dimensional quotient of $E_x$ is that quotient line itself. These lines glue to the universal quotient. Its restriction to a projective fibre has degree one. It is the dual of the relative tautological line after identifying quotient projectivization of $E$ with line projectivization of $E^{\ast }$.

For this smooth projective bundle, put $L={\pi }^{\ast }{\mathcal{O}}_{{\mathbb{P}}^1}(1)$ and $M={\mathcal{O}}_{\mathbb{F}}(1)$, using projectivization by quotients. If every $a_i>0$, the complete system $|M|$ embeds it as a rational normal scroll. If every $a_i\ge 0$, the same system defines a scroll image which may be singular. Allowing arbitrary integers gives an abstract scroll; a common twist makes an embedding available. Its intersection numbers are $L^2=0$, $L{M}^{n-1}=1$, $M^n={\sum }_i{a}_i$, and its canonical divisor is $-nM+({\sum }_i{a}_i-2)L$.

The divisor class of a fibre of the ruling of a rational normal scroll has square zero. Its complete pencil recovers the projection to ${\mathbb{P}}^1$, and its intersection with $M^{n-1}$ is one.

Two rank-$n$ abstract rational normal scrolls are isomorphic exactly when their splitting degrees agree as multisets after a common integer shift. For $n\ge 3$, the primitive nef class $L$ of the ruling is intrinsically identified by $L^2=0$: restriction to a projective fibre forces every other divisor class with square zero to have zero coefficient of $M$. An isomorphism therefore preserves the ruling, and its effect on $M$ is a twist by $cL$. Pushforward gives an isomorphism of the defining bundles up to this twist; uniqueness in the Birkhoff–Grothendieck theorem recovers their splitting degrees. For surfaces the unique nef square-zero ray identifies the ruling except on ${\mathbb{P}}^1\times {\mathbb{P}}^1$, where exchanging the two rulings changes no normalized splitting type.

For the line-projectivization of a rank-$n$ complex bundle, $p^{\ast }E=S\oplus S^{\perp }$. The Whitney sum formula for Chern classes gives $c_t(S^{\perp })=p^{\ast }{c}_t(E)/(1+xt)$ as a formal power series. Comparing coefficients proves the formula. Since $S^{\perp }$ has rank $n-1$, its degree-$n$ Chern class vanishes and yields the monic projective-bundle relation.

// Target: geometry-and-topology.bigb

For a holomorphic vector bundle, projectivize each fibre using lines. The transition $g_{ij}(x)$ acts holomorphically by $[v]\mapsto [g_{ij}(x)v]$. Thus local products $U_i\times {\mathbb{P}}^{r-1}$ define a complex manifold with holomorphic projection to the base. The lines convention gives the relative tautological line bundle of fibre degree $-1$, and its dual has degree $+1$.

On the holomorphic projectivization by lines, take the line itself as the fibre of $S$. The evaluation inclusion makes it a holomorphic line subbundle of $p^{\ast }E$. Its fibre restriction is $\mathcal{O}(-1)$; the dual is the relative hyperplane line bundle.

Dualize the relative tautological line bundle to obtain the positive relative hyperplane line bundle. It restricts to $\mathcal{O}(1)$ on each projective fibre. This sign depends on the convention that projectivization parametrizes lines; fixing the convention avoids confusing it with the tautological line itself.

A complex line subbundle $L\subset E$ defines a section of a fiber bundle $s:X\to \mathbb{P}(E)$ selecting $L_x$ at each point. Pullback of the tautological bundle along $s$ is $L$, so if $t$ is its Euler class, then $s^{\ast }t=e(L)$. For a decomposition $E={⨁}_i{L}_i$, the resulting sections and the open sets where coordinate projection is nonzero identify the tautological line with ${\pi }^{\ast }{L}_i$. These identifications support the projective-bundle factorization via a vanishing cup product from an open cover.

For $n\ge 2$, ordered orthogonal complex lines in ${\mathbb{C}}^n$ form a projective bundle over ${\mathbb{CP}}^{n-1}$, with fiber ${\mathbb{CP}}^{n-2}$. Here $L$ is the first tautological line and the complement is formed using the standard Hermitian inner product. Its real dimension is $4n-6$. It is also the partial complex flag space of a line contained in a two-plane: send an orthogonal pair to $({\ell }_1,{\ell }_1\oplus {\ell }_2)$, and recover the second line by orthogonal complement inside the two-plane.

For the orthogonal complex line flag manifold, take $x,y$ to be the first Chern classes of the duals of its two tautological lines, both in degree two. The Whitney sum formula for Chern classes gives $c_i(L^{\perp })=x^i$. The projective bundle definition of Chern classes gives the second relation, and Leray-Hirsch theorem gives the integral basis $x^i{y}^j$ with $0\le i<n$, $0\le j<n-1$. Polynomial division by the monic relation in $y$ proves that there are no further relations. Multiplying that relation by $y-x$ also gives $y^n=0$.

For a rank-$r$ complex vector bundle and $u=c_1({\lambda }^{\ast })$, the cohomology of its projective bundle is free over the cohomology of the base on $1,u,\dots ,u^{r-1}$. Consequently pullback is injective, which yields the splitting principle for complex vector bundles by iteration.