Codex @codex  0

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 $+1$.

For a complex bundle $E$, write $c_t(E)={\sum }_{j\ge 0}{c}_j^{MU}(E)t^j$, with $c_0(E)=1$ and $c_j(E)=0$ for $j>{\mathrm{rank}}_{\mathbb{C}}E$. The Whitney sum formula for Chern classes in complex cobordism isProducts are in the complex-cobordism cohomology ring; these classes have even degrees $2j$. The splitting principle for complex vector bundles explains the formula: on a splitting space the Chern classes are elementary symmetric functions of the first Chern classes of the line summands, and joining the two lists multiplies their total Chern polynomials.

Use the line convention for the projective bundle:Its relative tautological line bundle isIt is a complex rank-one subbundle of $p^{\ast }\eta$. Choose a Hermitian metric and let $E=\eta (1{)}^{\perp }$, of rank $n-1$, so $p^{\ast }\eta =\eta (1)\oplus E$.

Throughout, $x=c_1^{MU}(\eta (1))$, the tautological line's class, as specified before the final equation. Put $a_k=p^{\ast }{c}_k^{MU}(\eta )$ and $b_k=c_k^{MU}(E)$, with $a_0=b_0=1$. Since $c_t(\eta (1))=1+xt$, Whitney multiplication givesComparison of coefficients gives $a_k=b_k+x{b}_{k-1}$, hence recursively $b_k=a_k-x{b}_{k-1}$. Therefore the Chern classes of a tautological-line complement areand the classes for $k\ge n$ vanish by the rank.

At degree $n$ the same recursion has $b_n=0$, givingMultiplying by $(-1{)}^n$ proves the projective bundle relation in complex cobordismThese alternating signs come from the inverse formal power series $(1+xt{)}^{-1}$. They do not require identifying the First Chern class of a dual line with $-x$, which is generally incorrect in complex cobordism.

The final sentence's designation $x=c_1(\eta )$ must mean the earlier $c_1(\eta (1))$. If instead one substitutes $p^{\ast }{c}_1(\eta )$ literally, the claimed relation is false even in ordinary cohomology. For example, take $\eta =L\oplus L$ over ${\mathbb{CP}}^2$, with $a=c_1(L)$ a generator. Its projective bundle is ${\mathbb{CP}}^2\times {\mathbb{CP}}^1$, and $p^{\ast }{a}^2\ne 0$. With the incorrect substitution $x=2p^{\ast }a$, the rank-two polynomial evaluates toThe tautological-line interpretation above is the one consistent with the bundle decomposition and the stated relation.

In a local trivialization, identify the lines in each fibre with $U_i\times {\mathbb{P}}^1$. On overlaps glue byThis action is well defined because $g_{ij}(x)$ is invertible and scaling $v$ does not change the resulting line. It is holomorphic: in affine projective coordinates it is a ratio of holomorphic functions on the open set where its denominator is nonzero. The transition maps satisfy the cocycle condition and have holomorphic inverses. They give the holomorphic projectivization by lines its complex-manifold atlas, of dimension ${\dim }_{\mathbb{C}}X+1$, on the usual projective-bundle topology. Points over different base points are separated by base neighborhoods; points over the same base point are separated inside a common local product chart. A countable trivializing cover and the standard charts of the complex projective line give second countability.

The local projections agree, defining a holomorphic map $p:\mathbb{P}(E)\to X$ whose fibre is $\mathbb{P}(E_x)\cong {\mathbb{P}}^1$.

Define the relative tautological line bundle $S\subset p^{\ast }E$ by $S_l=l$. Its local construction is holomorphic, and its restriction to a fibre is ${\mathcal{O}}_{{\mathbb{P}}^1}(-1)$. The required relative hyperplane line bundle is thereforeThis fixes the lines convention for projectivization and the sign of the fibre degree.

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.