Complex projective bundle and tautological complex line

Assume AC. Let $E\to B$ be a numerable complex vector bundle of rank $n\ge 1$ over a paracompact Hausdorff CW complex $B$, with zero section $0_B$ and total space $E$. Its projective bundle is the quotient $P(E):=(E\setminus 0_B(B))\mathrm{/}{\mathbb{C}}^{\times },$ where $\lambda \in {\mathbb{C}}^{\times }$ acts fiberwise by $v\mapsto \lambda v$; write $[\mathrm{\ell }]$ for the class of a nonzero vector. The projection $p:P(E)\to B$, $p([v])=\pi (v)$, is well defined because scaling preserves the base point.

$P(E)$ is a fiber bundle over $B$ with fiber ${\mathbb{C}\mathbb{P}}^{n-1}$: over a complex linear chart $U\times {\mathbb{C}}^n$ of $E$ the quotient is $U\times {\mathbb{C}\mathbb{P}}^{n-1}$. On an overlap, the transition matrix $g_{UV}(b)\in {\mathrm{GL}}_n(\mathbb{C})$ induces $(b,[v])\mapsto (b,[g_{UV}(b)v])$; this is a homeomorphism with inverse induced by $g_{VU}$, depends continuously on $b$, and the cocycle identities descend unchanged to projective classes. These quotient charts therefore form a fiber-bundle atlas with fiber ${\mathbb{C}\mathbb{P}}^{n-1}$. Under the identification ${\mathbb{C}\mathbb{P}}^{n-1}={\mathrm{Gr}}_1({\mathbb{C}}^n)$ supplied by Stiefel spaces, Grassmannians, and tautological bundles, a point of the fiber over $b\in B$ is a complex line $\mathrm{\ell }\subseteq E_b$.

The tautological complex line ${\gamma }_E\subseteq p^{\ast }E$ is the subbundle whose fiber over $\mathrm{\ell }\subseteq E_b$ is $\mathrm{\ell }$ itself, with the complex structure induced from $E$; its transition functions are the projectivized linear maps restricted to the selected line, so it is a complex line bundle over $P(E)$.

Here is the base and orientation justification needed to define its Euler class. The numeration for $E$ also numerates the displayed projective charts. The CW complex $B$ is CGWH. The fiber ${\mathbb{C}\mathbb{P}}^{n-1}$ is compact Hausdorff and a finite CW complex (with one cell in dimensions $0,2,\dots ,2n-2$). Thus Compact-fibre bundle totals preserve paracompactness, and CW type under CW-type hypotheses applies and makes $P(E)$ paracompact Hausdorff, CGWH, and of CW type. The tautological line is locally trivial: in a projective coordinate chart $v_j\mathrm{\ne }0$, choose the unique representative with $v_j=1$ and write each vector on the line as its scalar multiple. These charts, combined with the charts of $E$, give linear trivializations. Under AC (hence DC), Under choice and dependent choice, every open cover of a paracompact Hausdorff space admits a locally finite subordinate partition of unity numerates their open cover.

Orient the underlying real line bundle by the frame $(v,iv)$ in each such complex trivialization. Changing $v$ to $(a+ib)v$ has real matrix $\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)$, of determinant $a^2+b^2>0$. Hence these orientations agree on overlaps by Oriented real bundles and oriented frame bundles. This direct rank-one construction uses no CW structure on $P(E)$ itself. The resulting numerable oriented real rank-two bundle is in the general Thom scope, so Euler class by zero-section pullback of the Thom class defines $x=x_E:=e(({\gamma }_E{)}_{\mathbb{R}})\in H^2(P(E);\mathbb{Z})\mathrm{.}$ Defining $x$ by the Euler class of the tautological line avoids any circular use of Chern classes, which are introduced only afterwards on this page. For the zero bundle of rank $0$ we set $P(0_B):=\mathrm{\varnothing }$, and $x$ is not defined there; for a line bundle $L$ the map $P(L)\to B$ is a homeomorphism over $B$ and ${\gamma }_L$ corresponds to $L$ under it.