Complex vector bundle

Let $M$ be a smooth manifold. A smooth complex vector bundle of rank $r$ over $M$ is a smooth surjective map $\pi:E\to M$ with the following properties:

For each $x\in M$ , the fiber $E_x:=\pi^{-1}(x)$ is a finite-dimensional complex vector space.

There is an open cover $\{U_\alpha\}$ of $M$ and smooth maps (local trivializations) $\Phi_\alpha:\pi^{-1}(U_\alpha)\to U_\alpha\times \mathbb C^r$ such that:

$\mathrm{pr}_1\circ \Phi_\alpha=\pi$ on $\pi^{-1}(U_\alpha)$ , and
for each $x\in U_\alpha$ , the induced map $(\Phi_\alpha)_x:E_x\to \{x\}\times\mathbb C^r\cong \mathbb C^r$ is complex linear.

The integer $r$ is the complex rank. More generally, allowing different ranks on different connected components gives a locally constant rank function (see rank of a vector bundle).

Equivalent characterizations

Equivalently, on overlaps $U_\alpha\cap U_\beta$ the transition maps

\Phi_\alpha\circ \Phi_\beta^{-1}:(U_\alpha\cap U_\beta)\times \mathbb C^r\to (U_\alpha\cap U_\beta)\times \mathbb C^r

have the form $(x,v)\mapsto (x,g_{\alpha\beta}(x)v)$ for a smooth map $g_{\alpha\beta}:U_\alpha\cap U_\beta\to \mathrm{GL}(r,\mathbb C)$ .

Examples

Trivial bundle. For any $r\ge 1$ , the projection $M\times\mathbb C^r\to M$ is a complex vector bundle with the obvious trivializations.

Complexified tangent and cotangent bundles. The bundles $TM\otimes_\mathbb R \mathbb C$ and $T^*M\otimes_\mathbb R \mathbb C$ are complex vector bundles obtained by complexifying the tangent bundle and cotangent bundle.

Bundle of complex-valued $k$ -forms. The exterior power $\Lambda^k(T^*M\otimes\mathbb C)$ is a complex vector bundle whose smooth sections are complex-valued differential k-forms.