# Line Bundles

Simply speaking, a line bundle is a complex vector bundle such that each fibre $F_x$ is a one-dimensional complex vector space i.e. one-dimensional vector space over the complex field $\mathbb{C}$. More specifically,

Definition. A complex line bundle over a manifold $M$ is a manifold $L$ and a smooth onto map $\pi: L\longrightarrow M$ such that

1. For each $m\in M$, $\pi^{-1}(m)=L_m$ is a one-dimensional complex vector space.
2. For each $m\in M$, there exists an open neighborhood $U(m)\subset M$ such that $\pi^{-1}(U(m))\stackrel{\varphi}{\cong}U(m)\times\mathbb{C}$ (here $\cong$ means “is homeormorphic to” as usual) and $\varphi(L_m)\subset\{m\}\times\mathbb{C}$. Moreover, $\varphi|_{L_m}:L_m\longrightarrow\{m\}\times\mathbb{C}$ is a linear isomorphism.

Example. The Trivial Bundle $M\times\mathbb{C}$.

Example. If $u\in S^2$, the tangent plane at $u$ is identified with
$$T_uS^2=\{v\in\mathbb{R}^3:\langle u,v\rangle=0\}.$$
We can make this 2-dimensional real vector space a 1-dimensional complex vector space by defining
$$(a+i\beta)v:=\alpha v+\beta\cdot u\times v.$$
So, the tangent bundle $TS^2$ is a line bundle. $TS^2$ as a complex line bundle is called the mini-twistor space and it plays an important role in the study of BPS monopoles in physics.

Example. Let $\Sigma\subset\mathbb{R}^3$ be a surface. If $x\in\Sigma$ and $\hat n_x$ is a unit normal, then $T_x\Sigma=\hat n_x^\perp$ (the orthogonal complement of $\hat n_x$). We make this a 1-dimensional complex vector space by defining
$$(\alpha+i\beta)v=\alpha v+\beta\hat n_x\times v.$$
So, the tangent bundle $T\Sigma$ is a line bundle.

Example. [Hopf Bundle] Let $\mathbb{C}P^1$ be the set of all lines through the origin in $\mathbb{C}^2$. Denote the line through the vector $z=(z^0,z^1)$ by $[z]=[z^0,z^1]$. Define two open sets $U_i$, $i=0,1$ by
$$U_i=\{[z^0,z^1]:z^i\ne 0\},\ i=0,1$$
and $\psi_i:U_i\longrightarrow\mathbb{C}$ by
$$\psi_0([z])=\frac{z^1}{z^0},\ \psi_1([z])=\frac{z^0}{z^1}.$$
Then $\mathbb{C}P^1$ is a complex manifold of dimension 1. As a manifold $\mathbb{C}P^1$ is diffeomorphic to $S^2$. An explicit diffeomorphism $S^2\longrightarrow\mathbb{C}P^1$ is given by
$$(x^1,x^2,x^3)\longmapsto[x^1+ix^2,1-x^3].$$
Define a line bundle $H\subset\mathbb{C}^2\times\mathbb{C}P^1$ over $\mathbb{C}P^1$ by
$$H=\{(\omega,[z]): \omega=\lambda z\ \mbox{for some}\ \lambda\in\mathbb{C}\setminus\{0\}\}.$$
Define a projection $\pi:H\longrightarrow\mathbb{C}P^1$ by $\pi(\omega,[z])=[z]$. The fibre $H_{[z]}=\pi^{-1}([z])$ is the set $\{(\lambda z,[z]):\lambda\in\mathbb{C}\setminus\{0\}\}$ which is identified with the line $[z]$ through the vector $z$. The fibre $H_{[z]}$ can be made to a 1-dimensional complex vector space by
\begin{align*}
\alpha(\omega,[z])+\beta(\omega’,[z])&:=(\alpha\omega+\beta\omega’,[z]),\ \alpha,\beta\in\mathbb{C}\setminus\{0\},\\
0(\omega,[z])&:=(0,0).
\end{align*}

References:

 M. Murray, Notes on Line Bundles