I am going through Nakahara's "Geometry, topology and physics" by myself and I got stuck on a calculation.
I have the total space $$L=\{(p,v)\in\mathbb{C}P^n\times\mathbb{C}^{n+1}|v=ap,a\in\mathbb{C} \} $$ and the projection map $\pi:L\rightarrow\mathbb{C}P^n$ as $\pi(p,v)=p.$
So far I have that the fibre is then $\mathbb{C}^{n+1}$ over $\mathbb{C}P^n$ and that the trivialization is the map $\phi_i^{-1}:\pi(p,v)\rightarrow U_k\times\mathbb{C}^{n+1}$ where $\{U_k\}$ is a n open cover of $\mathbb{C}P^n\simeq\mathbb{C}$ with coordinates $(z_0/z_1,z_1/z_2\dots z_n/z_0)$.
So I thought that the trivialization is then $\phi_i(v)=(p,\frac{z_k}{|z_{k+1}|})$ and the transition function is $t_{ij}=\frac{z_k|z_{j+1}|}{|z_{k+1}|z_j} $.
the group structure is $G\supset \{t_{kj}\}$.
I am a bit out of my depth here so any help is greatly appreciated.
$\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}$First, a small note on terminology: Nowadays it's common to call the line bundle $L$ the tautological bundle. The term "canonical" bundle almost universally connotes the top exterior power of the cotangent bundle.
The standard trivialization of the tautological line bundle over $U_{k}$ is given by the (non-vanishing local holomorphic) section $$ \sigma_{k}[z_{0} : \dots : z_{n}] = \bigl(\underbrace{[z_{0} : \dots : z_{n}]}_{\in \Cpx\Proj^{n}}, \underbrace{z_{0}/z_{k}, \dots, z_{n}/z_{k}}_{\in \Cpx^{n+1}}\bigr). $$ The $\Cpx^{n+1}$ components are ratios of homogeneous coordinates, and therefore well-defined local functions; together they span the line represented by $[z_{0} : \dots : z_{n}]$. The transition function is $t_{ij} = z_{i}/z_{j}$.
The diagram below (taken from Flag manifold to Complex Projective line) shows the two local sections for the projective line.