I'm reading through a computation of the Chern number $c_1$ of the complex tautological line bundle $$ \tau = \{(l, p) \in \mathbb{CP}^1 \times \mathbb C^2 \mid p \in l\}\\ \downarrow \\ \mathbb{CP}^1. $$
One of the first steps is to notice that removing the zero section $\sigma_0$ gives $$ \tau - \sigma_0 \cong \mathbb C^2 - \{0\}, $$ but I'm having trouble seeing this. I'd appreciate any help with the intuition.
$\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}$Projection to the second factor ($\pi_{2}:\Cpx\Proj^{1} \times \Cpx^{2} \to \Cpx^{2}$) induces a biholomorphism from the complement of the zero section of $\tau$ to $\Cpx^{2} \setminus\{(0, 0)\}$. Geometrically, every point $p$ of $\Cpx^{2} \setminus\{(0, 0)\}$ defines a unique complex line $\ell_{p} = \{tp : t \in \Cpx\}$, and $\ell_{p}$ is precisely the fibre $\tau_{p}$ of the tautological bundle over the point $p$.
