I have a definition of the projective line given in my lecture notes as:
$ \mathbb{P}^1=(\mathbb{C}^2 $ \ $ \{0\})/\mathbb{C}^\times $
my notes then state that this is the space of lines through the origin in $\mathbb{C}^2$ and is isomorphic to the Riemann Sphere.
I can imagine how the space of lines through the origin is isomorphic to the Riemann sphere, however I cannot see/imagine from the definition given above of $ \mathbb{P}^1 $ how this definition corresponds to the space of lines through the origin in $\mathbb{C}^2$.
Is there a good way to see this directly from the definition as given? Or a good way to visualise this definition that makes it easier?
Thank you in advance for any help.
I think that the problem might be that the notion here is one of complex lines through the origin. Such a line consists of all points $c (z, w)$, where $(z, w)$ is a fixed non-origin point in $\Bbb C^2$, and $c$ ranges over the complex numbers. So a complex line "looks just like" a real plane.
The definition you were given says to take all non-origin pairs $(z, w)$ and declare two of them "equal" if one is a multiple of the other. So if our pairs are $(z_1, w_1)$ and $(z_2, w_2)$, and they're "equal" then there's some non-zero complex number $c$ with $$ c(z_1, w_1) = (z_2, w_2), $$ which exactly says that $(z_2, w_2)$ is on the line spanned by $(z_1, w_1)$.
As for the correspondence with $S^2$, that's not so bad either. For any point $$ (z, w) \in \Bbb C^2 $$ you can say that it's "the same" (i.e., on the same complex line through the origin as) the point $$ (\frac{z}{w}, 1) $$ by picking $c = 1/w$. (This fails if $w = 0$ --- more on that in a moment.)
Thus for (almost) each complex line through the origin, we get a pair $$ (q, 1) $$ where $q$ is a complex number. And for each complex number $q$, there's an associated line spanned by $(q, 1)$. So the set of "almost all lines-through-the-origin-in-$\Bbb C^2$" corresponds in a 1-1 way to $\Bbb C$.
There's exactly one complex line we haven't handled by this method --- the one spanned by $(1, 0)$. And that one ends up corresponding to the "point at infinity", so that we get the whole Riemann sphere. (To make sense of this, there's some work to do: you have to show that lines closer and closer to the span of $(1, 0)$ correspond to points of the complex plane that are farther and farther from the origin, so that the span of $(1, 0)$ really does correspond to a 'point at infinity', but that's actually not too difficult.)