Constructing a complex structure on $S^2$

Question

By definition of complex manifold, a complex manifold is a manifold with holomorphic charts $U \to D^2 \subseteq \mathbb C$.

I want to define a complex structure on $S^2$.

Can you tell me if this is correct?

Let $D^+$ and $D^-$ denote $S^2-S$ and $S^2 -N$ respectively where $S,N$ are the north and south pole. Define charts $f_+$ and $f_-$ in the obvious way: map $D^+$ and $D^-$ homeomorphically to the open unit disk. Then $\{(D^+,f_+), (D^-, f_-)\}$ is a complex atlas (complex structure) for $S^2$.

Answer 1

I'm going to suggest that you choose the chart that sends $P = (x, y, t)$ in $S^2 - S$ to the intersection $z$ of the segment $SP$ with the $t = 0$ plane. Do the same for the $S^2 - N$, but throw in a conjugate. Those are your charts $f_{\pm}$. Now write out the transition function, which should end up being something like $z \mapsto \frac{1}{z}$.

Answer 2

Look up the standard homeomorphism you get by removing the "point at infinity" in $S^2$ and the complex plane. Two charts using this homeomorphism are enough to give you the complex manifold charts.