I'm having trouble finding a definition for a meromorphic function from the Riemann sphere to itself. Denoting the sphere $\hat{\mathbb{C}}$ we have that
$$f:\hat{\mathbb{C}}\rightarrow \hat{\mathbb{C}}$$
is holomorphic if for any coordinates $(U_1,\phi_1)$ and $(U_2,\phi_2)$ the function
$$\phi_2\circ f\circ \phi_1^{-1}:\mathbb{C}\supset \phi_1(U_1)\rightarrow \phi_2^{-1}(U_2)\subset\mathbb{C}$$
is holomorphic. How would i define a meromorphic function?
Best Answer
For two complex manifolds $f:M \to N$ is analytic iff $\psi^{-1}\circ f\circ \phi$ is analytic in the usual sense for enough charts.
$f:M \to \Bbb{C}$ is meromorphic iff $f\circ \phi$ is meromorphic (ie. $f\circ \phi= u/v$ with $u,v$ analytic and $v\ne 0$) for enough charts $\phi$.
Moreover in dimension $1$ meromorphic on $M$ is the same as analytic $M \to \Bbb{P^1(C)}$ (in dimension $\ge 2$ it is the same as analytic away from a codimension $\ge 2$ set)
There is no meaning of meromorphic $M \to N$ because we need a field structure on $N$ to define $u/v$ in a way compatible with analyticity.