It might seem a silly question but I'm asking the following:
Take the complex projective line, are the inhomogeneous coordinates sufficient to have an atlas where the transition functions are holomorphic? the property should be independent of the choice of the coordinates, so when using homogeneous coordinates it seems to me that the manifold is no longer analytic. I'm really confused.
The projective complex line is the quotient of $C^2-0$ by the relation $(x,y)\simeq (x',y')$ if $(x',y')=c(x,y)$.
$P^1C$ is the union of $U$ defined by $[x,y], x\neq 0$ and $V$ defined by $[x,y], y\neq 0$
You have $f:C\rightarrow P^1C$ which sends x to [1,x] and g(x) =[x,1] The image of f is U and the image of g is V
$f\circ g^{-1}(x) =1/x$ is holomorphic on C-0