I would like to ask for some clarifications in the following questions about complex curves.
- My first question is if I correctly understand what the complex curve in $\mathbb CP^2$ is. Is it only a set which is locally given as the nullset of some analytic function with non-zero gradient at each zero? If this is correct, what is the complex curve with boundary?
- If I understand the definition of complex curve correctly, then the complex curve needn't to be a Riemann surface in general, it may have finite number of points in which it fails to be a smooth manifold. With these points excluded it is a Riemann surface. Right?
- I know the definition of the normalization of a complex curve (A holomorphic map from a Riemann surface to this complex curve such that ...), but in one article the authors used in this definition meromorphic map instead of holomorphic one. What is the meromorphic map in this context? Does one call by a meromorphic map from a Riemann surface to $\mathbb C^2$ a holomorphic map from this Riemann surface to $\mathbb CP^2$?