I have a question that is purely on vocabulary. My native language is not english, so I would like to know the usual convention for the following.
When people say "let $f: X \to Y$ be an analytic map", where $X,Y$ are, say, complex manifolds, do they allow for singularities on $X$ ? If yes, what about essential singularities?
For example, consider the following statements:
All analytic maps $f: \mathbb{P}^1(\mathbb{C}) \to \mathbb C$ are constants.
The exponential function $\exp: \mathbb{P}^1(\mathbb{C}) \to \mathbb C$ is analytic.
Which of those statements would be true, according to usual convention? i.e. can we say that the exponential function is analytic on the projective line, but with an essential singularity ?
No, an analytic map has no singularities. "Holomorphic" is a synonym in English, in all cases of which I'm aware. "Meromorphic" functions may have poles, but no essential singularities. I don't know a word for a function which is analytic at most points, but with arbitrary singularities permitted.