Let $X$ and $Y$ be smooth quasi-projective varieties defined over $\mathbf{C}$ and let $$ f:X(\mathbf{C})\rightarrow Y(\mathbf{C}) $$ be a holomorphic map (not necessarily regular=algebraic). Then it is natural to ask what are additional conditions that one can impose on the data $(f,X,Y)$ in order to force $f$ to be algebraic. Let me give 3 examples of such conditions:

1) Assume that $f$ is finite, unramified and that $X(\mathbf{C})$ has only one algebraic structure. Then a combination of Grauert-Remmert and GAGA implies that $f$ is algebraic. Note that (a postiori) the finiteness assumption on $f$ is essential since one has for example the exponential map $exp:\mathbf{C}\rightarrow\mathbf{C}^{\times}$ which is not algebraic but satisfy all the other assumptions (except the finiteness). Moreover, in general, it is also essential to assume that $X(\mathbf{C})$ has only one algebraic structure since there are examples of complex manifolds with at least 2 non-equivalent algebraic structures.

2) If $X$ is compact then from GAGA we ge automatically that $f$ is algebraic

3) Say that $X$ is a curve and $Y=\mathbb{P}^1(\mathbf{C})-\{0,1,\infty\}$. Then Picard's theorem (+removable singularity result) imply that $f$ is meromorphic on the compactification of $X$ and therefore $f$ is algebraic. (If I remember correctly, I think that there is some kind of generalization of Picard's result to higher dimension from the work of Kwack).

So with these 3 examples in mind, here is my question:

**Q**: what is known in the litterature about additional conditions that one may impose on the data $(f,X,Y)$ in order to force $f$ to be algebraic?

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