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Le $X_1$, $X_2$ and $Z$ be smooth quasi-projective connected varieties defined over $\mathbf{C}$. Let $p_1:X_1\rightarrow Z$ and $p_2:X_2\rightarrow Z$ be finite etale maps. Assume that $f:X_1\rightarrow X_2$ is an analytic isomorphism such that $p_2\circ f=p_1$.

Q

Q1: Does it follow that $f$ is regular?

Note that once we know that if the answer to Q1 is positive then because of the symmetry of the problem $f$ is regular then using automatically biregular. A positive answer to Q1 would give "in some sense" a strengthening of Proposition 9 on p. 13 of GAGAwe automatically get . This proposition says that if one has a regular map $f:X_1\rightarrow X_2$ such that $f$ is an analytic isomorphism then $f$ is biregular.

So basically, I'm asking if it is possible to replace the regularity assumption on $f$ by the weaker data of two finite etale maps over a base $Z$ which are compatible with $f$ in order to be able to deduce that $f$ is biregular.

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# Analytic isomorphisms above two etale maps

Le $X_1$, $X_2$ and $Z$ be smooth quasi-projective connected varieties defined over $\mathbf{C}$. Let $p_1:X_1\rightarrow Z$ and $p_2:X_2\rightarrow Z$ be finite etale maps. Assume that $f:X_1\rightarrow X_2$ is an analytic isomorphism such that $p_2\circ f=p_1$.

Q: Does it follow that $f$ is regular?

Note that once we know that $f$ is regular then using Proposition 9 on p. 13 of GAGA we automatically get that $f$ is biregular.