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 $p\_2\circ f=p\_1$.


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


Note that if the answer to Q1 is positive then because of the symmetry of the problem $f$ is automatically biregular. A positive answer to Q1 would give "in some sense" a strengthening of Proposition 9 on p. 13 of GAGA. 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.



Analytic isomorphisms above two etale maps
This (from SGA 1) is the proof that the functor is fully faithful: We may suppose
that $X$ is connected. To give an $X$-morphism $Y\rightarrow Y^{\prime}$ is to
give a section to $Y\times\_{X}Y^{\prime}\rightarrow Y$, which is the same as
to give a connected component $\Gamma$ of $Y\times\_{X}Y^{\prime}$ such that
the morphism $\Gamma\rightarrow X$ induced by the projection $Y\times\_{X}Y^{\prime}\rightarrow Y$ is an isomorphism. But the connected components of $Y\times\_{X}Y^{\prime}$
coincide with the connected components of $Y^{an}\times\_{X^{an}}Y^{\prime an}$, and if $\Gamma$ is a connected component of
$Y\times\_{X}Y^{\prime}$, then the projection $\Gamma\rightarrow X$ is an
isomorphism if and only if $\Gamma^{an}\rightarrow Y^{an}$ is an isomorphism.


