Given a birational proper morphism $f\colon X \rightarrow Y$ ( Assume $X$ and $Y$ irreducible ) of complex algebraic varieties. It is always true that $f^* \colon \pi^{et}_1 (X)\rightarrow \pi^{et}_1(Y)$ is an isomorphism? I think that this follows from (SGA.1 exp X. Corollary 1.4) But im not totally sure.

Thanks, in advance.


1 Answer 1


This is true if $X$ and $Y$ are smooth, see SGA I, exp. X, Cor. 3.4. But certainly not in general: if $Y$ is a plane cubic with one node and $f$ its normalization, $X$ is simply connected while $\pi _1^{et}(Y)\cong \hat{\mathbb{Z}}$, cf. same SGA, exp. IX, example 5.5.


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