Let $X$ be a normal proper variety with rational singularities (or terminal if that is necessary) and $X_{\text{reg}} \to X$ the regular locus. Let $\pi : \tilde{X} \to X$ be a resolution of singularities. Since $\pi$ is an isomorphism over $X_{\text{reg}}$ we get maps,
$$ \pi_1^{\text{ét}}(X_{\text{reg}}) \to \pi_1^{\text{ét}}(\tilde{X}) \to \pi_1^{\text{ét}}(X). $$
I know that the composition $\pi_1^{\text{ét}}(X_{\text{reg}}) \to \pi_1^{\text{ét}}(X)$ is only surjective, not an isomorphism, in general (even if $X$ has terminal singularities). I am confused as to whether $\pi_1^{\text{ét}}(X_{\text{reg}}) \to \pi_1^{\text{ét}}(\tilde{X})$ is expected (known) to be an isomorphism?
Some reasons I suspected this might be true:
(1) Kebekus and Schnell proved in Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities that you can extend forms from $X_{\text{reg}}$ to $\tilde{X}$ and hence the “torsion-free abelian part” of $\pi_1$ should agree.
(2) There are some results on extending representations of $\pi_1$ from $X_{\text{reg}}$ to $\tilde{X}$ in the guise of stable vector bundles or Higgs bundles e.g. the work of Greb (see Greb, Kebekus, Peternell, and Taji – Harmonic metrics on Higgs sheaves and uniformization of varieties of general type) or Lu (see Lu and Taji – A characterization of finite quotients of Abelian varieties).
However, Kollár proved (Theorem 7.8 of Shafarevich maps and plurigenera of algebraic varieties) that $\pi_1^{\text{ét}}(\tilde{X}) \to \pi_1^{\text{ét}}(X)$ is an isomorphism when $X$ is log terminal. Therefore, it seems impossible for the first map to be an isomorphism in general but the above suggests that something weaker might be true.
Q. Is it true that the map of abelianizations
$$ \pi_1(X_{\text{reg}})^{\text{ab}} \to \pi_1(\tilde{X})^{\text{ab}} $$
is an isomorphism?
Greb, Kebekus, and Peternell also showed in Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties that there exists a quasi-etale cover $X' \to X$ such that $\pi_1(X'_{\text{reg}}) = \pi_1(X')$. This is close to giving a result I would be very interested in:
for any étale cover $Y \to X_{\text{reg}}$ there is an étale refinement $Y' \to Y$ such that $Y'$ extends to an étale cover $Y'' \to \tilde{X}$.
Does anyone know if this is true in lieu of an isomorphism $\pi_1^{\text{ét}}(X_{\text{reg}}) \to \pi_1^{\text{ét}}(\tilde{X})$?
Best Answer
As indicated by @Angelo, this is often false.
For example, let $X = (C \times C) / \left< \iota \times \iota \right>$ where $C$ is a hyperelliptic curve and $\iota : C \to C$ the hyperelliptic involution.
Then $\pi_1(\tilde{X}) \cong \pi_1(X) \cong 0$. However, $\pi_1(X_{\text{reg}})$ is an extension of $\mathbb{Z} / 2 \mathbb{Z}$ by $\pi_1(C \times C \setminus \text{ramification}) = \pi_1(C \times C)$ which is nontrivial.