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Let $X$ be a connected complex-analytic space. , $G = \pi_1(X)$ the fundamental group of $X$ , $\hat{G}=\varprojlim G/N$ its profinite completion.

Let $\beta\in H^2(X,\mathcal{O}_X^\times).$, say with $n\beta=0$, and choose a class $\alpha\in H^2(X,\mu_n)$ mapping to $\beta$.

suppose that : $$ H^2(G,\mu_n(\mathbb{C}) ) \simeq H^2(\hat{G},\mu_n(\mathbb{C}))\simeq \varinjlim H^2(G/N,\mu_n(\mathbb{C})). $$ and $$ H^2(G,\mu_n(\mathbb{C}) )\simeq H^2(X,\mu_n) $$ Then there is an finite étale covering $g:X'\rightarrow X$ with $g^*(\beta)=0$, and in particular $g^*(\alpha)=0$.

I couldn't realize how can we get this finite étale covering $g :X'\rightarrow X$ from the previous assertions and why $g^*(\beta)=0$?.

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  • $\begingroup$ There has to be some $N$ such that $\beta$ is in the H2 of G/N. Pick the corresponding cover and by inflation restriction or something like that this element maps to zero in H2(N,..). I think this works... $\endgroup$ – Asvin Aug 11 at 4:47
  • $\begingroup$ Thank dear Asvin, may this work for the vanishing of β, but still don't understand why the corresponding cover is finite and etale $\endgroup$ – Moutand Mohammed yesterday
  • $\begingroup$ The cover is automatically finite etale, by definition. $\endgroup$ – Asvin yesterday

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