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Let $X$ be an integral Noetherian separated scheme. Can there exist two morphisms of finite type $X\rightarrow \mathrm{Spec}\,\mathbb{C}$ such that the corresponding complex-analytic spaces are not weakly homotopy equivalent?

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    $\begingroup$ If one changes the structure morphism by an automorphism of $\mathrm{Spec} \mathbb{C}$ then the (topological) fundamental group can change (as first shown by Serre). $\endgroup$ – ulrich May 19 at 11:40
  • $\begingroup$ Probably related: meta.mathoverflow.net/questions/4200/flood-of-new-users $\endgroup$ – YCor May 31 at 14:16

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