# Does the structure morphism matter in GAGA?

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?

• 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). – ulrich May 19 '19 at 11:40
• Probably related: meta.mathoverflow.net/questions/4200/flood-of-new-users – YCor May 31 '19 at 14:16