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    $\begingroup$ You might have a look at this question and the answers there. $\endgroup$ Commented May 19, 2022 at 16:37
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    $\begingroup$ @tota: The function field of $Y$ is not algebraically closed. This is the relevant field, not $\mathbb{C}$. $\endgroup$ Commented May 19, 2022 at 19:19
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    $\begingroup$ For principal G-bundles, this is related to G being a "special group" in the sense of Serre; the original reference is here: numdam.org/item/SB_1951-1954__2__305_0 $\endgroup$ Commented May 19, 2022 at 19:23
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    $\begingroup$ Name of @DanielLitt's reference: Serre - Éspaces fibrés algébriques. $\endgroup$ Commented May 19, 2022 at 21:09
  • $\begingroup$ What do you mean by "fibre bundle in usual complex topology": a topologically locally trivial fibre bundle, or a holomorphically locally trivial fibre bundle? In the former case, the fibres need not even be isomorphic as complex manifolds (see e.g. Ehresmann's theorem, which implies that any smooth projective morphism of (smooth) varieties is a $C^\infty$ fibre bundle, but of course the holomorphic/algebraic structure can vary). $\endgroup$ Commented May 19, 2022 at 23:07