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Let $X$ and $Y$ be algebraic varieties defined over the complex numbers. Let $f:X\to Y$ be a morphism of algebraic varieties such that $f$ is a locally trivial fibre bundle in usual complex topology. When is $f$ Zariski locally trivial?

Any comments, suggestions on how to think about the question will be extremely helpful.

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    $\begingroup$ You might have a look at this question and the answers there. $\endgroup$
    – abx
    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 $\mathbb{C}$. $\endgroup$
    – Daniel Loughran
    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 $\endgroup$
    – Daniel Litt
    Commented May 19, 2022 at 19:23
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    $\begingroup$ Name of @DanielLitt's reference: Serre - Éspaces fibrés algébriques. $\endgroup$
    – LSpice
    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). $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 19, 2022 at 23:07

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Since you asked for "comments, suggestions" here are two thoughts off the top of my head:

  1. In Torsion Homologique et Sections Rationnelles by Grothendieck (Séminaire Chevalley, 1958), the case for principal bundles is worked out. In particular, for $G$-principal bundles over $\mathbb{C}$ where $G$ is isomorphic to a product of groups of type $\operatorname{SL}_n$ or $\operatorname{Sp}_n$, if the (algebraic) bundle is locally trivial in the analytic topology then it will be locally trivial in the Zariski topology.
  2. To learn how to "work around" the difference for cohomological computations, see the propositions in Section 2 in Hodge polynomials of $\operatorname{SL}(2,\mathbb{C})$-character varieties for curves of small genus by Logares, Muñoz, and Newstead.
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    $\begingroup$ Thanks @Sean Lawton for your answer. $\endgroup$
    – tota
    Commented May 20, 2022 at 6:07

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