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Let $X$ be a variety and $E$ be a principal $G$ bundles, where $G$ is a semisimple group. Is there a variety $f: \tilde{X}\rightarrow X$ such that $f^*E$ is trivial $G$ bundle?

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  • $\begingroup$ Are you interested in an online study group or a group blogging?? I am reading some differential geometry, category theory, algebriac geometry $\endgroup$
    – Praphulla Koushik
    Commented Apr 9, 2018 at 3:29

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Yes, a principal $G$-bundle is trivial if and only if it has a section, and if you take $\tilde{X} = E$, then $f^*E = E \times_X E$ has a section given by the diagonal map $E \to E \times_X E$.

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    $\begingroup$ Another way to see that $f^*E$ is trivial is that its classifying map $E \to BG$ factors through $EG$ which is contractible. $\endgroup$
    – Michael Albanese
    Commented Apr 8, 2018 at 22:15

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