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I am looking for the fundamental group of the moduli space of parabolic bundles with fixed determinant over a smooth projective curve.

I know that the fundamental group of the moduli space of vector bundles with fixed determinant is trivial as it is Fano (Unirationality and Picard number one implies Fano). But I am not sure whether the same is true for the parabolic case.

Also, I wonder if the higher homotopy groups of the moduli space of parabolic bundles are known. Any help would be appreciable.

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    $\begingroup$ The moduli space is still rational (not just unirational), hence simply connected (in fact, all separably rationally connected, smooth projective varieties over an algebraically closed field are simply connected -- char 0 by Campana, char p by Koll'ar and Debarre). In characteristic 0, by Hurewicz, the map from $\pi_2$ to $H_2$ is an isomorphism, and these are free Abelian groups of rank $1$ (if no parabolic structure). For $\pi_3$, already the Jacobian of the curve intercedes. I believe this is worked out in the textbook of Griffiths-Morgan. $\endgroup$
    – Jason Starr
    Commented Apr 30, 2023 at 16:42
  • $\begingroup$ Thank you for your response. So since the moduli space of parabolic bundles is rational, it is simply connected. Is the isomorphism from $\pi_2$ to $H_2$ true for the parabolic case (in characteristic zero)? $\endgroup$
    – yors
    Commented May 1, 2023 at 2:21
  • $\begingroup$ That follows from the Hurewicz Isomorphism. $\endgroup$
    – Jason Starr
    Commented May 1, 2023 at 10:56
  • $\begingroup$ Thank you so much. I got it. $\endgroup$
    – yors
    Commented May 1, 2023 at 15:20

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