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Let $X$ be a smooth projective surface over an algebraically closed field of char $p > 0$. Suppose that $\pi_{1}^{et}(X) = \{1\}$. Can Nori fundamental group scheme of $X$ be non-trivial?

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    $\begingroup$ A non-classical Enriques surface in characteristic 2? $\endgroup$
    – Piotr Achinger
    Commented Jan 21, 2014 at 1:11
  • $\begingroup$ Thanks for noticing, but unfortunately, I don't understand why it should be the case. $\endgroup$
    – AlekseiG
    Commented Jan 21, 2014 at 2:14
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    $\begingroup$ Because it admits a non-trivial $\alpha_2$ or $\mu_2$-torsor, hence the fundamental group scheme should be either $\alpha_2$ or $\mu_2$. $\endgroup$
    – Piotr Achinger
    Commented Jan 21, 2014 at 4:20

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