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In "Vektorbündel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe." Math. Z. 156 (1977), no. 1, 73–83. by Herbert and Ulrich, the authors consider a complete smooth variety $X$ defined over an algebraically closed field $k$ of positive characteristic.

They show that: for a vector bundle $E$ of rank $n$ over $X$, the following three conditions are equivalent: (1) $E$ is induced by a continuous representation $π_1(X) \rightarrow GL(n,k)$ (where $GL(n,k)$ is endowed with the discrete topology); (2) $E$ becomes trivial over some étale finite covering of $X$; (3) the pull-back of $E$ by a suitable power of the Frobenius morphism is isomorphic to $E$. Here $\pi_1(X)$ is the algebraic fundamental group in the paper.

I don't know much German, can anyone explain the idea between equivalence of (2) and (3)? And how do we classify Frobenius-unstable vector bundles?

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    $\begingroup$ There is a mistake in the paper in that (2) does not always imply (3) but does if E is stable. See the paper of Biswas and Ducrohet "An analogue of a Theorem of Lange and Stuhler" $\endgroup$ – ulrich Jan 29 at 13:05

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