# Vector bundles on complete smooth variety $X$ in char $p$ and Frobenius

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?

• 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" – ulrich Jan 29 at 13:05