For a smooth projective variety over an algebraically closed field of positive characteristic, any representation of its fundamental group induces a D-module on the variety. But it is difficult to construct a representation of fundamental group for a given D-module , for example, Gieseker made a conjecture that there is no nontrivial D-module on the variety if its fundamental group is trivial, which was proved by Esnault and Mehta (Invent. Math. 181 (2010), 449-465).
In this paper, by generalizing Simpson’s construction, we construct a moduli space of semi-stable bundles with frames at a fixed points at first. Then, for a given D-module, we construct a subvariety of the moduli spaces with a dominate self-rational map induced by Frobenius pull-back. Then the periodic points of the rational map provide representations of the fundamental group. The main result of this paper implies immediately a relative version of Gieseker's conjecture, which generalies the main theorem of Esnault -Mehta in [1].
[1] H. Esnault, V. Mehta, Simply connected projective manifolds in characteristic p >0 have no nontrivial stratified bundles, Invent. Math. 181 (2010) 449–465.
