Simply connected projective manifolds in characteristic \(p>0\) have no nontrivial stratified bundles. (English) Zbl 1203.14029

Let \(X\) be a smooth projective variety defined over an algebraically closed field of characteristic \(p>0\). Let \(F\) denote the absolute Frobenius morphism on \(X\). A stratified bundle on \(X\) is a sequence of locally free sheaves \(\{E_n\} _{n\in {\mathbb N}}\) together with isomorphisms \(\sigma _n: F^*E_{n+1}\to E_n\). D. Gieseker [Ann. Sc. Norm. Super. Pisa 2, 1–31 (1975; Zbl 0322.14009)] conjectured that if the étale fundamental group of \(X\) vanishes then \(X\) has only trivial stratified bundles. The main aim of the paper under review is proof of this conjecture.
In characteristic zero the above conjecture is analogous to the fact that if étale fundamental group of \(X\) vanishes then a pro-algebraic completion of \(X\) also vanishes (this was known due to Grothendieck and, independently, Malcev).
Roughly speaking, the proof follows from application of E. Hrushovski’s theorem on periodic points [arXiv:math/0406514]) to a Verschiebung morphism on the moduli space of semistable vector bundles on \(X\) (see the reviewer’s paper in [Ann. Math. 159, 251–276 (2004; Zbl 1080.14014)]).


14G17 Positive characteristic ground fields in algebraic geometry
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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