Evidence for a generalization of Gieseker's conjecture on stratified bundles in positive characteristic (Q2437871)

In this carefully written paper, the author studies stratified bundles on a smooth connected variety \(X\) over an algebraically closed field of characteristic \(p>0\). In the case when \(X\) is projective, \textit{H. Esnault} and \textit{V. Mehta} [Invent. Math. 181, No. 3, 449--465 (2010; Zbl 1203.14029)] proved that if \(X\) has trivial étale fundamental group, every stratified bundle on \(X\) is trivial. The paper under review investigates possible extensions to the non-proper case. One of the main results concerns stratified line bundles: the author shows that if \(X\) is only assumed to be smooth and its prime-to-\(p\) abelianized fundamental group is trivial, then all stratified line bundles on \(X\) are trivial. The proof proceeds by investigating the \(p\)-completion of the Picard group of \(X\) via the theory of Albanese varieties. There is also a relative version of this result, assuming the existence of good compactifications. Concerning stratified bundles of higher rank, the author proves that if the tame fundamental group of \(X\) is trivial (actually a bit weaker assumption suffices), every regular singular stratified bundle on \(X\) with abelian monodromy is trivial. Here regular singularity is an analogue of a notion due to \textit{D. Gieseker} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 2, 1--31 (1975; Zbl 0322.14009)] worked out in positive characteristic by the author. Due to the assumption on abelian monodromy, the proof proceeds by reduction to the rank 1 case, but the difficult result of Esnault and Mehta mentioned above is also needed during the verification that there are no nontrivial extensions of rank 1 stratified bundles. The paper concludes with further observations concerning regular singular stratified bundles; in particular, it is proven that all regular singular stratified bundles on affine \(n\)-space are trivial.