Simply connected varieties in characteristic $p>0$

TL;DR

This paper proves the non-existence of non-trivial stratified bundles on certain simply connected varieties over finite fields, under specific compactification conditions, and establishes strong Lefschetz properties.

Contribution

It introduces new non-existence results for stratified bundles in positive characteristic and proves strong Lefschetz theorems for these varieties.

Findings

No non-trivial stratified bundles on certain simply connected varieties over finite fields.

Establishment of strong Lefschetz properties in this context.

Conditions involving compactification and boundary codimension are crucial.

Abstract

We show that there are no non-trivial stratified bundles over a smooth simply connected quasi-projective variety over the algebraic closure of a finite field, if the variety admits a normal projective compactification with boundary locus of codimension $\ge 2$. In the appendix, various strong forms of the Lefschetz LEF are proven. Final version, to appear in Compositio.