Specialization map between stratified bundles and pro-\'etale fundamental group

TL;DR

This paper constructs a specialization functor linking representations of the pro-étale fundamental group of a fiber to stratified bundles on the generic fiber, extending Grothendieck's specialization map via Tannakian duality.

Contribution

It introduces a new specialization functor between categories of representations and stratified bundles, and shows it lifts Grothendieck's étale fundamental group specialization map.

Findings

Constructed a functor between fundamental group representations and stratified bundles.

Proved the functor induces a morphism between affine group schemes.

Demonstrated the morphism lifts Grothendieck's specialization map.

Abstract

Given a projective family of semi-stable curves over a complete discrete valuation ring of characteristic p with algebraically closed residue field, we construct a specialization functor between the category of continuous representations of the pro-\'etale fundamental group of the closed fibre and the category of stratified bundles on the geometric generic fibre. By Tannakian duality, this functor induces a morphism between the corresponding affine group schemes. We show that this morphism is a lifting of the specialization map, constructed by Grothendieck, between the \'etale fundamental groups.