Isomorphism of \'etale fundamental groups lifts to isomorphism of stratified fundamental group

TL;DR

The paper proves that an isomorphism of étale fundamental groups between smooth projective varieties implies an isomorphism of their stratified fundamental groups, under certain conditions.

Contribution

It establishes that isomorphism of étale fundamental groups lifts to an isomorphism of stratified fundamental groups for smooth projective varieties.

Findings

Étale fundamental group isomorphism implies stratified fundamental group isomorphism.

The result applies to finite generically smooth morphisms.

Provides a link between étale and stratified fundamental groups in algebraic geometry.

Abstract

It is shown that if a finite generically smooth morphism $f\,:\,Y\,\longrightarrow\, X$ of smooth projective varieties induces an isomorphism of the \'etale fundamental groups, then the induced map of the stratified fundamental groups $\pi_1^{str}(f)\, :\, \pi_1^{str}(Y,\, y)\,\longrightarrow\, \pi_1^{str}(X,\, f(y))$ is also an isomorphism.