Abelianization of the F-divided fundamental group scheme

TL;DR

This paper investigates the relationship between the abelianization of the F-divided fundamental group scheme of a smooth proper variety and the F-divided fundamental group of its Albanese variety, establishing a surjective homomorphism with finite kernel.

Contribution

It proves that the homomorphism from the abelianized F-divided fundamental group scheme to the Albanese's F-divided fundamental group is surjective with finite kernel, and describes this kernel.

Findings

The homomorphism is surjective.

The kernel of the homomorphism is finite.

The kernel is explicitly described.

Abstract

Let $(X ,x_0)$ be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for $(X ,x_0)$ produces a homomorphism from the abelianization of the $F$-divided fundamental group scheme of $X$ to the $F$-divided fundamental group of the Albanese variety of $X$. We prove that this homomorphism is surjective with finite kernel. The kernel is also described.