Fundamental Group Schemes of Generalized Kummer Variety

TL;DR

This paper investigates the fundamental group schemes of generalized Kummer varieties derived from abelian surfaces over algebraically closed fields of characteristic greater than 3, extending understanding of their algebraic fundamental groups.

Contribution

It determines the S-fundamental and Nori's fundamental group schemes for generalized Kummer varieties associated to abelian surfaces in characteristic p > 3.

Findings

Computed the S-fundamental group scheme of generalized Kummer varieties.

Determined Nori's fundamental group scheme for these varieties.

Extended the theory of fundamental group schemes in positive characteristic.

Abstract

Let $k$ be an algebraically closed field of characteristic $p > 3$. Let $A$ be an abelian surface over $k$. Fix an integer $n \geq 1$ such that $p \nmid n$ and let $K^{[n]}$ be the $n$-th Generalized Kummer Variety associated to $A$. In this article we aim to find the $S$-fundamental group scheme and Nori's fundamental group scheme of $K^{[n]}$.