The Homotopy Sequence of Nori's Fundamental Group

TL;DR

This paper studies the conditions for the exactness of Nori's fundamental group sequence, applying them to classical theorems and providing counterexamples to illustrate when conditions fail.

Contribution

It establishes necessary and sufficient conditions for the exactness of Nori's fundamental group sequence and explores their implications in various contexts.

Findings

Identified conditions under which the homotopy sequence is exact

Reproved classical theorems using these conditions

Provided a counterexample where conditions are not met

Abstract

In this paper, we investigate the necessary sufficient conditions for the exactness of the homotopy sequence of Nori's fundamental group and apply these to various special situations to regain some classical theorems and give a counter example to show the conditions are not always satisfied. This work partially bases on the earlier work of H.Esnault, P.H.Hai, E.Viehweg.