The Hodge theoretic fundamental group and its cohomology

TL;DR

This paper investigates a nonabelian Hodge theoretic approach to the fundamental group of algebraic varieties, compares it with existing methods, and provides criteria for varieties to be Hodge theoretic K(pi,1) spaces.

Contribution

It introduces a new perspective on nonabelian Hodge structures on fundamental groups and establishes criteria for identifying Hodge theoretic K(pi,1) varieties.

Findings

Comparison of nonabelian Hodge structures with existing approaches

Criteria for a variety to be a Hodge theoretic K(pi,1)

Insights into cohomology of variations of mixed Hodge structure

Abstract

In this paper, we explore a notion of nonabelian Hodge structure on the fundamental group of an algebraic variety. This is approach is compared to some alternative approaches due to Morgan, Hain and others. We also give criteria for a variety to be a Hodge theoretic K(pi,1), which roughly means that the cohomology of variations of mixed Hodge structure can be determined from the group.