Homomorphisms between fundamental groups of K\"ahler manifolds

TL;DR

The paper constructs a group morphism between fundamental groups of Kähler manifolds that cannot be realized by a holomorphic map between smooth projective varieties, highlighting differences in geometric realizability.

Contribution

It provides an explicit example of a fundamental group morphism realizable in Kähler geometry but not in projective geometry, and proves no such example exists for abelian groups.

Findings

Existence of a fundamental group morphism realizable in Kähler but not in projective settings.

No such non-realizable morphism exists between abelian groups.

Highlights differences between Kähler and projective manifold fundamental groups.

Abstract

A group morphism is constructed, which can be realized as the induced morphism of fundamental groups from a holomorphic map between compact Kahler manifolds, but can not be realized by a holomorphic map between smooth projective varieties. And it is also proved that there exists no such example between abelian groups.