Homomorphisms between Kaehler groups

TL;DR

This paper surveys known results and open problems about Kaehler groups, introduces a new notion of Kaehler homomorphism induced by holomorphic maps, and discusses obstructions including a key cohomology vanishing result.

Contribution

It introduces the concept of Kaehler homomorphisms and extends decomposition theorems to Kaehler orbifolds, advancing understanding of their algebraic and geometric properties.

Findings

Identification of obstructions to Kaehler homomorphisms

Main result on vanishing of a cohomology class

Extension of decomposition theorem to Kaehler orbifolds

Abstract

This is partly a survey and partly a research article. Some known results and open problems about Kaehler groups (fundamental groups of compact Kaehler manifolds) are discussed. A new notion of Kaehler homomorphism is introduced. This is a homomorphism induced by a holomorphic map between these types of manifolds. Some obstructions for a homomorphism to be Kaehler are discussed. Among these is the main result on the vanishing of a certain cohomology class associated to such map. This is reduced to the decomposition theorem for perverse sheaves suitably extended to Kaehler orbifolds.