  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.
