A Morphism of Intersection Homology and Hard Lefschetz

We consider a possibility of the existence of intersection homology morphism, which would be associated to a map of analytic varieties. We assume that the map is an inclusion of codimension one. Then the existence of a morphism follows from Saito's decomposition theorem. For varieties with conical singularities we show, that the existence of intersection homology morphism is exactly equivalent to the validity of Hard Lefschetz Theorem for links. For varieties with arbitrary analytic singularities we extract a remarkable property, which we call Local Hard Lefschetz.