  In our previous work "Characterization of certain homorphic geodesic cycles
on Hermitian locally symmetric manifolds of the noncompact type" in "Modern
methods in Complex Analysis" Annals of Math. Studies 138 (1995) 85-118, we
formulated a conjecture: the so called "gap phenomenon".
  The purpose of the article is two-fold. We give a counterexample to the gap
phenomenon in the most general situation. We give new examples of situations in
which the gap phenomenon holds and a unified conceptual, hopefully definitive,
presentation of these situations. In the last section of the article, we survey
some open problems connected to the gap phenomenon.
