We study many properties concerning weak K\"ahlerianity on compact complex
manifolds which admits a holomorphic submersion onto a K\"ahler or a balanced
manifold. We get generalizations of some results of Harvey and Lawson (the
K\"ahler case), Michelson (the balanced case), Popovici (the sG case) and
others.Comment: 14 pages; second revised version; in press, Tohoku Math. J. vol 68,
n.4 (2016
