  To seek for the useful numerical analogues to the Iitaka dimension, various
numerical Iitaka dimensions have been defined from a number of different
perspectives. It has been accepted that all the known numerical Iitaka
dimensions coincide with each other until the recent discovery of a
counterexample constructed by Lesieutre. In this paper, we prove that many of
them still coincide with the numerical Iitaka dimension introduced by
Boucksom-Demailly-P\u{a}un-Peternell. On the other hand, we show that some
other numerical Iitaka dimensions introduced by Nakayama and Lehmann can be
arbitrarily larger than the rest of numerical Iitaka dimensions. We also study
some properties of abundant divisors.
