  Context: Local reductions of the stellar velocity dispersion in the central
regions of galaxies are known as sigma-drops. Knowing the origin of these
features can lead to better understanding of inner galactic dynamics.
  Aims: We present a sample of 20 sigma-drop galaxies matched with a control
sample of galaxies without sigma-drop in order to search for correlations
between sigma-drops and the properties, primarily morphological, of the nuclear
zones and discs of their host galaxies.
  Methods: We study the dust and Halpha distribution at 0.1 arcsec scale, using
Hubble Space Telescope imaging, in the circumnuclear zones of the two samples
of galaxies, searching for differences and trying to establish a link between
the nuclear kinematics and the host morphology. We have also considered the CO
and HI emission of the galaxies and their luminosity profiles.
  Results: We classify the two samples following both morphological parameters
and the luminosity profiles. We find a larger fraction of nuclear dust spirals
and Halpha rings in the sigma-drop sample. We also find that the fraction of
Seyfert galaxies in the sigma-drop sample is bigger than that of LINERs and
that the reverse is true for the control sample.
  Conclusions: Our findings are evidence that a $\sigma$-drop is very probably
due to inflow-induced star formation in a dynamically cool disc, or in a gas
ring, shock focused by an inner Lindblad resonance above a certain critical
density level. The same mechanism that feeds the nuclear ring or the nuclear
disc is probably reponsible for the higher rate of Seyfert galaxies among the
$\sigma$-drop hosts.
