  We study the core mass -- halo mass relation of bosonic dark matter halos, in
the form of self-gravitating Bose-Einstein condensates, harbouring a
supermassive black hole. We use the ``velocity dispersion tracing'' relation
according to which the velocity dispersion in the core $v_c^2\sim GM_c/R_c$ is
of the same order as the velocity dispersion in the halo $v_h^2\sim GM_h/r_h$
(this relation can be justified from thermodynamical arguments) and the
approximate analytical mass-radius relation of the quantum core in the presence
of a central black hole obtained in our previous paper [P.H. Chavanis, Eur.
Phys. J. Plus 134, 352 (2019)]. For a given minimum halo mass $(M_h)_{\rm
min}\sim 10^8\, M_{\odot}$ determined by the observations, the only free
parameter of our model is the scattering length $a_s$ of the bosons (their mass
$m$ is then determined by the characteristics of the minimum halo). For
noninteracting bosons and for bosons with a repulsive self-interaction, we find
that the core mass $M_c$ increases with the halo mass $M_h$ and achieves a
maximum value $(M_c)_{\rm max}$ at some halo mass $(M_h)_{*}$ before
decreasing. The whole series of equilibria is stable. For bosons with an
attractive self-interaction, we find that the core mass achieves a maximum
value $(M_c)_{\rm max}$ at some halo mass $(M_h)_{*}$ before decreasing. The
series of equilibria becomes unstable above a maximum halo mass $(M_h)_{\rm
max}\ge (M_h)_{*}$. In the absence of black hole $(M_h)_{\rm max}=(M_h)_{*}$.
At that point, the quantum core (similar to a dilute axion star) collapses. We
perform a similar study for fermionic dark matter halos. We find that they
behave similarly to bosonic dark matter halos with a repulsive
self-interaction, the Pauli principle for fermions playing the role of the
repulsive self-interaction for bosons.
