  The growth of a supermassive black hole (BH) is determined by how much gas
the host galaxy is able to feed it, which in turn is controlled by the cosmic
environment, through galaxy mergers and accretion of cosmic flows that time how
galaxies obtain their gas, but also by internal processes in the galaxy, such
as star formation and feedback from stars and the BH itself. In this paper, we
study the growth of a 10^12 Msun halo at z=2, which is the progenitor of al
group of galaxies at z=0, and of its central BH by means of a high-resolution
zoomed cosmological simulation, the Seth simulation. We study the evolution of
the BH driven by the accretion of cold gas in the galaxy, and explore the
efficiency of the feedback from supernovae (SNe). For a relatively inefficient
energy input from SNe, the BH grows at the Eddington rate from early times, and
reaches self-regulation once it is massive enough. We find that at early cosmic
times z>3.5, efficient feedback from SNe forbids the formation of a settled
disc as well as the accumulation of dense cold gas in the vicinity of the BH
and starves the central compact object. As the galaxy and its halo accumulate
mass, they become able to confine the nuclear inflows provided by major mergers
and the BH grows at a sustained near-to-Eddington accretion rate. We argue that
this mechanism should be ubiquitous amongst low-mass galaxies, corresponding to
galaxies with a stellar mass below <10^9 Msun in our simulations.
