  We analyze the basic properties of Brightest Cluster Galaxies (BCGs) produced
by state of the art cosmological zoom-in hydrodynamical simulations. These
simulations have been run with different sub-grid physics included. Here we
focus on the results obtained with and without the inclusion of the
prescriptions for supermassive black hole (SMBH) growth and of the ensuing
Active Galactic Nuclei (AGN) feedback. The latter process goes in the right
direction of decreasing significantly the overall formation of stars. However,
BCGs end up still containing too much stellar mass, a problem that increases
with halo mass, and having an unsatisfactory structure. This is in the sense
that their effective radii are too large, and that their density profiles
feature a flattening on scales much larger than observed. We also find that our
model of thermal AGN feedback has very little effect on the stellar velocity
dispersions, which turn out to be very large. Taken together, these problems,
which to some extent can be recognized also in other numerical studies
typically dealing with smaller halo masses, indicate that on one hand present
day sub-resolution models of AGN feedback are not effective enough in
diminishing the global formation of stars in the most massive galaxies, but on
the other hand they are relatively too effective in their centers. It is likely
that a form of feedback generating large scale gas outflows from BCGs
precursors, and a more widespread effect over the galaxy volume, can alleviate
these difficulties.
