  By means of zoom-in hydrodynamic simulations we quantify the amount of
neutral hydrogen (HI) hosted by groups and clusters of galaxies. Our
simulations, which are based on an improved formulation of smoothed particle
hydrodynamics (SPH), include radiative cooling, star formation, metal
enrichment and supernova feedback, and can be split in two different groups,
depending on whether feedback from active galactic nuclei (AGN) is turned on or
off. Simulations are analyzed to account for HI self-shielding and the presence
of molecular hydrogen. We find that the mass in neutral hydrogen of dark matter
halos monotonically increases with the halo mass and can be well described by a
power-law of the form $M_{\rm HI}(M,z)\propto M^{3/4}$. Our results point out
that AGN feedback reduces both the total halo mass and its HI mass, although it
is more efficient in removing HI. We conclude that AGN feedback reduces the
neutral hydrogen mass of a given halo by $\sim50\%$, with a weak dependence on
halo mass and redshift. The spatial distribution of neutral hydrogen within
halos is also affected by AGN feedback, whose effect is to decrease the
fraction of HI that resides in the halo inner regions. By extrapolating our
results to halos not resolved in our simulations we derive astrophysical
implications from the measurements of $\Omega_{\rm HI}(z)$: halos with circular
velocities larger than $\sim25~{\rm km/s}$ are needed to host HI in order to
reproduce observations. We find that only the model with AGN feedback is
capable of reproducing the value of $\Omega_{\rm HI}b_{\rm HI}$ derived from
available 21cm intensity mapping observations.
