  Active Galactic Nuclei (AGNs) release huge amounts of energy in their host
galaxies, which, if the coupling is sufficient, can affect the interstellar
medium (ISM). We use a high-resolution simulation ($\sim6$ pc) of a z $\sim2$
star-forming galaxy hosting an AGN, to study this not yet well-understood
coupling. In addition to the often considered small-scale thermal energy
deposition by the AGN, which is implemented in the simulation, we model
long-range photo-ionizing AGN radiation in post-processing, and quantify the
impact of AGN feedback on the ability of the gas to form stars. Surprisingly,
even though the AGN generates powerful outflows, the impact of AGN heating and
photo-ionization on instantaneous star formation is weak: the star formation
rate decreases by a few percent at most, even in a quasar regime
($L_{bol}=10^{46.5}$ erg s$^{-1}$). Furthermore, the reservoirs of atomic gas
that are expected to form stars on a 100 - 200 Myrs time scale are also
marginally affected. Therefore, while the AGN-driven outflows can remove
substantial amounts of gas in the long term, the impact of AGN feedback on the
star formation efficiency in the ISM of high-redshift galaxies is marginal,
even when long-range radiative effects are taken into account.
