  Feedback from AGN is thought to be key in shaping the life cycle of their
host galaxies by regulating star-formation activity. Therefore, to understand
the impact of AGN on star formation, it is essential to trace the molecular gas
out of which stars form. In this paper we present the first systematic study of
the CO properties of AGN hosts at z~2 for a sample of 27 X-ray selected AGN
spanning two orders of magnitude in AGN bolometric luminosity (Lbol=
10^44.7-10^46.9 erg/s) by using ALMA Band 3 observations of the CO(3-2)
transition (~1" angular resolution). To search for evidence of AGN feedback on
the CO properties of the host galaxies, we compared our AGN with a sample of
inactive (i.e., non-AGN) galaxies from the PHIBSS survey with similar redshift,
stellar masses, and SFRs. We used the same CO transition as a consistent proxy
for the gas mass for the two samples in order to avoid systematics involved
when assuming conversion factors. By adopting a Bayesian approach to take upper
limits into account, we analyzed CO luminosities as a function of stellar
masses and SFRs, as well as the ratio LCO(3-2)/M* (proxy for the gas fraction).
The two samples show statistically consistent trends in the LCO(3-2)-Lfir and
LCO(3-2)-M* planes. However, there are indications that AGN feature lower
CO(3-2) luminosities (0.4-0.7 dex) than inactive galaxies at the 2-3sigma level
when we focus on the subset of parameters where the results are better
constrained and on the distribution of the mean LCO(3-2)/M*. Therefore, even by
conservatively assuming the same excitation factor r31, we would find lower
molecular gas masses in AGN, and assuming higher r31 would exacerbate this
difference. We interpret our result as a hint of the potential effect of AGN
activity (e.g., radiation and outflows), which may be able to heat, excite,
dissociate, and/or deplete the gas reservoir of the host galaxies. (abridged)
