Obscured AGN represent a significant fraction of the entire AGN population, especially at high redshift (~70% at z=3--5). They are often characterized by the presence of large gas and dust reservoirs that are thought to sustain and possibly obscure vigorous star formation processes that make these objects shine at far-IR and sub-mm wavelengths. We exploit ALMA Cycle 4 observations of the continuum (~2.1mm) and high-J CO emission of a sample of six X-ray selected SMGs hosting an obscured AGN at z_spec>2.5 in the 7 Ms CDF-S. We measured the masses and sizes of the dust and molecular gas and we derived the gas density and column density on the basis of a uniform sphere geometry. Finally, we compared the measured column densities with those derived from the Chandra X-ray spectra. We detected both the continuum and line emission for three sources for which we measured both the flux density and size. For the undetected sources, we derived an upper limit on the flux density. We found that the detected galaxies are rich in gas and dust (molecular gas mass in the range <0.5 - 2.7 x 10^10 M_sun for {alpha}_CO=0.8 and up to ~2 x 10^11~M_sun for {alpha}_CO=6.5, and dust mass <0.9 - 4.9 x 10^8 M_sun) and compact (gas major axis 2.1-3.0 kpc, dust major axis 1.4-2.7 kpc). The column densities associated with the ISM are on the order of 10^(23-24) cm-2, which is comparable with those derived from the X-ray spectra. For the detected sources we also derived dynamical masses in the range 0.8 - 3.7 x 10^10 M_sun. We conclude that the ISM of high redshift galaxies can substantially contribute to nuclear obscuration up to the Compton-thick (>10^24 cm-2) regime. In addition, we found that all the detected sources show a velocity gradient reminding one rotating system, even though two of them show peculiar features in their morphology that can be associated with a chaotic, possibly merging, structure.
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