  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.
