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We present ALMA observations of the CO(6-5) and [CII] emission lines and the sub-millimeter continuum of the z ∼ 6 quasi-stellar object (QSO) SDSS J231038.88+185519.7. Compared to previous studies, we have analyzed a synthetic beam that is ten times smaller in angular size, we have achieved ten times better sensitivity in the CO(6-5) line, and two and half times better sensitivity in the [CII] line, enabling us to resolve the molecular gas emission. We obtain a size of the dense molecular gas of 2.9 ± 0.5 kpc, and of 1.4 ± 0.2 kpc for the 91.5 GHz dust continuum. By assuming that CO(6-5) is thermalized, and by adopting a CO to H2 conversion factor αCO = 0.8 M⊙K-1 (km s)-1 pc2, we infer a molecular gas mass of M(H2) = (3.2±0.2)×1010 M⊙. Assuming that the observed CO velocity gradient is due to an inclined rotating disk, we derive a dynamical mass of Mdynsin2(i)=(2.4 ± 0.5)×1010 M⊙, which is a factor of approximately two smaller than the previously reported estimate based on [CII]. Regarding the central black hole, we provide a new estimate of the black hole mass based on the C IV emission line detected in the VLT/X-shooter spectrum: MBH (1.8 ± 0.5)×109 M⊙. We find a molecular gas fraction of μ = M(H2)/M∗ ∼ 4.4, where M∗ ≈ Mdyn - M(H2) - M(BH). We derive a ratio νrot/σ≈1-2 suggesting high gas turbulence, outflows/inflows and/or complex kinematics due to a merger event. We estimate a global Toomre parameter Q ∼ 0.2- 0.5, indicating likely cloud fragmentation. We compare, at the same angular resolution, the CO(6-5) and [CII] distributions, finding that dense molecular gas is more centrally concentrated with respect to [CII]. We find that the current BH growth rate is similar to that of its host galaxy.

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