We investigate if the active galactic nucleus (AGN) of Mrk 590, whose supermassive black hole was until recently highly accreting, is turning off due to a lack of central gas to fuel it. We analyse new sub-arcsecond resolution ALMA maps of the $^{12}$CO(3-2) line and 344 GHz continuum emission in Mrk 590. We detect no $^{12}$CO(3-2) emission in the inner 150 pc, constraining the central molecular gas mass to $M({rm H_2}) lesssim 1.6 times 10^5, {M_{odot}}$, no more than a typical giant molecular gas cloud, for a CO luminosity to gas mass conversion factor of $alpha_{rm CO}sim 0.8,{M_{odot},rm (K ,km,s^{-1},pc^{2}})^{-1}$. However, there is still potentially enough gas to fuel the black hole for another $2.6 times 10^5$ years assuming Eddington-limited accretion. We therefore cannot rule out that the AGN may just be experiencing a temporary feeding break, and may turn on again in the near future. We discover a ring-like structure at a radius of $sim 1$ kpc, where a gas clump exhibiting disturbed kinematics and located just $sim 200$ pc west of the AGN, may be refueling the centre. Mrk 590 does not have significantly less gas than other nearby AGN host galaxies at kpc scales, confirming that gas reservoirs at these scales provide no direct indication of on-going AGN activity and accretion rates. Continuum emission detected in the central 150 pc likely originates from warm AGN-heated dust, although contributions from synchrotron and free-free emission cannot be ruled out.
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