Multi-phase filamentary structures around Brightest Cluster Galaxies ( BCG ) are likely a key step of AGN-feedback .
We observed molecular gas in 3 cool cluster cores : Centaurus , Abell S1101 , and RXJ1539.5 and gathered ALMA ( Atacama Large Millimeter/submillimeter Array ) and MUSE ( Multi Unit Spectroscopic Explorer ) data for 12 other clusters .
Those observations show clumpy , massive and long , 3–25 kpc , molecular filaments , preferentially located around the radio bubbles inflated by the AGN ( Active Galactic Nucleus ) .
Two objects show nuclear molecular disks .
The optical nebula is certainly tracing the warm envelopes of cold molecular filaments .
Surprisingly , the radial profile of the H \alpha /CO flux ratio is roughly constant for most of the objects , suggesting that ( i ) between 1.2 to 7 times more cold gas could be present and ( ii ) local processes must be responsible for the excitation .

Projected velocities are between 100–400 km s ^ { -1 } , with disturbed kinematics and sometimes coherent gradients .
This is likely due to the mixing in projection of several thin ( as yet ) unresolved filaments .
The velocity fields may be stirred by turbulence induced by bubbles , jets or merger-induced sloshing .
Velocity and dispersions are low , below the escape velocity .
Cold clouds should eventually fall back and fuel the AGN .

We compare the filament ’ s radial extent , r _ { fil } , with the region where the X-ray gas can become thermally unstable .
The filaments are always inside the low-entropy and short cooling time region , where t _ { cool } /t _ { ff } ¡20 ( 9 of 13 sources ) .
The range t _ { cool } /t _ { ff } , 8-23 at r _ { fil } , is likely due to ( i ) a more complex gravitational potential affecting the free-fall time t _ { ff } ( sloshing , mergers… ) ; ( ii ) the presence of inhomogeneities or uplifted gas in the ICM , affecting the cooling time t _ { cool } .
For some of the sources , r _ { fil } lies where the ratio of the cooling time to the eddy-turnover time , t _ { cool } /t _ { eddy } , is approximately unity .