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For early-type galaxies, the ability to sustain a corona of hot, X-rayemitting gas could have played a key role in quenching their star-formationhistory. Yet, it is still unclear what drives the precise amount of hot gasaround these galaxies. By combining photometric and spectroscopic measurementsfor the early-type galaxies observed during the Atlas3D integral-field surveywith measurements of their X-ray luminosity based on X-ray data of both low andhigh spatial resolution we conclude that the hot-gas content of early-typegalaxies can depend on their dynamical structure. Specifically, whereas slowrotators generally have X-ray halos with luminosity L_X,gas and temperature Tvalues that are in line with what is expected if the hot-gas emission issustained by the thermalisaton of the kinetic energy carried by thestellar-mass loss material, fast rotators tend to display L_X,gas values thatfall consistently below the prediction of this model, with similar T valuesthat do not scale with the stellar kinetic energy as observed in the case ofslow rotators. Considering that fast rotators are likely to be intrinsicallyflatter than slow rotators, and that the few L_X,gas-deficient slow rotatorsalso happen to be relatively flat, the observed L_X,gas deficiency in theseobjects would support the hypothesis whereby flatter galaxies have a hardertime in retaining their hot gas. We discuss the implications that a differenthot-gas content could have on the fate of both acquired and internally-producedgaseous material, considering in particular how the L_X,gas deficiency of fastrotators would make them more capable to recycle the stellar-mass loss materialinto new stars than slow rotators. This is consistent with the finding thatmolecular gas and young stars are detected only in fast rotators in the Atlas3Dsample, and that fast rotators tend to dustier than slow rotators. [Abridged]

The Atlas3D project - XIX. The hot-gas content of early-type galaxies: fast versus slow rotators