We have carried out a survey for 12CO J=1-0 and J=2-1 emission in the 260 early-type galaxies of the volume-limited Atlas3D sample, with the goal of connecting their star formation and assembly histories to their cold gas content. This is the largest volume-limited CO survey of its kind and is the first to include many Virgo Cluster members. Sample members are dynamically hot galaxies with a median stellar mass 3times 10^{10} Msun; they are selected by morphology rather than colour, and the bulk of them lie on the red sequence. The overall CO detection rate is 56/259 = 0.22 error 0.03, with no dependence on K luminosity and only a modest dependence on dynamical mass. There are a dozen CO detections among the Virgo Cluster members; statistical analysis of their H_2 mass distributions and their dynamical status within the cluster shows that the clusters influence on their molecular masses is subtle at best, even though (unlike spirals) they seem to be virialized within the cluster. We suggest that the cluster members have retained their molecular gas through several Gyr residences in the cluster. There are also a few extremely CO-rich early-type galaxies with H_2 masses >= 10^9 Msun, and these are in low density environments. We do find a significant trend between molecular content and the stellar specific angular momentum. The galaxies of low angular momentum also have low CO detection rates, suggesting that their formation processes were more effective at destroying molecular gas or preventing its re-accretion. We speculate on the implications of these data for the formation of various sub-classes of early-type galaxies.
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