The ATLAS3D Project -- XXIII. Angular momentum and nuclear surface brightness profiles

[Abridged and Edited] We investigate nuclear light profiles in 135 ATLAS3D galaxies for which the Hubble Space Telescope (HST) imaging is available and compare them to the large scale kinematics obtained with the SAURON integral-field spectrograph. Specific angular momentum, lambda_R, correlates with the shape of nuclear light profiles, where cores are typically found in slow rotators and core-less galaxies are fast rotators. Cores are found only in massive galaxies and only in systems with the stellar mass M>8x10^10 Msun. Based on our sample, we, however, see no evidence for a bimodal distribution of nuclear slopes. The best predictor for finding a core is based on the stellar velocity dispersion within an effective radius, sigma_e, and specific angular momentum, where cores are found for lambda_R<0.25 and sigma_e>160 km/s. We estimate that only about 10% of nearby early-type galaxies contain cores. Furthermore, we show that there is a genuine population of fast rotators with cores. We also show that core fast rotators are morphologically, kinematically and dynamically different from core slow rotators. The cores of fast rotators could harbour black holes of similar masses to those in core slow rotators, but typically more massive than those found in core-less fast rotators. Core-less galaxies, and especially core-less fast rotators, are under-luminous in the diffuse X-ray emission, but the presence of a core does not imply high X-ray luminosities. We postulate a possible population of core-less galaxies among slow rotators, which can not be explained as face-on discs, but comprise a genuine sub-population of slow rotators. These galaxies are typically less massive and flatter than core slow rotators, and show evidence for dynamical cold structures and exponential photometric components. We discuss possible processes for formation of cores and their subsequent preservation.