  Neutrinos from a supernova (SN) might undergo fast flavor conversions near
the collapsed stellar core. We perform a detailed study of this intriguing
possibility, analyzing time-dependent state-of-the-art 3D SN models of 9 and 20
Msun. Both models were computed with multi-D three-flavor neutrino transport
based on a two-moment solver, and both exhibit the presence of the
lepton-number emission self-sustained asymmetry (LESA). The transport solution
does not provide the angular distributions of the neutrino fluxes, which are
crucial to track the fast flavor instability. To overcome this limitation, we
use a recently proposed approach based on the angular moments of the
energy-integrated electron lepton-number distribution. With this method we find
the possibility of fast neutrino flavor instability at radii <~20 km, which is
well interior to the neutrinosphere. Our results confirm recent observations in
a 2D SN model and in 2D/3D models with fixed matter background, which were
computed with Boltzmann neutrino transport. However, the flavor unstable
locations are not isolated points as discussed previously, but thin skins
surrounding volumes where electron antineutrinos are more abundant than
electron neutrinos. These volumes grow with time and appear first in the
convective layer of the proto-neutron star (PNS), where a decreasing electron
fraction (Ye) and high temperatures favor the occurrence of regions with
negative neutrino chemical potential. Since Ye remains higher in the LESA
dipole direction, where convective lepton-number transport out from the
nonconvective PNS core slows down the deleptonization, flavor unstable
conditions become more widespread in the opposite hemisphere. This interesting
phenomenon deserves further investigation, since its impact on SN modeling and
possible consequences for SN dynamics and neutrino observations are presently
unclear. (abridged)
