Role of dense matter in collective supernova neutrino transformations

7 pages, 2 figuresFor neutrinos streaming from a supernova (SN) core, dense matter suppresses self-induced flavor transformations if the electron density n_e significantly exceeds the neutrino density n_nu in the conversion region. If n_e is comparable to n_nu one finds multi-angle decoherence, whereas the standard self-induced transformation behavior requires that in the transformation region n_nu is safely above n_e. This condition need not be satisfied in the early phase after SN core bounce. Our new multi-angle effect is a subtle consequence of neutrinos traveling on different trajectories when streaming from a source that is not point-like