  We consider the minimal supersymmetric triplet seesaw model as the origin of
neutrino masses and mixing as well as of the baryon asymmetry of the Universe,
which is generated through soft leptogenesis employing a CP violating phase and
a resonant behavior in the supersymmetry breaking sector. We calculate the full
gauge--annihilation cross section for the Higgs triplets, including all
relevant supersymmetric intermediate and final states, as well as
coannihilations with the fermionic superpartners of the triplets. We find that
these gauge annihilation processes strongly suppress the resulting lepton
asymmetry. As a consequence of this, successful leptogenesis can occur only for
a triplet mass at the TeV scale, where the contribution of soft supersymmetry
breaking terms enhances the CP and lepton asymmetry. This opens up an
interesting opportunity for testing the model in future colliders.
