Stability of the normal vacuum in multi-Higgs-doublet models

We show that the vacuum structure of a generic multi-Higgs-doublet model shares several important features with the vacuum structure of the two and three Higgs-doublet model. In particular, one can still define the usual charge breaking, spontaneous CP breaking and normal (charge and CP preserving) stationary points. We analyse the possibility of charge or spontaneous CP breaking, by studying the relative depth of the potential in each of the possible stationary points.