  Conditions for CP violation in the scalar potential sector of general
N-Higgs-doublet models (NHDMs) are analyzed from a group theoretical
perspective. For the simplest two-Higgs-doublet model (2HDM) potential, a
minimum set of conditions for explicit and spontaneous CP violation is
presented. The conditions can be given a clear geometrical interpretation in
terms of quantities in the adjoint representation of the basis transformation
group for the two doublets. Such conditions depend on CP-odd pseudoscalar
invariants. When the potential is CP invariant, the explicit procedure to reach
the real CP-basis and the explicit CP transformation can also be obtained. The
procedure to find the real basis and the conditions for CP violation are then
extended to general NHDM potentials. The analysis becomes more involved and
only a formal procedure to reach the real basis is found. Necessary conditions
for CP invariance can still be formulated in terms of group invariants: the
CP-odd generalized pseudoscalars. The problem can be completely solved for
three Higgs-doublets.
