  Here, we resume and broaden the results concerned which appeared in
math.AG/0101098 and math.AG/0104021. We start from summing up our example of a
complex algebraic surface which is not deformation equivalent to its complex
conjugate and which, moreover, has no homeomorphisms reversing the canonical
class. Then, we construct several series of higher dimensional compact complex
manifolds having the same property. We end with discussing applications to the
Dif=Def problems, to the existence of diffeomorphic plane cuspidal curves non
equivalent under equisingular deformations and to the existence of
(deformation) non equivalent symplectic structures with opposite canonical
classes.
