  We prove that assuming the Colmez conjecture and the ``no Siegel zeros"
conjecture, the stable Faltings height of a CM abelian variety over a number
field is less than or equal to the logarithm of the root discriminant of the
field of definition of the abelian variety times an effective constant
depending only on the dimension of the abelian variety. In view of the fact
that the Colmez conjecture for abelian CM fields, the averaged Colmez
conjecture, and the ``no Siegel zeros" conjecture for CM fields with no complex
quadratic subfields are already proved, we prove unconditional analogues of the
result above. In addition, we also prove that the logarithm of the root
discriminant of the field of everywhere good reduction of CM abelian varieties
can be ``small".
