Smallness of Faltings heights of CM abelian varieties

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