  The present paper is devoted to the relations between Deligne's conjecture on
critical values of motivic $L$-functions and the multiplicative relations
between periods of arithmetically normalized automorphic forms on unitary
groups. In the first place, we combine the Ichino--Ikeda--Neal-Harris (IINH)
formula -- which is now a theorem -- with an analysis of cup products of
coherent cohomological automorphic forms on Shimura varieties to establish
relations between certain automorphic periods and critical values of
Rankin-Selberg and Asai $L$-functions of ${\rm GL}(n)\times{\rm GL}(m)$ over CM
fields. By reinterpreting these critical values in terms of automorphic periods
of holomorphic automorphic forms on unitary groups, we show that the
automorphic periods of holomorphic forms can be factored as products of
coherent cohomological forms, compatibly with a motivic factorization predicted
by the Tate conjecture. All of these results are conditional on a conjecture on
non-vanishing of twists of automorphic $L$-functions of ${\rm GL}(n)$ by
anticyclotomic characters of finite order, and are stated under a certain
regularity condition.
