Hodge cycles and quadratic relations between holomorphic periods on CM abelian varieties

In this paper, we prove the following result advocating the importance of monomial quadratic relations between holomorphic CM periods. For any simple CM abelian variety $A$, we can construct a CM abelian variety $B$ such that all non-trivial Hodge relations between the holomorphic periods of the product $A\times B$ are generated by monomial quadratic ones which are also explicit. Moreover, $B$ splits over the Galois closure of the CM field associated with $A$.