Abelian CM Varieties with Q(i)-multiplication of Type (2,1) and Singular Moduli

It is well-known that principally polarized abelian varieties with Q(i)- multiplication of type (2,1) are parametrized by a complex 2-dimensional ball. In this note we show that set of ball points parametrizing CM varieties coincides with the set of cubic Q(i)-singular moduli, i.e. with the set of fixed points of the unitary group U((2,1), Q(i)) acting on the ball which generate cubic extensions of the Gauss number field.