K3 surfaces with algebraic period ratios have complex multiplication

Let Omega be a non-zero holomorphic 2-form on a K3 surface S. Suppose that S is projective algebraic and is defined over (Q) over bar. Let P be the (Q) over bar -vector space generated by the numbers given by all the periods integral(gamma) Omega, gamma is an element of H-2 (S, Z). We show that, if dim((Q) over bar) P = 1, then S has complex multiplication, meaning that the Mumford-Tate group of the rational Hodge structure on H-2(S, Q) is abelian. This result was announced in [P. Tretkoff, Transcendence and CM on Borcea-Voisin towers of Calabi-Yau manifolds, J. Number Theory 152 (2015) 118-155], without a detailed proof. The converse is already well known.