Algebraic real multiplication on K3 surfaces of Picard number at least eight, and the Hodge conjecture for their powers and Hilbert schemes

For a complex projective K3 surface S whose transcendental Hodge structure T has real multiplication by a totally real field E, the Hodge conjecture for the powers of S comes down to the action of E on T and to the E-determinant classes on S^m, where m = dim_E T. Following Varesco, the Hodge conjecture for S × S implies it for S^n with n < m; using the decomposition of de Cataldo and Migliorini we show that it also implies it for every product of Hilbert schemes S^[n_1] × … × S^[n_k] with δ(n_1) + … + δ(n_k) < m, where δ(n) is the largest number of distinct parts of a partition of n. For E = Q the hypothesis holds, and for Picard number one this covers S^[n] for n ≤ 230. The determinant classes and E are algebraic if the Kuga–Satake correspondence is, as Floccari's work shows for E = Q. For E = Q(√5) we transport, by Varesco's method, the real multiplication of an eight-dimensional family from earlier work, using Buskin's theorem and a density argument: it is induced by algebraic cycles on every projective K3 surface of Picard number at least eight with Q(√5) among its Hodge endomorphisms, and for Picard numbers 6, 4 and 2 under explicit local conditions, which hold for the K3 surfaces with an automorphism of order five and their deformations. The same method gives algebraic real multiplication by the maximal real subfields of Q(ζ_9) and Q(ζ_11) from two families of van Geemen and Schütt, and by Q(√5) on hyper-Kähler varieties of K3^[n]-type, on cubic fourfolds, including an eight-dimensional family, the largest possible, and on Gushel–Mukai varieties with enough algebraic classes. The results for Q(√5) rest on the earlier construction, which has not been refereed and whose main theorem rests in part on computer calculations. Code and data are at github.com/sallerk/HC-k3-rm-powers.