K3 surfaces with real or complex multiplication

Abstract

Let $E$ be a totally real number field of degree $d$ and let $m \geqslant 3$ be an integer. We show that if $md \leqslant 21$ then there exists an $(m-2)$-dimensional family of complex projective $K3$ surfaces with real multiplication by $E$. Analogous results are proved for CM number fields and also for all known higher-dimensional hyperk\"ahler manifolds.