Rigidity of Lagrangian embeddings into symplectic tori and K3 surfaces

A Kähler-type form is a symplectic form compatible with an integrable complex structure. Let be either a torus or a K3-surface equipped with a Kähler-type form. We show that the homology class of any Maslov-zero Lagrangian torus in has to be nonzero and primitive. This extends previous results of Abouzaid and Smith (for tori) and Sheridan and Smith (for K3-surfaces) who proved it for particular Kähler-type forms on. In the K3 case, our proof uses dynamical properties of the action of the diffeomorphism group of on the space of the Kähler-type forms. These properties are obtained using Shah's arithmetic version of Ratner's orbit closure theorem.