Rigidity of Lagrangian embeddings into symplectic tori and K3 surfaces

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