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A Kähler-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 Kähler-type form. We show that the homology class of any Maslov-zero Lagrangian torus in   $M$   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   $M$  . In the K3 case, our proof uses dynamical properties of the action of the diffeomorphism group of   $M$   on the space of the Kähler-type forms. These properties are obtained using Shah’s arithmetic version of Ratner’s orbit closure theorem.

Rigidity of Lagrangian embeddings into symplectic tori and K3 surfaces