  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.
