Special Bohr–Sommerfeld Lagrangian submanifolds

We introduce a new notion in symplectic geometry, that of speciality for Lagrangian submanifolds satisfying the Bohr–Sommerfeld condition. We show that it enables one to construct finite-dimensional moduli spaces of special Bohr–Sommerfeld Lagrangian submanifolds with respect to any ample line bundle on an algebraic variety with a Hodge metric regarded as the symplectic form. This construction can be used to study mirror symmetry.