Lagrangian tori in the projective plane

We extend the discussion of the homological mirror symmetry for toric manifolds to the more general case of monotonic symplectic manifolds with real polarizations. We claim that the Hori–Vafa conjecture, proved for toric Fano varieties, can be verified in a much wider context. Then the Bohr–Sommerfeld notion regarding the canonical class Lagrangian submanifold appears and plays an important role. A bridge is thus manifested between the geometric quantization and homological mirror symmetry programs for the projective plane in terms of its Lagrangian geometry. This allows using standard facts from the theory of geometric quantization to obtain some results in the framework of the theory of homological mirror symmetry.