Lagrangian tori in the projective plane (Q735675)

Among several unknowns in a general study of Lagrangian tori, the Hori-Vafa conjecture was proved under certain assumptions in [\textit{C.-H. Cho, Y.-G. Oh}, Asian J. Math. 10, No.~4, 773--814 (2006; Zbl 1130.53055)]. The author's motivation are also some problems raised by himself in [ArXiv:math/9902027v1 (1999)], in order to relate mirror symmetry and geometric quantization. Related to these problems, the author states a series of five very interesting new conjectures. He proves the first one in the case of \(\mathbb CP^2\): If a smooth Lagrangian fiber is displaceable, then it is not Bohr-Sommerfield with respect to the canonical class, and if this fiber is Bohr-Sommerfield with respect to the canonical class, then it is also monotonic.