Special Bohr-Sommerfeld geometry

This survey sums up a cycle of papers devoted to the construction of finite-dimensional moduli spaces points in which are certain special Lagrangian submanifolds of compact complex simply connected algebraic varieties. The starting point for this construction was the idea, due to A. Tyurin, to treat Largrangian submanifolds (or equivalence classes of such submanifolds) as mirror counterparts of stable vector bundles. Our constructions are based on the programme of abelian Lagrangian algebraic geometry developed by A. Tyurin and Gorodentsev 25 years ago. Since this programme was in its turn based on the Bohr-Sommerfeld Lagrangian geometry known in geometric quantization, we call our construction special Bohr-Sommerfeld geometry. The definitions arising in the course of work turn out to be closely connected with the theory of Weinstein domains, Eliashberg's conjectures, and many other concepts in symplectic geometry. The core conjecture that arose in our work and is confirmed by the available examples states that each moduli space of this type is in its turn an algebraic variety. Bibliography: 13 titles.