According to Yu. I. Manin, mirror symmetry can be understood as a kind of duality between complex and symplectic geometries. In papers on mirror symmetry, the following realization of this duality was stated: Lagrangian submanifolds, or classes of Lagrangian submanifolds, must correspond to holomorphic vector bundles, since, unlike complex geometry, symplectic is not rigid. More than a quarter of a century ago, A. N. Tyurin, being a specialist in stable holomorphic vector bundles, laid the foundation for future work with Lagrangian submanifolds, namely, he, in a joint work with A. L. Gorodentsev, proposed varieties of moduli of Boron-Sommerfeld Lagrangian submanifolds. However, such moduli varieties were infinite-dimensional Fr茅chet-smooth real varieties, and in algebraic geometry the moduli varieties of stable vector bundles are always finite-dimensional. Developing the work of A. N. Tyurin, it turned out to be natural to introduce a suspension over his Bohr-Sommerfeld geometry, and such a suspension was called the special Bohr-Sommerfeld geometry. The main thing in this generalization is the possibility to define exactly finite-dimensional varieties of modules, and in known examples (of which, unfortunately, there are very few), such varieties of modules turn out to be algebraic varieties in themselves.
In this short course, we will present the details of the Bohr-Sommerfeld geometry and the basic constructions of its special version.
