Modified moduli space of special Bohr - Sommerfeld lagrangian submanifolds

In 1999 A. Tyurin and A. Gorodentsev presented a new program devoted to lagrangian geometry of algebraic varieties, which was called ALAG and based on Bohr - Sommerfeld lagrangian geometry. As the main result they constructed the moduli space of Bohr - Sommerfeld lagrangian cycles,which is infinite dimensional Frechet smooth variety. At the same time they expected that there is certain natural action on the moduli space and then it is possible to factorize the space with respect to this action, and it should lead to a finite dimensional moduli space. It would be a construction transversl in certain sense to the SpLAG programme of N. Hitchin.

Almost twenty years later I would like to present another elaboration of the Tyurin - Gorodentsev construction called Special Bohr - Sommerfeld geometry.In the talk one presents the construction of certain finite dimensional moduli space consisting of pairs (divisor from very ample complete linear system, class of D- exact smooth lagrangian submanifolds) which exists for any smooth compact simply connected algebraic variety, proves that this moduli space is Kahler manifold. If time permits, I will explain how this moduli space is related to pure Special Bohr - Sommerfeld geometry.