Special Bohr - Sommerfeld geometry (DEDICATED TO THE MEMORY OF ANDREY TYURIN)

One of the favorite ideas of Andrey Tyurin suggests certain duality between stable vector bundles and lagrangian submanifolds. But the moduli spaces of stable vector bundles are finite dimensional, while the spaces of lagrangian deformations are infinite dimensional. To make the situation finite one imposes certain speciality conditions - e.g. SpLag geometry defined N.Hitchin treatss the case of Calabi - Yau varieties. In 1999 A. Tyurin in the joint paper with Alexei Gorodentsev (first appeared as a MPIM -prepirnt) constructed the moduli space of Bohr - Sommerfeld lagrangian cycles. For these lagrangian cycles it is possible to impose new speciality condition with respect to sections of the prequantization bundle. For algebraic varieties it leads to the definition of finite dimensional moduli space of special Bohr - Sommerfeld lagrangian cycles. The theory is closely related to the theory of Kahler potential, plurisubharmonic functions etc. At the end we get certain geometric correspondence between divisors and lagrangian cycles in algebraic varieties. The talk is based on paper arXiv:1508.06804