Alladi Ramakrishnan Hall

Special Bohr - Sommerfeld geometry

Nikolai Tyurin

BLTPh, JINR (Dubna) and HSE (Moscow)

One of the favorite ideas of Andrey Tyurin suggests certain duality
between stable vector bundles and lagrangian submanifolds. But the moduli
space of stable vector bundles are finite while the space of Lagrangian
deformations is infinite. To make the situation finite one imposes certain
speciality conditions - f.e. SpLag geometry due to N.Hitchin concerns the
case of Calabi - Yau varieties. In 1999 A. Tyurin in the joint paper with
Alexei Gorodentsev (firstly appeared as a MPIM -prepirnt) constructed the
moduli space of Bohr - Sommerfeld lagrangian cycles. These moduli spaces
have been exploited in Geometric Quantization problem, what we discuss at
the 3d lecture. But for these lagrangian cycles
it is possible to define certain new speciality condition with respect to
sections of the prequantization bundle (L, a). 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 (if
time permits) we get certain geometric correspondence between divisors and
lagrangian cycles in algebraic varieties.

The talk is based on the papers arXiv:1508.06804
and arXiv:1601.05974