2018/07/30 by Nikolai Andreevich Tyurin, Tyurin, Nikolai A.
Mathematics · #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1807.11351
openalex publication_date 2018/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the previous papers we present a construction of the set \cal USBS in the direct product \cal BS × ℙ Γ(M, L) of the moduli space of Bohr - Sommerfeld lagrangian submanifolds of fixed topological type and the projectivized space of smooth sections of the prequantization bundle L → M over a given compact simply connected symplectic manifold M. Canonical projections p: \cal USBS → ℙ Γ(M, L) and q: \cal USBS → \cal BS are studied in the present text: first, we show that the differential \rm d p at a given point is an isomorphism, which implies that a natural complex structure can be defined on \cal USBS; second, the projection q: \cal USBS → \cal BS splits as the combination \cal USBS → T \cal BS → \cal BS such that the fibers of the first map are complex subsets in \cal USBS. This implies that an appropriate section of the first map should define a complex structure on T \cal BS; therefore it can be seen as a complexification of the moduli space \cal BS. The construction can be exploited in the Lagrangian approach to Geometric Quantization