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Working paper

On a complexification of the moduli space of Bohr - Sommerfeld lagrangian cycles

In the previous papers we present a construction of the set U_SBS in the direct product B_S×PΓ(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:U_SBS→PΓ(M, L) and q:U_SBS→ B_S are studied in the present text: first, we show that the differential dp at a given point is an isomorphism, which implies that a natural complex structure can be defined on U_SBS; second, the projection q:U_SBS→ B_S splits as the combination U_SBS→ TB_S→ B_S such that the fibers of the first map are complex subsets in U_SBS. This implies that an appropriate section of the first map should define a complex structure on TB_S; therefore it can be seen as a complexification of the moduli space B_S. The construction can be exploited in the Lagrangian approach to Geometric Quantization.