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

In the previous papers we present a construction of the set ${\mathcal{U}}_{SBS}$ in the direct product ${\mathcal{B}}_S\times \mathbb{P}\mathrm{\Gamma }(M,L)$ of the moduli space of Bohr - Sommerfeld lagrangian submanifolds of fixed topological type and the projectivized space of smooth sections of the prequantization bundle $L\to M$ over a given compact simply connected symplectic manifold $M$. Canonical projections $p:{\mathcal{U}}_{SBS}\to \mathbb{P}\mathrm{\Gamma }(M,L)$ and $q: {\cal U}_{SBS} \to {\cal B}_S$ are studied in the present text: first, we show that the differential $\mathrm{d}p$ at a given point is an isomorphism, which implies that a natural complex structure can be defined on ${\mathcal{U}}_{SBS}$; second, the projection $q:{\mathcal{U}}_{SBS}\to {\mathcal{B}}_S$ splits as the combination ${\mathcal{U}}_{SBS}\to T{\mathcal{B}}_S\to {\mathcal{B}}_S$ such that the fibers of the first map are complex subsets in ${\mathcal{U}}_{SBS}$. This implies that an appropriate section of the first map should define a complex structure on $T{\mathcal{B}}_S$; therefore it can be seen as a complexification of the moduli space ${\mathcal{B}}_S$. The construction can be exploited in the Lagrangian approach to Geometric Quantization