On the Kählerization of the moduli space of Bohr-Sommerfeld Lagrangian submanifolds (Q2192177)

Fix a compact simply connected symplectic manifold \((M^{2n}, \omega )\) with integer symplectic form \([\omega ]\in H^2(M, \mathbb{Z})\subset H^2(M, \mathbb{R})\). Consider also the prequantization data \((L, a)\) with a complex line bundle \(L\rightarrow M\) with fixed Hermitian structure \(h\) and a Hermitian connection \(a\in \mathcal{A}_h(L)\) with curvature \(2\pi i\omega \). Previously, the present author considered the moduli space \(\mathcal{B}_S\) of Bohr-Sommerfeld Lagrangian submanifolds of fixed topological type. In the paper under review, the notion of a special Bohr-Sommerfeld Lagrangian submanifold \(S\subset M\) is introduced and it results the subspace \(\mathcal{U}_{SBS}\subset \mathbb{P}\Gamma (M, L)\times \mathcal{B}_S\) on which there exists a weak Kähler form \(p^{\ast}\Omega _{FS}\) where \(p\) is the projection onto the first factor above and \(\Omega _{FS}\) the standard Fubini-Study Kähler form. The main result of this note is that a suitable subset of \(\mathcal{U}_{SBS}\) is Kähler with respect to \(p^{\ast}\Omega _{FS}\).