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On the Kählerization of the moduli space of Bohr-Sommerfeld Lagrangian submanifolds. (English. Russian original) Zbl 1443.53046
Math. Notes 107, No. 6, 1038-1039 (2020); translation from Mat. Zametki 107, No. 6, 945-947 (2020).
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 $$p^{\ast}\Omega _{FS}$$.
##### MSC:
 53D05 Symplectic manifolds, general 53D50 Geometric quantization
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##### References:
 [1] Gorodentsev, A. L.; Tyurin, A. N., Izv. Ross. Akad. Nauk Ser. Mat., 65, 3, 15 (2001) [2] A. Tyurin, Complexification of Bohr-Sommerfeld Condition, arXiv: math/9909094 (1999). [3] Tyurin, N. A., Izv. Ross. Akad. Nauk Ser. Mat., 80, 6, 274 (2016)
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