On the Kählerization of the moduli space of Bohr-Sommerfeld Lagrangian submanifolds. (English. Russian original) Zbl 1443.53046

Tyurin, N. A.

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 \(p^{\ast}\Omega _{FS}\).

Reviewer: Mircea Crâşmăreanu (Iaşi)

MSC:

53D05	Symplectic manifolds, general
53D50	Geometric quantization

Keywords:

Bohr-Sommerfeld Lagrangian submanifold; Kähler structure

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)