Exact Lagrangian Submanifolds and the Moduli Space of Special Bohr–Sommerfeld Lagrangian Cycles

Abstract

In previous papers we introduced the notion of special Bohr–Sommerfeld Lagrangian cycles on a compact simply connected symplectic manifold with integer symplectic form, and presented the main interesting case: compact simply connected algebraic variety with an ample line bundle such that the space of Bohr–Sommerfeld Lagrangian cycles with respect to a compatible Kähler form of the Hodge type and holomorphic sections of the bundle is finite. The main problem appeared in this way is singular components of the corresponding Lagrangian shadows (or Weinstein skeletons) which are hard to distinguish or resolve. In this note we avoid this difficulty presenting the points of the moduli space of special Bohr–Sommerfeld Lagrangian cycles by exact compact Lagrangian submanifolds on the complements X\D_α modulo Hamiltonian isotopies, where D_α is the zero divisor of holomorphic section α. This correspondence is fair if the Eliashberg conjecture is true, stating that every smooth orientable exact Lagrangian submanifold is regular. In a sense our approach corresponds to the usage of gauge classes of Hermitian connections instead of pure holomorphic structures in the theory of the moduli space of (semi) stable vector bundles.