Towards the moduli space of special Bohr - Sommerfeld lagrangian cycles

TL;DR

This paper explores the moduli space of special Bohr-Sommerfeld Lagrangian cycles on algebraic varieties, proposing a new approach to handle singularities by using exact Lagrangian submanifolds modulo Hamiltonian isotopies.

Contribution

It introduces a novel method to parametrize the moduli space via exact Lagrangian submanifolds, avoiding singularity issues present in previous approaches.

Findings

Moduli space points correspond to exact Lagrangian submanifolds on complements of divisors.

The approach uses gauge classes of hermitian connections instead of holomorphic structures.

Provides a framework to distinguish and resolve singular components.

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 Kahler 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 sceletons of the corresponding Weinstein domains) which are hard to distinguish or resolve. In the present text 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 $X \backslash D_{\alpha}$ modulo Hamiltonian isotopies, where $D_{\alpha}$ is the zero divisor of holomorphic section $\alpha$. In a sense it 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.