The moduli space of $D$-exact Lagrangian submanifolds

This paper studies the Lagrangian geometry of algebraic varieties. Given a smooth compact simply-connected algebraic variety, we construct a family of finite-dimensional Kähler manifolds whose elements are the equivalence classes of Lagrangian submanifolds satisfying our new $D$-exactness condition. In connection with the theory of Weinstein structures, these moduli spaces turn out related to the special Bohr–Sommerfeld geometry constructed by the author previously. This enables us to extract from the moduli spaces some stable components and conjecture that they are not only Kähler but also algebraic.