On families of Lagrangian submanifold

We study the stability of a compact Lagrangian submanifold of a symplectic manifold under perturbation of the symplectic structure. If X is a compact manifold and the ωt are cohomologous symplectic forms on X, then by a well-known theorem of Moser there exists a family φt of diffeomorphisms of X such that ωt = φ*t(ω0). If L ⊂ X is a Lagrangian submanifold for (X, ω0), Lt φt-1 (L) is thus a Lagrangian submanifold for (X, ωt). Here we show that if we simply assume that L is compact and ωt|L is exact for every t, a family Lt as above still exists, for sufficiently small t. Similar results are proved concerning the stability of special Lagrangian and Bohr-Sommerfeld special Lagrangian submanifolds, under perturbation of the ambient Calabi-Yau structure.