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Let be the inclusion map of a compact reduced subspace of a complex manifold, and let be a subsheaf of the tangent bundle which is closed under the Lie bracket, but not necessarily a sheaf of -algebras. This paper discusses criteria to guarantee that infinitesimal deformations of which are induced by lift to positive-dimensional deformations of , where is deformed “along the sheaf ”.
In case where is complex-symplectic and the sheaf of locally Hamiltonian vector fields, this partially reproduces known results on unobstructedness of deformations of Lagrangian submanifolds. The proof is rather elementary and geometric, constructing higher-order liftings of a given infinitesimal deformation using flow maps of carefully crafted time-dependent vector fields.