Deformation along subsheaves, II

Let $f \\colon Y \\to X$ be the inclusion map of a compact reduced subspace of a complex manifold, and let $\\mathcal{F} \\subseteq T_X$ be a subsheaf of the tangent bundle which is closed under the Lie bracket, but not necessarily a sheaf of $\\mathcal{O}\_X$-algebras. This paper discusses criteria to guarantee that infinitesimal deformations of $f$ which are induced by $\\mathcal{F}$ lift to positive-dimensional deformations of $f$, where $f$ is deformed “along the sheaf $\\mathcal{F}$”. In case where $X$ is complex-symplectic and $\\mathcal{F}$ 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.