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Let Y be a compact reduced subspace of a complex manifold X, and let F be asubsheaf of the tangent bundle T_X which is closed under the Lie bracket. Thispaper discusses criteria to guarantee that infinitesimal deformations of theinclusion morphism Y -> X give rise to positive-dimensional deformationfamilies, deforming the inclusion map "along the sheaf F". In case where X iscomplex-symplectic and F is the sheaf of Hamiltonian vector fields, thispartially reproduces known results on unobstructedness of deformations ofLagrangian submanifolds.  Written for the IMPANGA Lecture Notes series, this paper aims at simplicityand clarity of argument. It does not strive to present the shortest proofs ormost general results available. The proof is rather elementary and geometric,constructing higher-order liftings of a given infinitesimal deformation usingflow maps of carefully crafted time-dependent vector fields.

Deformation along subsheaves, II