Lie description of higher obstructions to deforming submanifolds

To every morphism $\chi \text{\hspace{0.17em}⁣}:L\to M$ of differential graded Lie algebras we associate a functors of artin rings ${\text{Def}}_{\chi }$ whose tangent and obstruction spaces are respectively the first and second cohomology group of the cylinder of $\chi$. Such construction applies to Hilbert and Brill-Noether functors and allow to prove with ease that every higher obstruction to deforming a smooth submanifold of a Kaehler manifold is annihilated by the semiregularity map.