Lie description of higher obstructions to deforming submanifolds

To every morphism $χ\colon L\to M$ of differential graded Lie algebras we associate a functors of artin rings $\Def_χ$ whose tangent and obstruction spaces are respectively the first and second cohomology group of the cylinder of $χ$. 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.

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