Semiregularity and obstructions of complete intersections

We prove that, on a smooth projective variety over an algebraically closed field of characteristic 0, the semiregularity map annihilates every obstruction to embedded deformations of a local complete intersection subvariety with extendable normal bundle. The proof is based on the theory of L∞L∞-algebras and Tamarkin–Tsygan calculus on the de Rham complex of DG-schemes.