The semiregularity conjecture predicts that the semiregularity map, defined by Severi, Kodaira-Spencer and Bloch, annihilates every obstructions to embedded deformations of subvarieties. This conjecture had been partially proved by Bloch and more efforts toward it had been done by Ran, Buchweitz-Flenner, Manetti, Pridham and others. Recently, with an additional assumption, Iacono-Manetti proved this conjecture. Their method is to use DGLA to control deformations, which is known to Deligne, Drinfeld, Konsevich, Lurie et al.

Using Chern Character, we reconstruct Bloch’s semiregularity map. Combining with the theory of functor of Artin rings by Schlessinger and Fantechi-Manetti, we prove the semiregularity conjecture.