Dolbeault dga and L ∞ -algebroid of the formal neighborhood

Abstract We continue the study the Dolbeault dga of the formal neighborhood of an arbitrary closed embedding of complex manifolds previously defined by the author in [14] . The special case of the diagonal embedding has been analyzed in [13] . We describe here the Dolbeault dga of a general embedding explicitly in terms of the formal differential geometry of the embedding. Moreover, we show that the Dolbeault dga is the completed Chevalley–Eilenberg dga of an L ∞ -algebroid structure on the shifted normal bundle of the submanifold. This generalizes the result of Kapranov on the diagonal embedding and Atiyah class.