The dolbeault dga of a formal neighborhood

Inspired by a work of Kapranov (1999), we define the notion of a Dolbeault complex of the formal neighborhood of a closed embedding of complex manifolds. This construction allows us to study coherent sheaves over the formal neighborhood via a complex analytic approach, as in the case of usual complex manifolds and their Dolbeault complexes. Moreover, the Dolbeault complex as a differential graded algebra can be associated with a dg-category according to Block (2010). We show that this dg-category is a dgenhancement of the bounded derived category over the formal neighborhood under the assumption that the submanifold is compact. This generalizes a similar result of Block in the case of usual complex manifolds.