Dolbeault dga and $L_\infty$-algebroid of the formal neighborhood

TL;DR

This paper explores the Dolbeault differential graded algebra (dga) associated with the formal neighborhood of an arbitrary closed embedding of complex manifolds, linking it to formal differential geometry and $L_$-algebroid structures.

Contribution

It explicitly describes the Dolbeault dga in terms of formal differential geometry and establishes its structure as a completed Chevalley-Eilenberg dga with an $L_$-algebroid on the shifted normal bundle.

Findings

Dolbeault dga is explicitly described via formal differential geometry.

The Dolbeault dga is shown to be a completed Chevalley-Eilenberg dga.

Establishes an $L_$-algebroid structure on the shifted normal bundle.

Abstract

We continue the study the Dolbeault dga of the formal neighborhood of an arbitary closed embedding of complex manifolds previously defined by the author in \cite{DolbeaultDGA}. The special case of the diagonal embedding has been studied in \cite{Diagonal}. We describe the Dolbeault dga explicitly in terms of the formal differential geometry of the embedding. Moreover, we show that the Dolbeault dga is the completed Chevalley-Eilenberg dga an $L_\infty$-algebroid structure on the shifted normal bundle of the submanifold. This generlizes the result of Kapranov on the diagonal embedding and Atiyah class.