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

TL;DRAbstract

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.

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