Dolbeault dga of a formal neighborhood

TL;DR

This paper introduces a Dolbeault complex for formal neighborhoods of complex submanifolds, enabling the study of coherent sheaves via complex analytic methods and connecting to dg-categories, generalizing previous results.

Contribution

It defines a Dolbeault complex for formal neighborhoods and shows this complex forms a dg-enhancement of the derived category, extending Block's work to formal neighborhoods.

Findings

The Dolbeault complex of a formal neighborhood is a differential graded algebra.

This dg-algebra provides a dg-enhancement of the bounded derived category.

The construction generalizes Block's results from usual to formal neighborhoods.

Abstract

Inspired by a work of Kapranov, we define the notion of 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 complex analytic approach, as in the case of usual complex manifolds and their Dolbeault complexes. Moreover, our the Dolbeault complex as a differential graded algebra can be associated with a dg-category according to Block. We show this dg-category is a dg-enhancement 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.