[Paper Review] Dolbeault dga of a formal neighborhood

Shilin Yu|arXiv (Cornell University)|Jun 22, 2012

Algebraic structures and combinatorial models14 references4 citations

TL;DR

This paper introduces the Dolbeault differential graded algebra (dga) for the formal neighborhood of a compact complex submanifold in a complex manifold, enabling a complex-analytic approach to coherent sheaves on formal neighborhoods. It proves that the dg-category of perfect modules over this dga provides a dg-enhancement of the bounded derived category of coherent sheaves on the formal neighborhood, generalizing Block's result for smooth manifolds.

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.

Motivation & Objective

• To extend the Dolbeault complex construction from smooth complex manifolds to formal neighborhoods of compact complex submanifolds.
• To provide a complex-analytic framework for studying coherent sheaves on formal neighborhoods via differential graded algebras.
• To generalize Block's dg-enhancement of the derived category for smooth manifolds to the case of formal neighborhoods.
• To establish that the dg-category of perfect modules over the Dolbeault dga is equivalent to the bounded derived category of coherent sheaves on the formal neighborhood.

Proposed method

• Define the Dolbeault dga of a formal neighborhood as the completed tensor product of the sheaf of smooth (0,•)-forms with the structure sheaf of the formal neighborhood.
• Use the flatness of the Dolbeault sheaf over the formal structure sheaf to construct resolutions of coherent sheaves via completed tensor products.
• Apply Stein theory and the Mittag-Leffler condition to ensure exactness of projective limits of resolutions over Stein opens.
• Construct a dg-category of perfect modules over the Dolbeault dga, leveraging the triangulated and homotopy-theoretic structure of differential graded categories.
• Prove that the homotopy category of this dg-category is equivalent to the bounded derived category of coherent sheaves on the formal neighborhood.
• Utilize the compactness of the submanifold to ensure finiteness and coherence conditions required for the resolution and limit arguments.

Experimental results

Research questions

• RQ1 Can the Dolbeault complex be generalized to formal neighborhoods of compact complex submanifolds in complex manifolds?
• RQ2 Does the Dolbeault dga of a formal neighborhood provide a dg-enhancement of the derived category of coherent sheaves on that neighborhood?
• RQ3 How does the structure of the Dolbeault dga relate to the geometry of the normal bundle in the formal neighborhood?
• RQ4 What is the role of the $L_∞$-algebroid structure on the shifted normal bundle in this construction?
• RQ5 Can the resolution of coherent sheaves on the formal neighborhood be constructed using completed tensor products with the Dolbeault dga?

Key findings

• The Dolbeault dga of a formal neighborhood is defined as $(\mathcal{A}^{\bullet}(\hat{Y}), \overline{\partial})$, where $\mathcal{A}^{\bullet}(\hat{Y})$ is the completed tensor product of the sheaf of smooth (0,•)-forms with the formal structure sheaf.
• The Dolbeault dga is flat over the formal structure sheaf $\mathscr{O}_{\hat{Y}}$, ensuring compatibility with coherent sheaf resolutions.
• For any coherent $\mathscr{O}_{\hat{Y}}$-module $\mathscr{F}$, the completed tensor product $\mathscr{F} \hat{\otimes}_{\hat{\mathscr{O}}} \mathscr{A}^{\bullet}_{\hat{Y}}$ yields a resolution of $\mathscr{F}$ by $\mathscr{A}^{\bullet}_{\hat{Y}}$-modules.
• The dg-category $\mathcal{P}_A$ of perfect modules over the Dolbeault dga $A$ has a homotopy category equivalent to $\mathcal{D}^{b}_{coh}(\hat{Y})$, the bounded derived category of coherent sheaves on $\hat{Y}$.
• The construction reveals an $L_\infty$-algebroid structure on the shifted normal bundle $N[-1]$, with an $\infty$-anchor map to the tangent bundle of the submanifold.
• The result generalizes Block's dg-enhancement of the derived category for smooth manifolds to the case of formal neighborhoods, under the assumption of compactness of the submanifold.