Dolbeault Complex in Complex Geometry

• Dolbeault Complex is a fundamental structure capturing anti-holomorphic differentials in complex manifolds, enabling computation of sheaf cohomology and Hodge decomposition.
• It extends to singular, formal, quantum, and exceptional holonomy spaces by adapting analytic methods for resolutions and local corrections.
• The complex underpins advanced geometric analysis, index computations, and deformation theory, providing actionable insights into resolving coherent sheaves and duality constructions.

The Dolbeault complex is a fundamental analytic and topological structure in the study of complex and almost complex manifolds, singular spaces, and their generalizations, serving as the primary vehicle for encoding the anti-holomorphic differential structure and for calculating sheaf cohomology, including connections to index theory, noncommutative geometry, and deformation theory. In its classical form, it provides a fine resolution of the structure sheaf on complex manifolds via $(0,q)$-forms and the operator $\bar\partial$, but it admits deep extensions to singular spaces, formal neighborhoods, quantum and braided spaces, and manifolds of exceptional holonomy.

1. Classical Dolbeault Complex: Foundations and Resolutions

Given a complex manifold $X$ of complex dimension $n$, the Dolbeault complex is the sequence

$$0 \to \Omega_X^p \xrightarrow{i} A_X^{p,0} \xrightarrow{\bar\partial} A_X^{p,1} \xrightarrow{\bar\partial} \dots \xrightarrow{\bar\partial} A_X^{p,n} \to 0,$$

where $A_X^{p,q}$ denotes smooth $(p,q)$-forms, and $\Omega_X^p$ is the sheaf of holomorphic $p$-forms. Here, $\bar\partial$ is the $(0,1)$-component of the de Rham differential, satisfying $\bar\partial^2=0$.

The Dolbeault cohomology $H^{p,q}_{\bar\partial}(X)$ is defined as the $q$-th cohomology group of this complex. The fundamental isomorphism

$H^q(X, \Omega_X^p) \cong H^{p,q}_{\bar\partial}(X)$

relates sheaf cohomology to Dolbeault cohomology, with the latter computed analytically via forms. This identification underpins the analytic approach to complex geometry and is central to the proof of the Hodge decomposition on compact Kähler manifolds (Tardini, 2018).

The Dolbeault complex also features prominently in derived category theory, where it enables the construction of fine resolutions for coherent sheaves and the development of dg-enhancements for the bounded derived categories of coherent analytic sheaves, including over formal neighborhoods (Yu, 2012).

2. Dolbeault Complex on Singular and Formal Spaces

On singular complex spaces, the classical Dolbeault resolution requires significant modification. For compact Hermitian complex spaces $X$ with isolated singularities, one constructs a Dolbeault–Hilbert complex $(T^*,d)$ in which

$$T^q = \operatorname{Dom}(\bar\partial_{0,q,s}) \oplus A_q,$$

where $A_q$ are finite-dimensional corrections encoding local cohomology at singular points, and $\bar\partial_s$ is the minimal closed extension acting on $L^2_{0,q}$-forms over the regular locus $X_\text{reg}$ (Lott, 2019). The differential $d$ acts as $\bar\partial_s$ on forms and via projection to the skyscraper cohomology spaces on the singular locus.

For a closed embedding $X \hookrightarrow Y$ of complex manifolds, the Dolbeault complex of the formal neighborhood $\hat X_Y$ is constructed as an inverse system of quotient complexes of the ambient Dolbeault complex, encoding vanishing of Lie derivatives tangent to $X$ (Yu, 2012). The resulting Fréchet dga resolves the structure sheaf $\mathcal{O}_{\hat X_Y}$ and provides a dg-enhancement of the derived category of coherent sheaves on $\hat X_Y$.

A technical comparison is provided in Table 1:

Context	Underlying Sheaf	Resolution Type
Smooth complex	$\mathcal{O}_X$	Fine via $(0,q)$-forms and $\bar\partial$
Singular (isolated)	$\mathcal{O}_X$	Hilbert complex $(T^*,d)$ with local corrections
Formal neighborhood	$\mathcal{O}_{\hat X_Y}$	Inverse limit dga, Fréchet fine resolution

These constructions guarantee that even for singularities and formal thickenings, analytic methods via $(0,q)$-forms and their extensions retain control of the sheaf cohomology.

3. Noncommutative, Quantum, and Exceptional Holonomy Variants

Recent generalizations address Dolbeault complexes in noncommutative and quantum settings, as well as in real manifolds with exceptional holonomy.

On quantum spaces such as the $q$-deformation of full flag manifolds and Nichols–Woronowicz algebras, a Dolbeault-type bicomplex $\bigoplus_{p,q}\Omega^{p,q}$ is defined where forms are constructed from braided exterior algebras and the anti-holomorphic differential $\bar\partial$ is determined via quantum root vectors or braided derivations (Beggs et al., 9 Sep 2024, Buachalla et al., 2023). These complexes are often factorisable, *-compatible, and permit the construction of quantum analogues of Chern connections and metrics, with concrete realizations on the quantum plane and integer lattice.

On Riemannian manifolds with holonomy $G_2$ or $\mathrm{Spin}(7)$, so-called Dolbeault-type complexes are obtained by projecting the de Rham complex onto irreducible $G$-submodules of the exterior algebra. The analytic and algebraic structure of such complexes, including ellipticity criteria, Hodge isomorphisms, and harmonic representatives, is classified for $G_2$ and $\mathrm{Spin}(7)$-manifolds (Zhang, 2021).

4. Deformation Theory, Formality, and Poisson Structures

The Dolbeault complex admits further enhancement on holomorphic Poisson manifolds $(M, \pi)$. Here, the bigraded algebra $A_M^{p,q}$ of smooth forms is enriched by the Koszul–Brylinski operator $\partial_\pi$, defining a double complex $(A_M^{*,*}, \bar\partial, \partial_\pi)$ (Chen, 2022). The interplay of the Dolbeault operator and the Poisson differential leads to a differential graded Lie algebra (DGLA) structure via the Gerstenhaber bracket $[\cdot, \cdot]_{\partial_\pi}$. Under the $\partial_\pi \bar\partial$-lemma, this DGLA is formal, making Dolbeault cohomology sufficient to recover the homology of the total complex.

Maurer–Cartan solutions in this DGLA encode formal deformations of the complex structure, linking Dolbeault cohomology classes to families of Beltrami differentials. The canonical identification

$$H_k(A_M^{*,*}, \partial_\pi) \cong \bigoplus_{p-q=n-k} H^{p,q}_{\bar\partial}(M)$$

demonstrates that Dolbeault cohomology determines the Koszul–Brylinski homology under formality assumptions.

5. Extensions: Almost Complex Structures and Witten Deformations

The classical theory assumes integrability, but the Dolbeault complex extends to the almost complex case by considering operators induced by the Nijenhuis tensor. Even when $\bar\partial^2\neq0$, one constructs a spectral sequence whose $E_1$-page is Dolbeault-type cohomology, converging to de Rham cohomology. Harmonic theory for $(M, J)$ extends partially, with injectivity results for $\bar\partial$-harmonic forms into Dolbeault cohomology. The edge cases for bidegree yield finite-dimensional results even in absence of integrability (Cirici et al., 2018).

In the presence of a Kähler manifold $(M, g, J)$ and a primitive $(1,0)$-form $\omega$ with $\partial \omega = 0$, the Dolbeault complex admits a Witten–Novikov-type deformation. The perturbed differential $$\bar\partial_\omega = \bar\partial + \operatorname{ext}(\overline{\omega})$$ still yields an elliptic complex, and the index is preserved but the local index density acquires a universal factor dependent on $\omega$ (López et al., 2020).

6. Computations, Index Theorems, and Applications

In both classical and singular contexts, Dolbeault complexes compute invariants such as the holomorphic Euler characteristic and arithmetic genus. On singular curves, for example, the Dolbeault–Hilbert complex detects contributions from local $\delta$-invariants at singularities, matching topological calculations (Lott, 2019). Analytic K-homology classes constructed using Dolbeault complexes agree with classes defined by the Baum-Fulton-MacPherson formalism and provide analytic proofs of Riemann–Roch theorems.

On noncompact and nonintegrable models, supersymmetric quantum mechanical constructions compute the spectrum and index of the Dolbeault Laplacian, as in the cases of punctured spheres $S^4 \setminus \{\infty\}$ and $S^6 \setminus \{\infty\}$, with explicit enumeration of zero modes and calculation of the Dolbeault index (Smilga, 2011).

The table summarizes select advanced contexts for Dolbeault-type complexes:

Setting	Operator Structure	Index / Cohomology Interpretation
Singular space (isolated)	$(T^*, d)$ with correction terms	$H^q(T^*,d) \cong H^q(X;\mathcal{O}_X)$ (Lott, 2019)
Holomorphic Poisson manifold	$(A_M^{*,*}, \bar\partial, \partial_\pi)$	Formality, deformations, Koszul–Brylinski homology (Chen, 2022)
Quantum flag manifold	$q$-Dolbeault complex from root vectors	Quantum Borel–Weil theorem (Buachalla et al., 2023)
Exceptional holonomy manifold	Projections of de Rham complex	Harmonic representatives in irreducible summands (Zhang, 2021)

7. Related Structures: Dualities, Relative Cohomology, and Exact Sequences

The Dolbeault complex participates in a wide array of dualities and functorial operations. Čech–Dolbeault double complexes compute sheaf cohomology via open coverings and support the construction of relative Dolbeault cohomology for pairs $(X, A)$, giving rise to exact sequences important in blow-up formulas, injectivity theorems, and relative invariants (Tardini, 2018). These exact sequences function in both the domain of forms and of currents, with canonical quasi-isomorphisms relating them.

In noncommutative and braided settings, *-structures and factorization properties allow for the consistent development of quantum analogues of classical Dolbeault theory, including the construction of Chern connections and compatible metrics (Beggs et al., 9 Sep 2024).

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The Dolbeault complex, in its many guises, thus enables a flexible and robust analytic machinery, underpinning advances in complex geometry, singularity theory, Hodge theory, deformation, noncommutative geometry, and mathematical physics. Its formulations across classical, singular, formal, quantum, and exceptional holonomy settings emphasize the unity of the $\bar\partial$-complex framework and its centrality in modern geometric analysis.