Dolbeault CohomologyEdit
Dolbeault cohomology is a central tool in complex geometry that encodes the ways in which complex-analytic data on a manifold can vary. It arises from the Dolbeault operator, which acts on smooth differential forms of type (p,q) on a complex manifold and measures how far a given form is from being holomorphic in the anti-holomorphic directions. The resulting bi-graded groups H^{p,q}_{∂̄}(X) carry rich information about the complex structure of X and interact with several other cohomological theories, notably sheaf cohomology through the Dolbeault isomorphism.
On a technical level, Dolbeault cohomology is defined by the Dolbeault complex, a long sequence of differential operators
0 → Ω^{p,0} → Ω^{p,1} → Ω^{p,2} → ...,
where Ω^{p,q} denotes the space of smooth differential forms of type (p,q) and ∂̄ is the anti-holomorphic exterior derivative. The Dolbeault cohomology groups are
H^{p,q}_{∂̄}(X) = ker(∂̄: Ω^{p,q} → Ω^{p,q+1}) / im(∂̄: Ω^{p,q-1} → Ω^{p,q}).
These groups do not depend on the choice of a metric or connection; they depend only on the underlying complex structure of X. For compact complex manifolds, each H^{p,q}_{∂̄}(X) is finite-dimensional, reflecting a robust algebraic structure behind the analytic data.
Dolbeault cohomology
The Dolbeault operator and the Dolbeault complex
The central object is the anti-holomorphic exterior derivative ∂̄, which increases the second index of a form. Acting on smooth forms of type (p,q), ∂̄ maps to type (p,q+1). The cohomology groups measure obstructions to solving ∂̄-equations and thus detect the presence of holomorphic data in higher dimensions. For a more sheaf-theoretic perspective, one can view Dolbeault cohomology as computing the derived functors of the global section functor applied to the sheaf of holomorphic p-forms Ω^p.
Dolbeault cohomology groups and the Dolbeault isomorphism
There is a canonical comparison between analytic and algebraic viewpoints: for a complex manifold X, the Dolbeault cohomology groups H^{q}(X, Ω^p) are isomorphic to the analytic groups H^{p,q}_{∂̄}(X). This is the Dolbeault isomorphism, which provides a bridge between the differential-geometric description and the sheaf-theoretic description of holomorphic data. The isomorphism is particularly useful because it allows the use of powerful algebraic techniques from sheaf cohomology in the study of complex-analytic problems.
Finite-dimensionality and Hodge theory on compact manifolds
When X is compact, the spaces H^{p,q}{∂̄}(X) are finite-dimensional. In this setting, one can introduce the ∂̄-Laplacian Δ{∂̄} and characterize cohomology classes by harmonic representatives. On compact Kähler manifolds, this harmonic theory yields the Hodge decomposition of de Rham cohomology, linking the Dolbeault groups to the real topology of X via
H^k_{dR}(X, ℂ) ≅ ⊕{p+q=k} H^{p,q}{∂̄}(X).
The numbers h^{p,q} = dim H^{p,q}_{∂̄}(X) are usually referred to as the Hodge numbers of X and provide a concise numerical snapshot of the complex-geometric structure.
Examples
Riemann surfaces (complex curves): If X is a compact Riemann surface of genus g, then h^{1,0} = h^{0,1} = g. This reflects the correspondence between holomorphic 1-forms and anti-holomorphic deformations of the complex structure.
Complex projective space: For complex projective space CP^n, the Dolbeault cohomology is sparse: H^{p,q}{∂̄}(CP^n) vanishes unless p = q, in which case H^{p,p}{∂̄}(CP^n) ≅ ℂ for 0 ≤ p ≤ n. This reflects the rigidity of CP^n’s holomorphic geometry.
Complex tori: For a complex torus T^n, the Dolbeault cohomology is determined by the holomorphic and anti-holomorphic one-forms, giving h^{p,q} = binom(n,p) binom(n,q). This mirrors the simple product structure of holomorphic data on a quotient of a vector space by a lattice.
Non-Kähler phenomena
Not all complex manifolds support the same tidy Hodge picture as Kähler manifolds. In non-Kähler geometry, the Hodge decomposition of de Rham cohomology can fail, and the dimensions h^{p,q} may exhibit behaviors not seen in the Kähler setting. Classic non-Kähler examples illustrate that the ∂∂-lemma (a key property in Kähler geometry) can fail, leading to richer and more varied Dolbeault cohomology. These phenomena are of ongoing interest in complex geometry and mirror the broader theme that complex structure can be flexible and diverse outside the Kähler world.
Applications and relations to other invariants
Dolbeault cohomology sits at the crossroads of several major topics in mathematics. It provides invariants that help distinguish complex structures, informs deformation theory through H^{0,1}_{∂̄}(X, T_X) and related groups, and connects to algebraic geometry via the Dolbeault isomorphism to H^{q}(X, Ω^p). In the study of holomorphic vector bundles, one can generalize Dolbeault cohomology to coefficients in a holomorphic bundle, yielding groups H^{q}(X, Ω^p ⊗ E) that track more refined geometric information. The theory also interacts with classical cohomology theories, such as de Rham cohomology, and with global analytic techniques that leverage the spectrum of the ∂̄-Laplacian.