nLab Dolbeault cohomology

under construction

Idea

Dolbeault cohomology of a complex manifold $X$ is the chain cohomology of the Dolbeault complex of $X$ (see there for more).

Properties

Dolbeault theorem

By the Dolbeault theorem (see there) Dolbeault cohomology is equivalently the abelian sheaf cohomology $H^q(X;\Omega_X^p)$, of the abelian sheaf $\Omega_X^p$ which is the Dolbeault complex of holomorphic p-forms.

Hodge isomorphism

For $X$ a Hermitian manifold write $\mathcal{H}^{p,q}(X)$ for the space of $(p,q)$-harmonic differential forms and write $H^{p,q}$ for its Dolbeault cohomology in the bidegree.

Proposition

There is a canonical homomorphism

$$\mathcal{H}^{p,q}(X) \longrightarrow H^{p,q}(X) \,.$$

If $X$ is compact, then this is an isomorphism, the Hodge isomorphism

Also

$$\mathcal{H}^k(X,\mathbb{C}) \longrightarrow H^k_{dR}(X,\mathbb{C})$$

is an isomorphism

(e.g. Maddock, prop. 4.2.7)

Serre duality

Serre duality: On a Hermitian manifold $X$ of complex dimension $dim_{\mathbb{C}}(X) = n$ the Hodge star operator induces isomorphisms

$$\mathcal{H}^{p,q}(X)\simeq \mathcal{H}^{n-p, n-q}(X) \,.$$

(e.g. Maddock, prop. 4.2.8)

Hodge decomposition

If $X$ is a compact Kähler manifold then

$$\mathcal{H}^k(X) \simeq \underset{p+q = k}{\oplus} \mathcal{H}^{p,q}(X)$$

(e.g. Maddock, prop. 4.2.9)

This is called the Hodge decomposition.

Hodge symmetry

Over complex manifolds $X$, Hodge symmetry is the property that the Dolbeault cohomology groups $H^{p,q}(X)$ are taken into each other under complex conjugation followed by switching the bidegree:

$$H^{p,q}(X) \simeq \overline{H^{q,p}(X)} \,.$$

In particular this means that the dimension of the cohomology groups in degree $(p,q)$ coincides with that in bidegree $(q,p)$.

Dolbeault resolution

Given any holomorphic vector bundle $E$, one can form the Dolbeault resolution $E \otimes \Omega^{0,q}$, where $\Omega^{0,q}$ is the sheaf of $C^\infty$ $(0,q)$-forms. This is an acyclic resolution of $E$ and hence computes its sheaf cohomology.

(…)

Double complex and Frölicher spectral sequence

The Dolbeault complexes naturall fit into a double complex

$$\array{ \Omega^{p,0} &\stackrel{\bar \partial}{\to}& \Omega^{p-1,1} &\stackrel{\bar \partial}{\to}& \cdots \\ \downarrow^{\mathrlap{\partial }} && \downarrow^{\mathrlap{\partial }} \\ \Omega^{p+1,0} &\stackrel{\bar \partial}{\to}& \Omega^{p,1} &\stackrel{\bar \partial}{\to}& \cdots \\ \downarrow^{\mathrlap{\partial }} && \downarrow^{\mathrlap{\partial }} \\ \vdots && \vdots } \,.$$

The corresponding spectral sequence of a double complex is called the Frölicher spectral sequence. On a Kähler manifold it exhibits the Hodge filtration.

Related concepts

• Hodge number

• Kodaira vanishing theorem

• de Rham cohomology

• Bott-Chern cohomology

References

• Zachary Maddock, Dolbeault cohomology (pdf)