# nLab holomorphic de Rham complex

Contents

complex geometry

### Examples

#### Differential cohomology

differential cohomology

# Contents

## Idea

On a complex analytic space $\Sigma$, the holomorphic de Rham complex is the restriction of the de Rham complex to holomorphic differential forms $\Omega^p$

$\Omega^\bullet_\Sigma \coloneqq \left( \mathcal{O}_\Sigma \stackrel{\partial}{\longrightarrow} \Omega^1_\Sigma \stackrel{\partial}{\longrightarrow} \Omega^2_\Sigma \longrightarrow \cdots \right) \,.$

Notice here that since $\Omega^k$ denotes the holomorphic forms, this is the kernel of the Dolbeault operator $\bar \partial$ and hence the de Rham differential $\mathbf{d} = \partial + \bar \partial$ restricts to the holomorphic component $\partial$ as above.

Often considered is a version of this with singularities (“logarithmic de Rham complex”) where one consideres instead meromorphic differential forms which are holomorphic in the bulk with logarithmic singularities towards compactification boundaries of $X$.

## Properties

### Relation to complex cohomology

For $\Sigma$ a complex manifold then the ordinary cohomology of $\Sigma$ with coefficients in the complex numbers is naturally isomorphic to the hyper-abelian sheaf cohomology with coefficients in the holomorphic de Rham complex:

$H^k(\Sigma,\mathbb{C}) \simeq H^k(\Sigma, \Omega_\Sigma^\bullet)$

(e.g. Voisin 02, theorem 8.1).

### Filtering and Relation to Hodge filtation

The holomorphic de Rham complex is naturally filtered by degree with the $p$th filtering stage being

$F^p \Omega^\bullet_X \coloneqq (0 \to \cdots \to \Omega^p_X \stackrel{\partial}{\longrightarrow} \Omega^{p+1} \stackrel{\partial}{\longrightarrow} \cdots) \,.$

Notice that here $\Omega^p$ is still regarded as sitting in degree $-p$, one just replaces by 0 in the holomorphic de Rham complex the groups of differential forms of degree less than $p$.

With this, the Hodge filtration on $H^\bullet(X,\mathbb{C})$ is defined to be the filtration with $p$th stage the image of the hyper-abelian sheaf cohomology with coefficients in the $p$th filtering stage of the holomorphic de Rham complex inside that with coefficients the full de Rham complex:

$F^p H^k(X,\mathbb{C}) \coloneqq im \left( H^k(X, F^p \Omega^\bullet_X) \to H^k(X, \Omega^\bullet_X) \right)$

(e.g. Voisin 02, def. 8.2).

If $X$ happens to be a Kähler manifold then the Frölicher spectral sequence shows that this coincides with the more traditional definition via harmonic differential forms (e.g. Voisin 02, remark 8.29).

## References

Last revised on June 5, 2014 at 00:23:43. See the history of this page for a list of all contributions to it.