nLab holomorphic de Rham complex

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).

Related concepts

• differential form with logarithmic singularities

• Dolbeault cohomology

• Hodge filtration

• Hodge cohomology

• Hodge-filtered differential cohomology

References

• Claire Voisin, section 8.2 of Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3

• Alexander Beilinson, Higher regulators and values of L-functions, J. Soviet Math. 30 (1985), 2036-2070 (reviewed in Esnault-Viehweg 88) (doi)

• Hélène Esnault, Eckart Viehweg, section 2.5 of: Deligne-Beilinson cohomology, in: Michael Rapoport, Norbert Schappacher, Peter Schneider (eds.), Beilinson's Conjectures on Special Values of L-Functions, Perspectives in Mathematics 4, Academic Press, Inc. (1988) [ISBN:978-0-12-581120-0, pdf]

On holomorphic de Rham cohomology of Stein manifolds (where it coincides with ordinary de Rham cohomology, see there):

• Jean-Pierre Serre: Quelques problemes globaux relatifs aux varietes de Stein, Colloque sur les fonctions de plusieurs variables (1953) [doi:10.1007/978-3-642-39816-2_23]

• Jean-Pierre Serre: §2.1 in: Cohomologie et fonctions de variables complexes, Séminaire Bourbaki 2 71 (1954) [numdam:SB_1951-1954__2__213_0]