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holomorphic de Rham complex

Contents

Context

Complex geometry

Differential cohomology

Contents

Idea

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

Ω Σ (𝒪 ΣΩ Σ 1Ω Σ 2). \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 Ω k\Omega^k denotes the holomorphic forms, this is the kernel of the Dolbeault operator ¯\bar \partial and hence the de Rham differential d=+¯\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 XX.

Properties

Relation to complex cohomology

For Σ\Sigma a complex manifold then the ordinary cohomology of Σ\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(Σ,)H k(Σ,Ω Σ ) 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 ppth filtering stage being

F pΩ X (0Ω X pΩ p+1). 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 Ω p\Omega^p is still regarded as sitting in degree p-p, one just replaces by 0 in the holomorphic de Rham complex the groups of differential forms of degree less than pp.

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

F pH k(X,)im(H k(X,F pΩ X )H k(X,Ω X )) 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 XX 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.