geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
On a complex analytic space $\Sigma$, the holomorphic de Rham complex is the restriction of the de Rham complex to holomorphic differential forms $\Omega^p$
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 $X$.
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:
(e.g. Voisin 02, theorem 8.1).
The holomorphic de Rham complex is naturally filtered by degree with the $p$th filtering stage being
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:
(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).
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 Rapoport, Schappacher, Schneider (eds.) Beilinson’s Conjectures on Special Values of L-Functions . Perspectives in Math. 4, Academic Press (1988) 43 - 91 (pdf)
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