Hodge–de Rham spectral sequence

In mathematics, the Hodge–de Rham spectral sequence, also known as the Frölicher spectral sequence computes the cohomology of a complex manifold.

The spectral sequence is as follows:

where X is a complex manifold, ${\displaystyle H^{p+q}(X,\mathbf {C} )}$ is its cohomology with complex coefficients and the left hand term, which is the ${\displaystyle E_{1}}$-page of the spectral sequence, is the cohomology with values in the sheaf of holomorphic differential forms. The existence of the spectral sequence as stated above follows from the Poincaré lemma, which gives a quasi-isomorphism of complexes of sheaves

together with the usual spectral sequence resulting from a filtered object, in this case the Hodge filtration

of ${\displaystyle \Omega ^{*}}$.

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