Hodge–de Rham spectral sequence

Description of the spectral sequence

The spectral sequence is as follows:

where X is a complex manifold, H p + q ( X , C ) is its cohomology with complex coefficients and the left hand term, which is the 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 Ω ∗ .

Degeneration

The central theorem related to this spectral sequence is that for a compact Kähler manifold X, for example a projective variety, the above spectral sequence degenerates at the E 1 -page. In particular, it gives an isomorphism referred to as the Hodge decomposition

The degeneration of the spectral sequence can be shown using Hodge theory. A purely algebraic proof by means of reduction to positive characteristic was given by Deligne and Illusie. An extension of this degeneration in a relative situation, for a proper smooth map f : X → S , was also shown by Deligne.