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.

Description of the spectral sequence

The spectral sequence is as follows:

$H^p(X, \Omega^q) \Rightarrow H^{p+q}(X, \mathbf C)$

where X is a complex manifold, $H^{p+q}(X, \mathbf C)$ 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

$\mathbf C \rightarrow \Omega^* := [\Omega^0 \stackrel d \to \Omega^1 \stackrel d \to \cdots \to \Omega^{\dim X}],$

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

$F^p \Omega^* := [\cdots \to 0 \to \Omega^p \to \Omega^{p+1} \to \cdots ]$

of $\Omega^*$.

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

$\bigoplus_{p+q=n} H^p(X, \Omega^q) = H^{n}(X, \mathbf C).$

Proofs

The degeneration of the spectral sequence can be shown using Hodge theory.^[1][2] A purely algebraic proof by means of reduction to positive characteristic was given by Deligne and Illusie.^[3]

See also

• Frölicher spectral sequence

References

1. See for example Griffiths, Harris Principles of algebraic geometry
2. Deligne, P. (1968). "Théorème de Lefschetz et Critères de Dégénérescence de Suites Spectrales". Publications Mathématiques de l'Institut des Hautes Études Scientifiques (in French) 35 (1): 107–126. doi:10.1007/BF02698925. ISSN 0073-8301.
3. P. Deligne, L. Illusie, "Relèvements modulo p² et décomposition du complexe de de Rham", Invent. Math. 89, 247–270 (1987)