In mathematics, the Hodge–de Rham spectral sequence (named in honor of W. V. D. Hodge and Georges de Rham) or Frölicher spectral sequence (named after Alfred Frölicher, who actually discovered it; Frölicher (1955)) is a spectral sequence that describes the precise relationship between the Dolbeault cohomology and the de Rham cohomology of a general complex manifold. On a compact Kähler manifold, the sequence degenerates, thereby leading to the Hodge decomposition of the de Rham cohomology. It is a tool in the theory of complex manifolds for expressing the potential failure of the results of cohomology theory that are valid in general only for Kähler manifolds. A spectral sequence is set up, the degeneration of which would give the results of Hodge theory and Dolbeault's theorem.
Description of the spectral sequence The spectral sequence is as follows:
H q ( X , Ω p ) ⇒ H p + q ( X , C ) {\displaystyle H^{q}(X,\Omega ^{p})\Rightarrow H^{p+q}(X,\mathbf {C} )}
where X is a complex manifold, H p + q ( X , C ) {\displaystyle H^{p+q}(X,\mathbf {C} )} is its cohomology with complex coefficients and the left hand term, which is the E 1 {\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
C → Ω ∗ := [ Ω 0 → d Ω 1 → d ⋯ → Ω dim X ] , {\displaystyle \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 Ω ∗ := [ ⋯ → 0 → Ω p → Ω p + 1 → ⋯ ] {\displaystyle F^{p}\Omega ^{*}:=[\cdots \to 0\to \Omega ^{p}\to \Omega ^{p+1}\to \cdots ]}
of Ω ∗ {\displaystyle \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 {\displaystyle E_{1}} -page. In particular, it gives an isomorphism referred to as the Hodge decomposition
⨁ p + q = n H p ( X , Ω q ) = H n ( X , C ) . {\displaystyle \bigoplus _{p+q=n}H^{p}(X,\Omega ^{q})=H^{n}(X,\mathbf {C} ).}
The degeneration of the spectral sequence can be shown using Hodge theory. An extension of this degeneration in a relative situation, for a proper smooth map f : X → S {\displaystyle f:X\to S} , was also shown by Deligne.
Purely algebraic proof For smooth proper varieties over a field of characteristic 0, the spectral sequence can also be written as
H q ( X , Ω p ) ⇒ H p + q ( X , Ω ∗ ) , {\displaystyle H^{q}(X,\Omega ^{p})\Rightarrow H^{p+q}(X,\Omega ^{*}),}
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