8.1 Hodge complexes

Let A be as in §0.3. Then a Hodge A-complex K of weight n consists of

a complex K_A\in\operatorname{Ob}\mathrm{D}^+(A) such that \operatorname{H}^k(K_A) is an A-module of finite type for all k;
a filtration F on K_A\otimes\mathbb{C}, i.e. a filtered complex (K_\mathbb{C},F)\in\operatorname{Ob}\mathrm{D}^+F(\mathbb{C}), and an isomorphism \alpha\colon K_\mathbb{C}\simeq K_A\otimes\mathbb{C} in \mathrm{D}^+(\mathbb{C}).

The following axioms have to be satisfied:

The differential d of K_\mathbb{C} is strictly compatible with the filtration F; in other words, the spectral sequence defined by (K_\mathbb{C},F) degenerates at E_1 (i.e. E_1=E_\infty) (1.3.2).

For all k, the filtration F on \operatorname{H}^k(K_\mathbb{C})\simeq\operatorname{H}^k(K_A)\otimes\mathbb{C} defines a Hodge A-structure of weight n+k on \operatorname{H}^k(K_A), i.e. the filtration F is (n+k)-opposite to its complex conjugate (which makes sense, since A\subset\mathbb{R}).

Let X be a topological space. Then a cohomological Hodge A-complex K of weight n on X consists of:

a complex K_A\in\operatorname{Ob}\mathrm{D}^+(X,A),
a filtered complex (K_\mathbb{C},F)\in\operatorname{Ob}\mathrm{D}^+F(X,\mathbb{C}),
an isomorphism \alpha\colon K_\mathbb{C}\simeq K_A\otimes\mathbb{C} in \mathrm{D}^+(X,\mathbb{C}).

The following axiom has to be satisfied:

The triple (\mathrm{R}\Gamma(K_A),\mathrm{R}\Gamma((K_\mathbb{C},F)),\mathrm{R}\Gamma(\alpha)) is a Hodge complex of weight n.

When A=\mathbb{Z}, we simply speak of Hodge complexes and cohomological Hodge complexes.

The following statement reformulates a part of classical Hodge theory (cf. (2.2.1)).

Let X be a complex Kähler variety. Let K_\mathbb{Z} be the complex \mathbb{Z}[0] consisting of the constant sheaf \mathbb{Z} in degree 0, and let K_\mathbb{C} be the holomorphic de Rham complex \Omega_X^\bullet, with F its stupid filtration. We have \alpha\colon K_\mathbb{Z}\otimes\mathbb{C}\simeq\mathbb{C}[0]\xrightarrow{\approx}\Omega_X^\bullet (the Poincaré lemma), and (K_\mathbb{Z},(K_\mathbb{C},F),\alpha) is a cohomological Hodge complex of weight 0.

If K is a Hodge complex (resp. cohomological Hodge complex) of weight n, then K[m] is a Hodge complex (resp. cohomological Hodge complex) of weight n+m.

A mixed Hodge A-complex K consists of:

a complex K_A\in\operatorname{Ob}\mathrm{D}^+(A) such that \operatorname{H}^k(K_A) is an A-module of finite type for all k;
a filtered complex (K_{A\otimes\mathbb{Q}},W)\in\operatorname{Ob}\mathrm{D}^+F(A\otimes\mathbb{Q}) (for an increasing filtration W) and an isomorphism K_{A\otimes\mathbb{Q}}\simeq K_A\otimes\mathbb{Q} in \mathrm{D}^+(A\otimes\mathbb{Q});
a bifiltered complex (K_\mathbb{C},W,F)\in\operatorname{Ob}\mathrm{D}^+F_w(\mathbb{C}) (for an increasing filtration W and a decreasing filtration F) and an isomorphism \alpha\colon(K_\mathbb{C},W)\simeq(K_{A\otimes\mathbb{Q}},W)\otimes\mathbb{C} in \mathrm{D}^+F(\mathbb{C}).

The following axiom must be satisfied:

For all n, the system consisting of the complex \operatorname{Gr}_n^W(K_{A\otimes\mathbb{Q}})\in\operatorname{Ob}\mathrm{D}^+(A\otimes\mathbb{Q}), the complex \operatorname{Gr}_n^W(K_\mathbb{C})\in\operatorname{Ob}\mathrm{D}^+F(\mathbb{C}) filtered by F, and the isomorphism \operatorname{Gr}_n^W(\alpha)\colon \operatorname{Gr}_n^W(K_{A\otimes\mathbb{Q}})\otimes\mathbb{C} \simeq \operatorname{Gr}_n^W(K_\mathbb{C}), is a Hodge (A\otimes\mathbb{Q})-complex of weight n.

A cohomological mixed Hodge A-complex K on a topological space X consists of:

a complex K_A\in\operatorname{Ob}\mathrm{D}^+(X,A) such that \operatorname{\mathbb{H}}^k(X,K_A) is an A-module of finite type for all k, and such that \operatorname{\mathbb{H}}(X,K_A)\otimes\mathbb{Q}\xrightarrow{\sim}\operatorname{\mathbb{H}}(X,K_A\otimes\mathbb{Q});
a filtered complex (K_{A\otimes\mathbb{Q}},W)\in\operatorname{Ob}\mathrm{D}^+F(X,A\otimes\mathbb{Q}) (for an increasing filtration W) and an isomorphism K_{A\otimes\mathbb{Q}}\simeq K_A\otimes\mathbb{Q} in \mathrm{D}^+(X,A\otimes\mathbb{Q});
a bifiltered complex (K_\mathbb{C},W,F)\in\operatorname{Ob}\mathrm{D}^+F_w(X,\mathbb{C}) (for an increasing filtration W and a decreasing filtration F) and an isomorphism \alpha\colon(K_\mathbb{C},W)\simeq(K_{A\otimes\mathbb{Q}},W)\otimes\mathbb{C} in \mathrm{D}^+F(X,\mathbb{C}).

The following axiom must be satisfied:

For all n, the system consisting of the complex \operatorname{Gr}_n^W(K_{A\otimes\mathbb{Q}})\in\operatorname{Ob}\mathrm{D}^+(X,A\otimes\mathbb{Q}), the complex \operatorname{Gr}_n^W(K_\mathbb{C})\in\operatorname{Ob}\mathrm{D}^+F(X,\mathbb{C}) filtered by F, and the isomorphism \operatorname{Gr}_n^W(\alpha)\colon \operatorname{Gr}_n^W(K_{A\otimes\mathbb{Q}})\otimes\mathbb{C} \simeq \operatorname{Gr}_n^W(K_\mathbb{C}), is a cohomological Hodge (A\otimes\mathbb{Q})-complex of weight n.

We can trivially show:

If K=(K_A,(K_{A\otimes\mathbb{Q}},W),(K_\mathbb{C},W,F)) is a cohomological mixed Hodge A-complex, then \mathrm{R}\Gamma K=(\mathrm{R}\Gamma K_A,\mathrm{R}\Gamma(K_{A\otimes\mathbb{Q}},W),\mathrm{R}\Gamma(K_\mathbb{C},W,F)) is a mixed Hodge A-complex.

Let U be the complement in a smooth proper scheme X, of a normal crossing divisor D, with j\colon U\hookrightarrow X the inclusion morphism. Let W be the canonical filtration on \mathrm{R}j_*\mathbb{Q}=\mathrm{R}j_*\mathbb{Z}\otimes\mathbb{Q}\colon W_n(\mathrm{R}j_*\mathbb{Q})=\tau_{\leqslant n}(\mathrm{R}j_*\mathbb{Q}). We have (3.1.8) (\mathrm{R}j_*\mathbb{Q})\otimes\mathbb{C} \xrightarrow{\approx} \mathrm{R}j_*\mathbb{C} \xrightarrow{\approx} j_*\Omega_U^\bullet \xleftarrow{\approx} \Omega_X^\bullet(\log D). Let W be the weight filtration of \Omega_+X^\bullet(\log D), and F its Hodge filtration (by the \sigma_{\geqslant p}). Then (3.1.8) gives an isomorphism (\mathrm{R}j_*\mathbb{Q},W)\otimes\mathbb{C}\simeq(\Omega_X^\bullet(\log D),W) in \mathrm{D}^+F(X,\mathbb{C}), and, by (3.1.5) and (3.1.9), (\mathrm{R}j_*\mathbb{Z},(\mathrm{R}j_*\mathbb{Q},W),(\Omega_X^\bullet(\log D), W, F)) is a cohomological mixed Hodge complex on X.

The following result is an “abstract” version of §3.2.

Let K be a mixed Hodge A-complex.

On the pages {}_WE_r^{pq} of the spectral sequence of (K_\mathbb{C},W), the recurrent filtration and the two direct filtrations defined by F all coincide.
The filtration W[n] () on \operatorname{H}^n(K_{A\otimes\mathbb{Q}})\simeq\operatorname{H}^n(K_A)\otimes\mathbb{Q} and the filtration F on \operatorname{H}^n(K_\mathbb{C})\simeq\operatorname{H}^n(K_A)\otimes_A\mathbb{C} define a mixed Hodge A-structure on \operatorname{H}^n(K_A) ((2.3.1) along with (0.4)).
The morphisms {}_Wd_1\colon{}_WE_1^{pq}\to{}_WE_1^{p+1,q} are compatible with the Hodge bigrading; in particular, they are strictly compatible with the Hodge filtration.
The spectral sequence of (K_{A\otimes\mathbb{Q}},W) degenerates at E_2 (i.e. {}_WE_2={}_WE_\infty).
The spectral sequence of (K_\mathbb{C},F) degenerates at E_1 (i.e. {}_FE_2={}_FE_\infty).
The spectral sequence of the complex \operatorname{Gr}_F^p(K_\mathbb{C}), endowed with the filtration induced by W, degenerates at E_2.

Proof. The proof of (i), (ii), (iii), (iv) is parallel to that of (3.2.5). The analogy of (3.2.6) and (3.2.7) has been taken as an axiom ((CH 1) and (CH 2)). The proofs of (3.2.8), (3.2.9), and (3.2.10) can be carried over as they are, and, as in (3.2.10), we thus deduce (i), (ii), (iii), and (iv). Claims (v) and (vi) then follow from (7.2.8).

We abbreviate “differential graded” to DG, and “bounded-below differential graded” to DG+. A DG complex of A-modules can be seen as a double complex of A-modules; the first degree is that of the complex, and the second degree is the one defined by the grading of DG modules. We denote by \mathrm{D}^+(A\mathsf{-mod}_{\mathsf{DG}^+}) the derived category of the category of bounded-below complexes of DG A-modules, with degrees uniformly bounded below.

A DG mixed Hodge A-complex K consists of:

a complex K_A\in\operatorname{Ob}\mathrm{D}^+(A\mathsf{-mod}_{\mathsf{DG}^+});
a filtered complex (K_{A\otimes\mathbb{Q}},W)\in\operatorname{Ob}\mathrm{D}^+F((A\otimes\mathbb{Q})\mathsf{-mod}_{\mathsf{DG}^+}), and is isomorphism K_A\otimes\mathbb{Q}\simeq K_{A\otimes\mathbb{Q}} in \mathrm{D}^+;
a bifiltered complex (K_\mathbb{C},F,W)\in\operatorname{Ob}\mathrm{D}^+F_2(\mathbb{C}\mathsf{-mod}_{\mathsf{DG}^+}), and an isomorphism K_{A\otimes\mathbb{Q}}\otimes\mathbb{C}\simeq K_\mathbb{C} in \mathrm{D}^+F.

We require that, for each n, the component (K^{\bullet n},F,W) of K whose second degree is n be a mixed Hodge A-complex.

We denote by L the filtration by the second degree of \mathrm{s}K. We denote by \operatorname{Dec}_1W the filtration of K_{A\otimes\mathbb{Q}} induced by shifting ((1.3.3)) W. We will not confuse the filtration that this induces on \mathrm{s}K_{A\otimes\mathbb{Q}} (also denoted by \operatorname{Dec}_1W) with the shifted filtration of the filtration induced on \mathrm{s}K_{A\otimes\mathbb{Q}} by W.

We similarly define cosimplicial (resp. r-cosimplicial) mixed Hodge A-complexes by replacing “DG” with “cosimplicial” (resp. r-cosimplicial) everywhere in the definition. The functor from cosimplicial A-modules to DG A-modules sends cosimplicial mixed Hodge A-complexes to DG mixed Hodge A-complexes (and similarly for r-cosimplicial).

Let X_\bullet be a simplicial topological space. A cohomological mixed Hodge A-complex K on X_\bullet consists of:

a complex K_A\in\operatorname{Ob}\mathrm{D}^+(X_\bullet,A);
a filtered complex (K_{A\otimes\mathbb{Q}},W)\in\operatorname{Ob}\mathrm{D}^+F(X_\bullet,A\otimes\mathbb{Q}), and an isomorphism K_{A\otimes\mathbb{Q}}\simeq K_A\otimes\mathbb{Q} in \mathrm{D}^+(X_\bullet,A\otimes\mathbb{Q});
a bifiltered complex (K_\mathbb{C},W,F)\in\operatorname{Ob}\mathrm{D}^+F_2(X_\bullet,\mathbb{C}), and an isomorphism (K_\mathbb{C},W)\simeq(K_{A\otimes\mathbb{Q}},W)\otimes\mathbb{C} in \mathrm{D}^+F(X_\bullet,\mathbb{C}).

The following axiom has to be satisfied:

The restriction of K to each of the X_n is a cohomological mixed Hodge A-complex.

For A=\mathbb{Z}, we simply speak of a cohomological mixed Hodge complex.

Let X_\bullet be a smooth proper scheme (over \mathbb{C}), and let Y_\bullet be a normal crossing divisor in X_\bullet. We set U_\bullet=X_\bullet\setminus Y_\bullet, and we denote by j the inclusion of U_\bullet into X_\bullet. The constructions in (8.1.8) can be performed “uniformly in X_n”, as (5.2.5) and (6.2.7) show. They thus give a cohomological mixed Hodge complex \big(\mathrm{R}j_*\mathbb{Z}, (\mathrm{R}j_*\mathbb{Q},\tau_{\leqslant}), (\Omega_{X_\bullet}^\bullet(\log Y_\bullet),W,F)\big) on X_\bullet.

Let K be a cohomological mixed Hodge A-complex on X_\bullet. Write A\mathsf{-mod}_\mathsf{cosimp} to mean the category of cosimplicial A-modules. If we apply the functor \mathrm{R}\Gamma^\bullet to K, then we obtain

a cosimplicial complex \mathrm{R}\Gamma^\bullet K_A\in\operatorname{Ob}\mathrm{D}^+(A\mathsf{-mod}_\mathsf{cosimp});
a cosimplicial filtered complex \mathrm{R}\Gamma^\bullet(K_{A\otimes\mathbb{Q}},W)\in\operatorname{Ob}\mathrm{D}^+F(A\otimes\mathbb{Q}\mathsf{-mod}_\mathsf{cosimp});
a cosimplicial bifiltered complex \mathrm{R}\Gamma^\bullet(K_\mathbb{C},W,F)\in\operatorname{Ob}\mathrm{D}^+F_2(\mathbb{C}\mathsf{-mod}_\mathsf{cosimp});
isomorphisms as in (b) and (c) of (8.1.10) between these objects.

It is clear that the \mathrm{R}\Gamma^\bullet(K,W,F) described above is a cosimplicial mixed Hodge A-module. It defines a DG mixed Hodge A-complex (8.1.10.1).

Let K be a DG mixed Hodge A-complex, and consider the spectral sequence abutting to the cohomology of \mathrm{s}K, which is defined by the filtration L. By hypothesis, the E_1 pages of this spectral sequence are endowed with mixed Hodge A-structures. The content of (8.1.15) is that every spectral sequence, including its abutment, is endowed with a mixed Hodge A-structure. In other words, c' is a spectral sequence in the abelian category of mixed Hodge A-structures. The structure on the abutment corresponds to a natural mixed Hodge A-complex “structure” on \mathrm{s}K.

Let K be a DG mixed Hodge A-complex.

With the notation of (8.1.10), (8.1.10.1), and (7.1.6), (\mathrm{s}K,\delta(W,L),F) is a mixed Hodge A-complex.
We have {}_{\delta(W,L)}E_1^{ab}(\mathrm{s}K\otimes\mathbb{Q}) = \sum_{\substack{\alpha=\gamma-\beta\\b=\alpha+\beta}} \operatorname{H}^\alpha(\operatorname{Gr}_\beta^W K^{\bullet\gamma}) \tag{8.1.15.1} where the complex ({}_{\delta(W,L)}E_1^{\bullet b},d_1) is the simple complex associated to the double complex of Hodge A\otimes\mathbb{Q}-structures of weight b given by \small \begin{CD} \operatorname{H}^{b-(\beta+1)}(\operatorname{Gr}_{\beta+1}^W K^{\bullet,\gamma+1}) @>{\partial}>> \operatorname{H}^{b-\beta}(\operatorname{Gr}_\beta^W K^{\bullet,\gamma+1}) @>{\partial}>> \operatorname{H}^{b-(\beta-1)}(\operatorname{Gr}_{\beta-1}^W K^{\bullet,\gamma+1}) \\@. @A{d''}AA @AA{d''}A \\@. \operatorname{H}^{b-\beta}(\operatorname{Gr}_\beta^W K^{\bullet,\gamma}) @>>{\partial}> \operatorname{H}^{b-(\beta-1)}(\operatorname{Gr}_{\beta-1}^W K^{\bullet,\gamma}) \end{CD} \tag{8.1.15.2}
On the {}_LE_r pages of the spectral sequence defined by (\mathrm{s}K_{A\otimes\mathbb{Q}},L), the recurrent filtration and the two direct filtrations defined by \operatorname{Dec}_1(W) agree; similarly, on the {}_LE_r(\mathrm{s}K_\mathbb{C}), the recurrent filtration and the two direct filtrations induced by F agree. We denoted these filtrations again by W and F (respectively).
For r\geqslant1, ({}_LE_r,W,F) is a mixed Hodge A-structure (0.4). The differentials d_r are morphisms of mixed Hodge A-structures.
The filtration L on \operatorname{H}^\bullet(\mathrm{s}K) is a filtration in the category of mixed Hodge structures, and \operatorname{Gr}_L^p\big( \operatorname{H}^{p+q}(\mathrm{s}K), \delta(W,L)[p+q], F \big) = ({}_LE_\infty^{pq},W,F). \tag{8.1.15.3}

In (8.1.15), the filtration \operatorname{Dec}_1(W) (8.1.10) plays a more important role than W. We have \operatorname{Dec}(\delta(W,L)) = \operatorname{Dec}_1(W) \qquad\text{on }\mathrm{s}K \tag{8.1.16.1} and (8.1.15.3) can be rewritten as \operatorname{Gr}_L(\operatorname{H}(K),\operatorname{Dec}_1(W,F)) = ({}_LE_\infty,W,F). \tag{8.1.16.2} In the same spirit, we note that, with the notation of (8.1.8), \mathrm{R}\Gamma\big( \mathrm{R}j_*\mathbb{Z}, (\mathrm{R}j_*\mathbb{Q},\operatorname{Dec}W), (\Omega_X^\bullet(\log D),\operatorname{Dec}W,F) \big) depends only on U, and not on the choice of compactification X.

We now prove (8.1.15). Claims (i) and (ii) follow from the fact that \begin{aligned} \operatorname{Gr}_n^{\delta(W,L)}(\mathrm{s}K\otimes\mathbb{Q}) &= \sum_{p-q=n} \operatorname{Gr}_p^W\operatorname{Gr}_L^q(\mathrm{s}K\otimes\mathbb{Q}) \\&= \sum_{p-q=n} (\operatorname{Gr}_p^W K^{\bullet q})[-q] \end{aligned} (this isomorphism is compatible with F, and we apply (8.1.4)). The proof of (8.1.15.2) is left to the reader.

Consider the claims:

(a{}_r) The morphisms d_r of {}_LE_r(\mathrm{s}K_{A\otimes\mathbb{Q}}) are strictly compatible with the recurrent filtration defined by \operatorname{Dec}_1(W).
(b{}_r) The morphisms d_r of {}_LE_r(\mathrm{s}K_\mathbb{C}) are strictly compatible with the recurrent filtration defined by F.
(c{}_r) {}_LE_r, endowed with the filtrations considered in (a) and (b) is a mixed Hodge A-structure.
(d{}_r) The d_r are morphisms of mixed Hodge A-structures.

We will prove (a{}_r) and (b{}_r) for r\geqslant0, and (c{}_r) and (d{}_r) for r\geqslant1, by simultaneous induction.

A. Proof of (a{}_0). We have \big( {}_LE_0^{p\bullet}(\mathrm{s}K_{A\otimes\mathbb{Q}}), \operatorname{Dec}_1 W \big) = (K_{A\otimes\mathbb{Q}}^{\bullet p}, \operatorname{Dec}W). By (iv) of (8.1.9), the spectral sequence of (K_{A\otimes\mathbb{Q}}^{\bullet p},W) degenerates at E_2. By (1.3.4), the spectral sequence of (K_{A\otimes\mathbb{Q}}^{\bullet p},\operatorname{Dec}W) degenerates at E_1, and this proves (a{}_0) (1.3.2).

B. Proof of (b{}_0). We have \big( {}_LE_0^{p\bullet}(\mathrm{s}K_\mathbb{C}),F \big) = (K_\mathbb{C}^{\bullet p}, F). We apply (v) of (8.1.9) and (1.3.2).

C. Proof of (c{}_1). We have to show that \operatorname{H}^n(K_A^{\bullet p}), endowed with the filtrations induced by \operatorname{Dec}W on \operatorname{H}^n(K_{A\otimes\mathbb{Q}}^{\bullet p})=\operatorname{H}^n(K_A^{\bullet p})\otimes\mathbb{Q} and by F on \operatorname{H}^n(K_\mathbb{C}^{\bullet p})=\operatorname{H}^n(K_A^{\bullet p})\otimes_A\mathbb{C}, is a mixed Hodge A-structure. But \operatorname{Dec}W induces on \operatorname{H}^n(K_{A\otimes\mathbb{Q}}^{\bullet p}) the same filtration as W[n], and so we conclude by (ii) of .

D. Proof of (c{}_r)+(d{}_r)\implies(a{}_r)+(b{}_r)+(c{}_{r+1}). This is the key point of the proof, and is a simple application of (2.3.5) (along with (0.4)).

E. Proof of (a{}_r){}_{r\leqslant r_0-2}+(b{}_r){}_{r\leqslant r_0-2}+(c{}_{r_0})\implies(d{}_{r_0}). By (1.3.16), the recurrent filtration and the direct filtrations induced by \operatorname{Dec}_1W (resp. F) on {}_LE_r agree for r\leqslant r_0, and we apply (i) of (1.3.13).

This finishes the proof by induction of (a), (b), (c), and (d).

Claim (iii) was proven in part (E) of the proof by induction, and the claims of (iv) are (c{}_r) and (d{}_r). We will prove (8.1.15.3) in the form of (8.1.16.2).

The fact that \operatorname{Gr}_L(\operatorname{H}(\mathrm{s}K_{A\otimes\mathbb{Q}}),\operatorname{Dec}_1W) = ({}_LE_\infty(\mathrm{s}K_{A\otimes\mathbb{Q}}),W) follows from (a{}_r) and (1.3.17). The fact that \operatorname{Gr}_L(\operatorname{H}(\mathrm{s}K_\mathbb{C}),F) = ({}_LE_\infty(\mathrm{s}K_\mathbb{C}),F) follows from (b{}_r) and (1.3.17).

We conclude by the following lemma, whose proof is left to the reader

Let H=(H_A,W,F) be a mixed Hodge A-structure. For a filtration L of H_A to come from a filtration of H, in the abelian category of mixed Hodge A-structures, it is necessary and sufficient that, for all n, (\operatorname{Gr}_L^n(H_A),\operatorname{Gr}_L^n(W),\operatorname{Gr}_L^n(F)) be a mixed Hodge A-structure.

We now apply (8.1.13) and (8.1.15) to (8.1.12).

We obtain a mixed Hodge complex consisting of (a), (b), and (c) below.

The complex \mathrm{R}\Gamma\mathrm{R}j_*\mathbb{Z}\simeq\mathrm{R}\Gamma(U_\bullet,\mathbb{Z}), whose cohomology groups are the \operatorname{H}^n(U_\bullet,\mathbb{Z}).
The filtration \delta(W,L) of \mathrm{R}\Gamma\mathrm{R}j_*\mathbb{Q}\simeq\mathrm{R}\Gamma(U_\bullet,\mathbb{Q}). With the notation of (3.1.4), we have \begin{aligned} \operatorname{Gr}_n^{\delta(W,L)}\mathrm{R}\Gamma(U_\bullet,\mathbb{Q}) &\underset{(7.1.6.5)}{\simeq} \bigoplus_m \operatorname{Gr}_{n+m}^W \mathrm{R}\Gamma(U_m,\mathbb{Q})[-m] \\&\simeq \bigoplus_m \mathrm{R}\Gamma(\widetilde{Y}_m^{n+m},\varepsilon_\mathbb{Q}^{n+m}[-n-m])[-m] \\&= \bigoplus_m \mathrm{R}\Gamma(\widetilde{Y}_m^{n+m},\varepsilon_\mathbb{Q}^{n+m})[-n-2m]. \end{aligned} The E_1 page of the corresponding spectral sequence is the sum of the groups of form \operatorname{H}^p(\widetilde{Y}_q^r,\varepsilon_\mathbb{Q}^r); this contributes to the \operatorname{H}^{p+q+r}(U_\bullet); it corresponds to \operatorname{Gr}_a^{\delta(W,F)} for p+2r=(p+q+r)+a: {}_{\delta(W,L)}E_1^{-a,b} = \bigoplus_{\substack{p+2r=b\\q-r=-a}} \operatorname{H}^p(\widetilde{Y}_q^r,\varepsilon_\mathbb{Q}^r) \Rightarrow\operatorname{H}^{-a+b}(U_\bullet,\mathbb{Q}). \tag{8.1.19.1}
The bifiltration (\delta(W,L),F) of \mathrm{R}\Gamma\mathrm{R}j_*\mathbb{C}\simeq\mathrm{s}\mathrm{R}\Gamma^\bullet\Omega_X^\bullet(\log D).

The filtration F endows {}_{\delta(W,L)}E_1^{-a,b} with a Hodge structure of weight b, and the differentials d_1 of {}_{\delta(W,L)}E are the morphisms of the Hodge \mathbb{Q}-structures. The complex {}_{\delta(W,L)}E_q^{\bullet b} is the simple complex associated the double complex of Hodge structures of weight b given by: \small \begin{CD} \operatorname{H}^{b-2(r+2)}(\widetilde{Y}_{q+1}^{r+1},\varepsilon_\mathbb{Q}^{r+1})[-r-1] @>{\mathrm{Gysin}}>> \operatorname{H}^{b-2r}(\widetilde{Y}_{q+1}^{r},\varepsilon_\mathbb{Q}^{r})[-r] @>{\mathrm{Gysin}}>> \operatorname{H}^{b-2(r-2)}(\widetilde{Y}_{q+1}^{r-1},\varepsilon_\mathbb{Q}^{r-1})[-r+1] \\ @. @A{\sum_i(-1)^i\delta_i}AA @AA{\sum_i(-1)^i\delta_i}A \\ @. \operatorname{H}^{b-2r}(\widetilde{Y}_q^r,\varepsilon_\mathbb{Q}^r)[-r] @>>{\mathrm{Gysin}}> \operatorname{H}^{b-2(r-2)}(\widetilde{Y}_q^{r-1},\varepsilon_\mathbb{Q}^{r-1})[-r+1] \end{CD} The indices in the terms in E_1^{ab} satisfy -a=q-r.

In particular, the \operatorname{H}^n(U_\bullet,\mathbb{Z}) are endowed with mixed Hodge structures (8.1.9).

Under the above hypotheses:

The mixed Hodge structure of \operatorname{H}^n(U_\bullet,\mathbb{Z}) is functorial in the pair (U_\bullet,X_\bullet).
In rational cohomology, the spectral sequence (8.1.19.1) degenerates at E_2 (i.e. E_2=E_\infty) and abuts to the weight filtration of \operatorname{H}^n(U_\bullet,\mathbb{Q}). The Hodge structure of E_2 induced by that of E_1 is also that of \operatorname{Gr}_W\operatorname{H}^n(U_\bullet,\mathbb{Q}).
The Hodge numbers of \operatorname{H}^n(U_\bullet,\mathbb{Z}) satisfy h^{pq}\neq0 \implies 0\leqslant p,q\leqslant n.
For Y_\bullet=\varnothing, the Hodge numbers of \operatorname{H}^n(X_\bullet,\mathbb{Z}) satisfy h^{pq}\neq0 \implies [0\leqslant p,q\leqslant n \quad\text{and}\quad p+q\leqslant n].

Proof. Both (i) and (ii) follow from the general theory (8.1.9) (). To prove (iii), we contemplate (8.1.19) and note that the Hodge structures \operatorname{H}^p(\widetilde{Y}_q^r,\varepsilon_\mathbb{Q}^r)[-r] (for p+q+r=n) satisfy (iii). To prove (iv), we note that the Hodge structures \operatorname{H}^p(X_q) (for p+q=n) satisfy (iv).

We define a cohomological mixed Hodge A-complex on an r-simplicial topological space by transposing (8.1.11). If K is such a complex, then we see, as in (8.1.13), that \mathrm{R}\Gamma^\bullet K is an r-cosimplicial mixed Hodge A-complex (8.1.10.1).

Let X_\bullet be a smooth and proper r-simplicial scheme. A normal crossing divisor D_\bullet on X_\bullet is a system of normal crossing divisors on the X_n such that X_\bullet\setminus D_\bullet=U_\bullet is an r-simplicial subscheme of X_\bullet. Let j\colon U_\bullet\hookrightarrow X_\bullet. We define, as in (8.1.12), a cohomological mixed Hodge complex \mathrm{R}j_*\mathbb{Z} on X_\bullet.

We can, as in (8.1.20), thus deduce a mixed Hodge structure on \operatorname{H}^\bullet(U_\bullet,\mathbb{Z}), but, since \operatorname{H}^\bullet(U_\bullet,\mathbb{Z}) = \operatorname{H}^\bullet(\delta U_\bullet,\mathbb{Z}), this does not teach us anything new (the above equality would be an identity between mixed Hodge structures).

Set r=2. For every bounded-below bicosimplicial complex K=(K^{pn_1n_2}) (where p is the differential degree, and n_i is the i-th simplicial degree), we denote by \mathrm{s}_1K the cosimplicial complex of simplicial degree n_2 given by (\mathrm{s}_1K)^{\bullet n_2} = \mathrm{s}K^{\bullet\bullet n_2}. We denote by L_1 the filtration on \mathrm{s}_1K that, on each \mathrm{s}K^{\bullet\bullet n_2}, induces the filtration by the simplicial degree n_1.

Let K be a bicosimplicial mixed Hodge A-complex. It follows from (8.1.15) that (\mathrm{s}_1K,\delta(W,L_1),F) is a cosimplicial mixed Hodge complex. Let L_2 be the filtration of \mathrm{s}K=\mathrm{s}\mathrm{s}_1K by the simplicial degree n_2. By (8.1.14) and (8.1.15), the spectral sequence defined by L_2 comes from a spectral sequence in the abelian category of mixed Hodge A-structures {}_{L_2}E_1^{pq} = \operatorname{H}^q(\mathrm{s}K^{\bullet\bullet p}) \Rightarrow\operatorname{H}^{p+q}(\mathrm{s}K). \tag{8.1.22.1}

In this spectral sequence, the mixed Hodge A-structure on E_1^{pq} is the mixed Hodge A-structure ((ii) of (8.1.9)) of the cohomology of the mixed Hodge A-complex (\mathrm{s}K^{\bullet\bullet p},\delta(W,L_1),F). The mixed Hodge A-structure of the abutment is the mixed Hodge A-structure of the cohomology of the mixed Hodge A-complex (\mathrm{s}K,\delta(\delta(W,L_1),L_2),F). The filtration \delta(\delta(W,L_1),L_2)\eqqcolon\delta(W,L_1,L_2) is given by \delta(W,L_1,L_2)_k(K^{pn_1n_2}) = W_{k+n_1+n_2}K^{pn_1n_2}.

In particular, a cohomological mixed Hodge A-complex on a bisimplicial topological space X_{\bullet\bullet} defines a spectral sequence of mixed Hodge A-structures E_1^{pq} = \operatorname{H}^q(X_{\bullet p},K) \Rightarrow\operatorname{H}^{p+q}(X_{\bullet\bullet},K). \tag{8.1.23.1}

In the particular case considered in (8.1.21), for r=2, we have A=\mathbb{Z}, and (8.1.23.1) can be written as E_1^{pq} = \operatorname{H}^q(U_{\bullet p},\mathbb{Z}) \Rightarrow\operatorname{H}^{p+q}(U_{\bullet\bullet},\mathbb{Z}). \tag{8.1.23.2}

If K' and K'' are mixed Hodge A-complexes, then then their product K'\otimes K'', defined by \begin{aligned} (K'\otimes K'')_A &= K'_A\overset{\mathrm{L}}{\otimes}K''_A \\((K'\otimes K'')_{A\otimes\mathbb{Q}},W) &= (K'_{A\otimes\mathbb{Q}},W')\otimes(K''_{A\otimes\mathbb{Q}},W'') \\((K'\otimes K'')_\mathbb{C},W,F) &= (K'_\mathbb{C},W',F')\otimes(K''_\mathbb{C},W'',F'') \end{aligned} is again a mixed Hodge A-complex.

Let U' (resp. U'') be the complement of a normal crossing divisor Y' (resp. Y'') in a smooth and proper scheme X' (resp. X''). Then U=U'\times U'' is the complement of a normal crossing divisor Y in X=X'\times X''. Let j (resp. j', j'') be the inclusion of U (resp. U', U'') into X (resp. X', X'').

We have a quasi-isomorphism \mathrm{R}j'_*\mathbb{Z}\overset{\mathrm{L}}{\boxtimes}\mathrm{R}j''_*\mathbb{Z} \coloneqq \operatorname{pr}_1^*\mathrm{R}j'_*\mathbb{Z}\overset{\mathrm{L}}{\otimes}\operatorname{pr}_2^*\mathrm{R}j''_*\mathbb{Z} \xrightarrow{\sim}\mathrm{R}j_*\mathbb{Z}, a filtered quasi-isomorphism (\mathrm{R}j'_*\mathbb{Q},\tau_\leqslant)\overset{\mathrm{L}}{\boxtimes}(\mathrm{R}j''_*\mathbb{Q},\tau_\leqslant) \xrightarrow{\sim}(\mathrm{R}j_*\mathbb{Q},\tau_\leqslant), and a bifiltered morphism (\Omega_{X'}^\bullet(\log Y'),W,F)\boxtimes(\Omega_{X''}(\log Y''),W,F) \to (\Omega_X^\bullet(\log Y),W,F).

Applying the functor \mathrm{R}\Gamma gives us, by the Künneth formula, and by its analogue for cohomology of coherent analytic sheaves, an isomorphism of mixed Hodge complexes \mathrm{R}\Gamma(U',\mathbb{Z})\overset{\mathrm{L}}{\otimes}\mathrm{R}\Gamma(U'',\mathbb{Z}) \xrightarrow{\sim}\mathrm{R}\Gamma(U,\mathbb{Z}) whence an isomorphism of mixed Hodge \mathbb{Q}-structures \operatorname{H}^\bullet(U',\mathbb{Q})\otimes\operatorname{H}^\bullet(U'',\mathbb{Q}) \xrightarrow{\sim}\operatorname{H}^\bullet(U,\mathbb{Q}).

Standard arguments, based on the Cartier–Eilenberg–Zilber theorem, give us the following simplicial variant of (8.1.24).

Let U'_\bullet (resp. U''_\bullet) be the complement of a normal crossing divisor in a smooth and proper simplicial scheme X'_\bullet (resp. X''_\bullet). Let U_\bullet=U'_\bullet\times U''_\bullet and X_\bullet=X'_\bullet\times X''_\bullet.

We have an isomorphism of mixed Hodge complexes \mathrm{R}\Gamma(U'_\bullet,\mathbb{Z})\overset{\mathrm{L}}{\otimes}\mathrm{R}\Gamma(U''_\bullet,\mathbb{Z}) \xrightarrow{\sim}\mathrm{R}\Gamma(U_\bullet,\mathbb{Z}) and, in particular, an isomorphism of mixed Hodge \mathbb{Q}-structures \operatorname{H}^\bullet(U'_\bullet,\mathbb{Q})\otimes\operatorname{H}^\bullet(U''_\bullet,\mathbb{Q}) \xrightarrow{\sim}\operatorname{H}^\bullet(U_\bullet,\mathbb{Q}).

Let U'_{\bullet\bullet} (resp. U''_{\bullet\bullet}) be the complement of a normal crossing divisor in a smooth and proper bisimplicial scheme X'_{\bullet\bullet} (resp. X''_{\bullet\bullet}). Let U_{\bullet\bullet}=U'_{\bullet\bullet}\times U''_{\bullet\bullet} and X_{\bullet\bullet}=X'_{\bullet\bullet}\times X''_{\bullet\bullet}. Standard arguments show that, in rational cohomology, the spectral sequence (8.1.23.2) for U_{\bullet\bullet} is the tensor product of the spectral sequences (8.1.23.2) for U'_{\bullet\bullet} and U''_{\bullet\bullet}.