Mixed Hodge complex cohomology and simplicial weights

(First page and pure strictness.) Put $G^p=W_{-p}$ and apply [F1] over $\mathbb{Q}$ and $\mathbb{C}$. Then $E_0^{p,q}={\mathrm{Gr}}_G^p{K}^{p+q}$, $E_1^{p,q}=H^{p+q}({\mathrm{Gr}}_{-p}^{W}K)$ and $E_{\mathrm{\infty }}^{p,q}={\mathrm{Gr}}_G^p{H}^{p+q}(K)$ for the image filtration. The axiom gives $E_1^{p,q}$ a pure rational Hodge structure of weight $q$. We will use the following explicit pure case of [F2]: a rational map preserving $F$ also preserves $\bar{F}$; between pure structures of the same weight it therefore preserves each type $F^a\cap {\bar{F}}^b$, so it is strict for $F$, and its kernels, images and cokernels have the induced pure structures. If the source has weight $q$ and the target has weight $q^{\mathrm{\prime }}<q$, it is zero: a source vector of type $(a,b)$ with $a+b=q$ maps into $F^a\cap {\bar{F}}^b$ of the target, which is zero because no target type of total $q^{\mathrm{\prime }}$ has both indices at least $a,b$. Thus no mixed Hodge structure on the abutment is being assumed.

(Strict representative correction.) For a complex $C$ with decreasing filtration $T$ by subcomplexes and $T$-strict differential, the sequence $0\to H^n(T^{a}C)\to H^n(C)\to H^n(C\mathrm{/}{T}^{a}C)\to 0$ is exact for every $a,n$. For injectivity, a cycle $x\in T^{a}C$ that is a boundary $dy$ in $C$ equals $d{y}^{\mathrm{\prime }}$ for some $y^{\mathrm{\prime }}\in T^{a}C$ by strictness. For surjectivity, a quotient cycle lifts to $x\in C$ with $dx\in T^{a}C$; strictness gives $y\in T^{a}C$ with $dy=dx$, so $x-y$ is a cycle with the same quotient class. If a cycle maps to a quotient boundary, subtract a lift of that boundary to obtain a cycle in $T^{a}C$, proving exactness at the middle. In particular $H(T^{a}C)$ identifies with the filtration induced on $H(C)$ by $T^a$. These corrections use only strictness of the differential on $C$.

(Scalar extension.) Tensoring rational vector spaces with $\mathbb{C}$ preserves exact sequences, hence kernels, images, quotients and cohomology. Here is a finite argument that needs no basis of an arbitrary vector space. The field $\mathbb{C}$ is the directed union of its finite-dimensional rational subspaces $U$; tensor elements and the relations equating them involve finitely many coefficients, so $V\otimes \mathbb{C}$ is the directed union, in the tensor presentation, of the systems $V\otimes U$. On a fixed $U$, a finite rational basis identifies tensoring with a finite direct sum of copies of the given sequence, which is exact. A tensor in a kernel is represented at some $U$, and its vanishing in the target is witnessed at a larger finite-dimensional $U^{\mathrm{\prime }}$; exactness there supplies a preimage. Surjectivity follows by lifting the finitely many vector coefficients of a tensor. Thus the original tensor sequence is exact; applying this to cycles and boundaries gives the cohomology assertion. The same subquotient argument identifies the rational spectral-sequence pages after scalar extension with the pages of $K_{\mathbb{Q}}\otimes \mathbb{C}$, compatibly with their differentials.

(Exact filtered pages through $E_2$.) Fix a Hodge index $a$, and apply [F1] with the intersection and quotient $G$ filtrations to $A=F^a{K}_{\mathbb{C}}$, $B=K_{\mathbb{C}}$ and $C=B\mathrm{/}A$. Their $E_0$ complexes form a short exact sequence $0\to E_0(A)\to E_0(B)\to E_0(C)\to 0$: the first image in $G^{p}B\mathrm{/}{G}^{p+1}B$ is $(F^{a}B\cap G^{p}B+G^{p+1}B)\mathrm{/}{G}^{p+1}B$, exactly $F^a{E}_0(B)$, and the last term is its quotient. The mixed Hodge complex axiom says $d_0$ on $E_0(B)={\mathrm{Gr}}^G{K}_{\mathbb{C}}$ is $F$-strict. Step 1.2 therefore gives a short exact sequence on $E_1$, with first image exactly $F^a{E}_1(B)$. Naturality of the page maps shows $d_1$ preserves this image for every $a$; its rational form then makes $d_1:E_1^{p,q}\to E_1^{p+1,q}$ a map of pure Hodge structures of weight $q$, strict by step 1.1. Apply step 1.2 again to the $E_1$ complexes to obtain exactness on $E_2$, with $E_2(A)$ identifying with the filtration induced on $E_2(B)=H(E_1(B),d_1)$. Denote this recursively induced filtration by $F_{\mathrm{r}\mathrm{e}\mathrm{c}}$. The pure type decomposition of $E_1$ descends through its type-preserving $d_1$, so $E_2^{p,q}(B)$ with $F_{\mathrm{r}\mathrm{e}\mathrm{c}}$ is pure of weight $q$.

(Higher differentials and exact stable pages.) Suppose the three page complexes are exact at page $r\ge 2$, with $E_r(A)$ identifying with $F_{\mathrm{r}\mathrm{e}\mathrm{c}}^a{E}_r(B)$ for every $a$, and $E_r^{p,q}(B)$ pure of weight $q$. Naturality shows $d_r$ preserves these images, hence $F_{\mathrm{r}\mathrm{e}\mathrm{c}}$; it also preserves the rational structures. Its target $E_r^{p+r,q-r+1}$ has lower weight, so step 1.1 forces $d_r=0$. This zero map is strict, and step 1.2 applied to the exact triple of $E_r$ complexes gives exactness at $E_{r+1}$ and the recursive filtration identification. Since $d_r(B)=0$, injectivity of $E_r(A)\to E_r(B)$ and surjectivity onto $E_r(C)$ also give $d_r(A)=d_r(C)=0$. Starting from step 2.1 proves this induction for all $r\ge 2$. Thus the weight sequence degenerates at $E_2$, and the three sequences for $A,B,C$ stabilize there; $E_{\mathrm{\infty }}^{p,q}(B)=E_2^{p,q}(B)$ is pure of weight $q$ for the recurrent filtration.

(Recurrent filtration equals the abutment filtration.) By convergence and naturality [F1], the injective stable-page map $E_{\mathrm{\infty }}(A)\to E_{\mathrm{\infty }}(B)$ is ${\mathrm{Gr}}_{G}H(A)\to {\mathrm{Gr}}_{G}H(B)$ for the image $G$ filtrations on cohomology. A filtered linear map between finite filtered spaces whose graded map is injective is injective and strict: for any nonzero $x$, choose its largest decreasing-filtration index $p$; its nonzero graded class has nonzero image, so the image has this same index. This proves injectivity and, if the image of $x$ belongs to $G^p$, forces $x\in G^p$, which is strictness. The cohomology $G$ filtrations here are finite in each degree because $G$ is finite on the degree-$n$ cochain group, so the argument applies to $H^n(F^{a}K)\to H^n(K)$. Consequently its graded image is the induced filtration of the subspace $F^a{H}^n(K)=\mathrm{im}(H^n(F^{a}K)\to H^n(K))$ on ${\mathrm{Gr}}_G{H}^n(K)$: in filtration degree $p$ the numerator is $(F^a{H}^n(K)\cap G^p{H}^n(K))+G^{p+1}{H}^n(K)$. By steps 2.1 and 3.1 that same image on $E_{\mathrm{\infty }}$ is $F_{\mathrm{r}\mathrm{e}\mathrm{c}}^a$. This proves exactly the recurrent-to-abutment comparison needed here, with the strict representative correction supplied in step 1.2.

(Mixed Hodge structure and shifted weights.) Write $W_m{H}^n(K)=\mathrm{im}(H^n(W_{m-n}K)\to H^n(K))$. Substitution $p=n-m$, $q=m$ in [F1] and step 3.1 gives ${\mathrm{Gr}}_m^W{H}^n(K)={\mathrm{Gr}}_G^{n-m}{H}^n(K)\cong E_2^{n-m,m}$. Step 4.1 makes this identification compatible with the Hodge filtration induced by $F$ on cohomology; the right side is pure of weight $m$ by step 2.1. By step 1.3, the rational comparison identifies scalar-extended rational pages with complex pages: it is an isomorphism on $E_1$ by the graded quasi-isomorphism axiom and hence on the stable pages. Its cohomology map is injective by the leading-term argument of step 4.1 and surjective by lifting a target graded class and correcting the remaining error through the finite filtration; thus it is an isomorphism. The finitely many nonzero stable terms in degree $n$ are finite-dimensional, so $H^n(K_{\mathbb{Q}})$ is finite-dimensional. The induced $W$ and $F$ are finite, and every ${\mathrm{Gr}}_m^W$ with its induced $F$ is pure of weight $m$, precisely the mixed Hodge axiom.

(Functoriality.) A morphism of cohomological mixed Hodge complexes induces rational and complex cohomology maps compatible with the comparisons. It maps $W_{m-n}K$ to $W_{m-n}L$ and $F^a{K}_{\mathbb{C}}$ to $F^a{L}_{\mathbb{C}}$, hence preserves the cohomology image filtrations $W_m$ and $F^a$. These maps are therefore morphisms of the mixed Hodge structures of step 5.1. Naturality of [F1] and of the exact-page construction makes the graded-piece identifications compatible with these maps and their Hodge filtrations.

(The cosimplicial case.) Form $s{K}^n={⨁}_{p+q=n}{K}^{p,q}$ on both rational and complex sides, with $F^a(s{K}^n{)}_{\mathbb{C}}={⨁}_{p+q=n}{F}^a{K}_{\mathbb{C}}^{p,q}$. On $K^{p,q}$ put $D=\delta +(-1{)}^{p}d$, where $\delta ={\sum }_{i=0}^{p+1}(-1{)}^i{\mathrm{\partial }}^i$ is the alternating coface map and $d$ is the internal differential. The cosimplicial identities pair the terms of ${\delta }^2$ with opposite signs, so ${\delta }^2=0$; the cofaces are chain maps, so $\delta d=d\delta$, and the cross terms in $D^2$ cancel because $\delta$ raises $p$. Thus $D^2=0$, and $D$ preserves $F$ because both its summands do. The filtration $L_{\mathrm{\ell }}s{K}^n={⨁}_{i\ge -\mathrm{\ell }}{K}^{i,n-i}$ is increasing, by subcomplexes, and finite on each degree because $0\le i\le n$. Its graded piece is ${\mathrm{Gr}}_{\mathrm{\ell }}^{L}sK\cong K^{-\mathrm{\ell }}[\mathrm{\ell }]$, so $H^r({\mathrm{Gr}}_{\mathrm{\ell }}^{L}sK)=H^{r+\mathrm{\ell }}(K^{-\mathrm{\ell }})$ is pure of weight $r+\mathrm{\ell }$, and its internal differential is strict for $F$ by the Hodge complex hypothesis. Levelwise rational comparisons and filtration-preserving coface maps assemble to the total comparison, which is a graded quasi-isomorphism; the total complex is bounded below and its $F$ filtration is degreewise finite because each diagonal is finite. Thus $sK$ is a rational cohomological mixed Hodge complex for $L$. Applying steps 1.1-6.1 gives $E_1^{p,q}=H^{p+q}({\mathrm{Gr}}_{-p}^{L}sK)=H^q(K^p)$, degeneration at $E_2$ and ${\mathrm{Gr}}_q^W{H}^n(sK)={\mathrm{Gr}}_{q-n}^L{H}^n(sK)\cong E_2^{n-q,q}$, with its induced Hodge filtration and functorial mixed structure. Here $W_q{H}^n=L_{q-n}{H}^n$ is the shifted cohomology filtration, not the unshifted $L$.

(Weights at most the degree.) Since the cosimplicial degree is nonnegative, $E_1^{p,q}=0$ for $p<0$, hence $E_2^{n-q,q}=0$ when $q>n$. Step 7.1 gives ${\mathrm{Gr}}_q^W{H}^n(sK)=0$ for all $q>n$; finiteness of the weight filtration then gives $W_n{H}^n(sK)=H^n(sK)$, proving the stated bound.