Mixed Hodge Structures And Strictness

A rational mixed Hodge structure consists of a $\mathbb{Q}$-vector space $H$ with a finite basis (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis) together with

• a finite increasing filtration $W$ of $H$, the weight filtration, and
• a finite decreasing filtration $F$ of $H_{\mathbb{C}}:=H{\otimes }_{\mathbb{Q}}\mathbb{C}$, the Hodge filtration (The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums),

such that the following opposition condition holds on every weight-graded piece. Here a filtration is finite when $W_{a}H=0$ for all sufficiently negative $a$ and $W_{a}H=H$ for all sufficiently positive $a$, and dually $F^a{H}_{\mathbb{C}}=H_{\mathbb{C}}$ for $a\ll 0$ and $F^a{H}_{\mathbb{C}}=0$ for $a\gg 0$ (Increasing and decreasing filtration of an object).

Complexification and conjugation. Complex scalars act by $c(h\otimes z)=h\otimes cz$; this respects the rational balancing relations. A finite rational basis of $H$ identifies $H_{\mathbb{C}}$ with the corresponding finite-dimensional coordinate space over $\mathbb{C}$. The $\mathbb{Q}$-linear involution $\sigma$ of $H_{\mathbb{C}}$, defined on generators by $\sigma (h\otimes z)=h\otimes \bar{z}$ and extended $\mathbb{Q}$-linearly, satisfies ${\sigma }^2=\mathrm{i}\mathrm{d}$ and $\sigma (cz)=\bar{c}\sigma (z)$ for $c\in \mathbb{C}$ (The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$). It preserves the subspace $W_a{H}_{\mathbb{C}}:=(W_{a}H){\otimes }_{\mathbb{Q}}\mathbb{C}$ for every $a$, so it descends to each graded piece ${\mathrm{Gr}}_a^W{H}_{\mathbb{C}}:=W_a{H}_{\mathbb{C}}\mathrm{/}{W}_{a-1}{H}_{\mathbb{C}}$ (Associated graded object of a filtered object). The conjugate Hodge filtration is ${\bar{F}}^q:=\sigma (F^q)$, a finite decreasing filtration of $H_{\mathbb{C}}$.

Opposition. Let $V$ be a finite-dimensional $\mathbb{C}$-vector space with finite decreasing filtrations $F$ and $\bar{F}$, and let $n$ be an integer. The two filtrations are $n$-opposed when the natural map ${⨁}_{p+q=n}(F^{p}V\cap {\bar{F}}^{q}V)⟶V,(v_{p,q}{)}_{p+q=n}⟼{\sum }_{p+q=n}{v}_{p,q},$ is an isomorphism. Applied to $V={\mathrm{Gr}}_n^W{H}_{\mathbb{C}}$ this condition is required for every $n$, with the induced filtrations: $F^p$ acts on ${\mathrm{Gr}}_n^W{H}_{\mathbb{C}}$ through the image of $F^p\cap W_n{H}_{\mathbb{C}}$, and ${\bar{F}}^q$ through the image of ${\bar{F}}^q\cap W_n{H}_{\mathbb{C}}$. Writing $H_n^{p,q}:=F^p{\mathrm{Gr}}_n^W{H}_{\mathbb{C}}\cap {\bar{F}}^q{\mathrm{Gr}}_n^W{H}_{\mathbb{C}},$ the requirement is therefore the direct sum decomposition ${\mathrm{Gr}}_n^W{H}_{\mathbb{C}}={⨁}_{p+q=n}{H}_n^{p,q}\text{for every }n\in \mathbb{Z}\mathrm{.}$ Since $\sigma$ interchanges the induced filtrations $F^p$ and ${\bar{F}}^p$ and preserves $W$, it maps $H_n^{p,q}$ onto $H_n^{q,p}$, so $\overline{H_n^{p,q}}=H_n^{q,p}$. Equivalently, the opposition condition requires both $F^{p}V\cap {\bar{F}}^{q}V=0$ whenever $p+q>n$ and $F^{p}V+{\bar{F}}^{n+1-p}V=V$ for every $p$, on $V={\mathrm{Gr}}_n^W{H}_{\mathbb{C}}$. The conjunction, rather than either condition alone, is equivalent to the displayed decomposition: the spanning condition and the modular law give $F^{p}V=F^{p+1}V+(F^{p}V\cap {\bar{F}}^{n-p}V)$, and the vanishing condition makes this sum direct; descending induction through the finite filtration gives $F^{p}V={⨁}_{a\ge p}(F^{a}V\cap {\bar{F}}^{n-a}V)$ and then the decomposition of $V$. Conversely, the decomposition gives these two conditions by grouping its summands by the first and second indices.

Morphisms. A morphism $f\text{\hspace{0.17em}⁣}:H\to K$ of rational mixed Hodge structures is a $\mathbb{Q}$-linear map with $f(W_{a}H)\subseteq W_{a}K$ for all $a$ and $f_{\mathbb{C}}(F^p{H}_{\mathbb{C}})\subseteq F^p{K}_{\mathbb{C}}$ for all $p$, where $f_{\mathbb{C}}=f\otimes \mathrm{i}\mathrm{d}$. Such an $f$ commutes with the conjugations $\sigma$, hence also satisfies $f_{\mathbb{C}}({\bar{F}}^p{H}_{\mathbb{C}})\subseteq {\bar{F}}^p{K}_{\mathbb{C}}$.

Substructures, pure structures, Tate. A subspace $H^{\mathrm{\prime }}\subseteq H$ with the induced filtrations $W_a{H}^{\mathrm{\prime }}=H^{\mathrm{\prime }}\cap W_{a}H$ and $F^p{H}_{\mathbb{C}}^{\mathrm{\prime }}=H_{\mathbb{C}}^{\mathrm{\prime }}\cap F^p{H}_{\mathbb{C}}$ is a substructure when the induced filtrations again satisfy the opposition condition on every graded piece, and a quotient $H\to Q$ with the quotient filtrations is a quotient structure under the same requirement; that the kernel, image and cokernel of a morphism do carry such structures is not part of the definition and is proved in the strictness lemma below. A structure is pure of weight $n$ when $W_{n-1}H=0$ and $W_{n}H=H$; the opposition condition then says that $H_{\mathbb{C}}={⨁}_{p+q=n}{H}^{p,q}$ with $H^{p,q}=F^p\cap {\bar{F}}^q$, and $\overline{H^{p,q}}=H^{q,p}$. The Tate structure $\mathbb{Q}(r)$, for $r\in \mathbb{Z}$, is the pure structure of weight $-2r$ with $\mathbb{Q}(r{)}_{\mathbb{C}}=\mathbb{C}$ concentrated in type $(-r,-r)$, that is $H^{-r,-r}=\mathbb{C}$ and $H^{p,q}=0$ for $(p,q)\mathrm{\ne }(-r,-r)$.

Tensor and dual conventions. The tensor product $H\otimes K$ carries $W_a(H\otimes K)={\sum }_{b+c=a}{W}_{b}H\otimes W_{c}K,F^p(H\otimes K{)}_{\mathbb{C}}={\sum }_{b+c=p}{F}^b{H}_{\mathbb{C}}\otimes F^c{K}_{\mathbb{C}},$ where the weight sum is taken over $\mathbb{Q}$ and the Hodge sum over $\mathbb{C}$, using $(H{\otimes }_{\mathbb{Q}}K{)}_{\mathbb{C}}\cong H_{\mathbb{C}}{\otimes }_{\mathbb{C}}{K}_{\mathbb{C}}$; the dual $H^{\ast }={\mathrm{Hom}}_{\mathbb{Q}}(H,\mathbb{Q})$ carries $W_a{H}^{\ast }=\{u:u(W_{-a-1}H)=0\},F^p{H}_{\mathbb{C}}^{\ast }=\{u:u(F^{1-p}{H}_{\mathbb{C}})=0\}\mathrm{.}$

Polarizability. A pure rational Hodge structure $H$ of weight $n$ is polarizable when it carries a rational bilinear form $Q$ of parity $(-1{)}^n$ that pairs only complementary Hodge types, i.e. $Q(H^{p,q},H^{p^{\mathrm{\prime }},q^{\mathrm{\prime }}})=0$ unless $p^{\mathrm{\prime }}=n-p$ and $q^{\mathrm{\prime }}=n-q$, and whose Weil-operator Hermitian form $h(u,v):=Q(Cu,\bar{v})$, where the Weil operator $C$ acts as $i^{p-q}$ on $H^{p,q}$, is positive definite.

No mixed structure in this definition is assumed to split over $\mathbb{R}$ or over $\mathbb{Q}$; the Deligne splitting constructed below is a decomposition over $\mathbb{C}$ only.

Deligne II (2.3.1)-(2.3.4) is the primary source; Voisin, Definition 1.22 and Definition 2.6 and Weber, (2.1.5) and (2.2.1) state the same datum in lattice and in vector-space form. The polarizability clause records the Hodge-Riemann convention of Deligne II (2.1.15)-(2.1.16); it is a convention about which structures may be called polarizable and asserts no existence theorem. The forward mentions of the strictness lemma and of the splitting lemma refer to the two consequences proved later on this page; they are explanatory and create no dependency.

Let $(H,W,F)$ be a rational mixed Hodge structure (Rational mixed Hodge structures) with complex conjugation $\sigma$ on $H_{\mathbb{C}}=H{\otimes }_{\mathbb{Q}}\mathbb{C}$, and write ${\bar{F}}^a=\sigma (F^a)$ for the conjugate Hodge filtration; all weight spaces below are the complexifications $W_a{H}_{\mathbb{C}}$. For integers $p,q$ put $m=p+q$ and $I^{p,q}=(F^p\cap W_m)\cap \left(\right({\bar{F}}^q\cap W_m)+{\sum }_{j\ge 1}({\bar{F}}^{q-j}\cap W_{m-j-1}\left)\right),$ the intersections and the sum taken in $H_{\mathbb{C}}$. Then:

(a) $I^{p,q}\subseteq W_m\cap F^p$, the intersection $I^{p,q}\cap W_{m-1}$ is zero, and the projection $W_m\to {\mathrm{Gr}}_m^W{H}_{\mathbb{C}}$ maps $I^{p,q}$ isomorphically onto the summand $H_m^{p,q}$ of the mixed Hodge definition;

(b) $W_m{H}_{\mathbb{C}}={⨁}_{p+q\le m}{I}^{p,q}$, $F^a{H}_{\mathbb{C}}={⨁}_{p\ge a}{I}^{p,q}$ for all integers $a,m$; in particular $H_{\mathbb{C}}={⨁}_{p,q}{I}^{p,q}$;

(c) every morphism $f\text{\hspace{0.17em}⁣}:H\to K$ of rational mixed Hodge structures satisfies $f_{\mathbb{C}}(I^{p,q}(H))\subseteq I^{p,q}(K)$.

The construction satisfies $I^{p,q}+W_{m-2}=\sigma (I^{q,p})+W_{m-2}$ inside $W_m$. It need not satisfy $\sigma (I^{p,q})=I^{q,p}$, and the same splitting need not split $\bar{F}$: its conjugate gives ${\bar{F}}^b{H}_{\mathbb{C}}={⨁}_{q\ge b}\sigma (I^{q,p})$. The splitting is the one pinned down in the sources by the displayed formula, not an arbitrary direct-sum decomposition.

The construction is Deligne's; the proof above is a complete reconstruction of Deligne II (1.2.5), (1.2.8)-(1.2.11) in the increasing-weight convention. Step 2.2 replaces Deligne's induction (1.2.8)/(1.2.9) by a direct descending correction through the finite weight filtration, and steps 1.1 and 2.1 supply the two-filtration lemma (1.2.5) in the forms used afterwards. Weber (2.2.4)-(2.2.5) and El Zein, Lemma 2.1 state the displayed formula and the partial-sum decompositions; Cirici-Horel, Lemma 3.3 was an independent cross-check of the formula. The formula is not conjugate-symmetric in general: step 5.2 proves the congruence $I^{p,q}\equiv \sigma (I^{q,p})<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>W_{m-2}$ (Weber (2.2.5)), and the note after Voisin's Theorem 2.8 records that the pieces cannot in general be chosen simultaneously conjugate-symmetric and compatible with the two filtrations. The proof uses the exact $W,F$ decompositions and the proved congruence, without asserting exact conjugate symmetry or an exact $\bar{F}$ decomposition by the same $I^{p,q}$.

A rational cohomological mixed Hodge complex consists of

(a) a bounded-below cochain complex $K_{\mathbb{Q}}$ of $\mathbb{Q}$-vector spaces with an increasing filtration $W$ by subcomplexes, finite on each cochain group;

(b) a cochain complex $K_{\mathbb{C}}$ of $\mathbb{C}$-vector spaces with an increasing filtration $W$ by subcomplexes and a decreasing filtration $F$ by subcomplexes, again finite on each cochain group;

(c) a morphism of complexes $\alpha \text{\hspace{0.17em}⁣}:K_{\mathbb{Q}}{\otimes }_{\mathbb{Q}}\mathbb{C}\to K_{\mathbb{C}}$ compatible with the weight filtrations, such that for every $w$ the induced map ${\mathrm{Gr}}_w^W\alpha$ is a quasi-isomorphism,

subject to the following axiom: for every integer $w$ the weight-graded complex ${\mathrm{Gr}}_w^{W}K$ has finite-dimensional cohomology, its differential (induced by $d_K$) is strict for the induced filtration $F$, and for every $r$ the induced filtration $F$ on $H^r({\mathrm{Gr}}_w^W{K}_{\mathbb{C}})$, together with the induced rational comparison $H^r({\mathrm{Gr}}_w^W{K}_{\mathbb{Q}}){\otimes }_{\mathbb{Q}}\mathbb{C}\to H^r({\mathrm{Gr}}_w^W{K}_{\mathbb{C}})$, is a pure Hodge structure of weight $r+w$ (Rational mixed Hodge structures).

Here a cochain complex is a complex in the sense of Cochain complex in an abelian category, a filtration by subcomplexes is a filtered complex in the sense of Filtered chain complex, a morphism compatible with the weight filtrations is a filtered chain map in the sense of Filtered chain map, quasi-isomorphism means Quasi-isomorphism, and finiteness on a cochain group $K^n$ means $W_a{K}^n=0$ for all sufficiently negative $a$ and $W_a{K}^n=K^n$ for all sufficiently positive $a$, with bounds depending on $n$; dually $F^p{K}^n=K^n$ for $p\ll 0$ and $F^p{K}^n=0$ for $p\gg 0$.

The weight filtration on cohomology is the shifted image filtration $W_m{H}^n(K):=\mathrm{im}(H^n(W_{m-n}K)\to H^n(K))\mathrm{.}$ The shift is essential: it is what makes the axiom's weight $r+w$ on the graded complexes compatible with the weight of $H^n$, and pole-count weight on a logarithmic complex is not the weight on its cohomology. The differential strictness in the axiom is Deligne's condition (GH1); the equivalent formulation that the $F$-spectral sequence of the graded complex degenerates at its first page is recorded in the sources but is neither asserted nor used here, so no degeneration theorem enters this definition.

Morphisms and the sheaf case. A morphism of cohomological mixed Hodge complexes is a pair of chain maps $K_{\mathbb{Q}}\to L_{\mathbb{Q}}$ and $K_{\mathbb{C}}\to L_{\mathbb{C}}$ compatible with both filtrations and with the comparisons; compatible filtered homotopies are permitted. The same definitions apply to bounded-below complexes of sheaves on a topological space when the complexes are replaced by their derived global sections and the resolutions used preserve the graded pieces; the existence of such resolutions for the concrete complexes used later is proved where those complexes are constructed, not here.

Deligne III (8.1.1)-(8.1.6) is the primary source; the concrete formulation above uses a concrete comparison map in the displayed direction to represent the filtered comparison in Deligne's derived-category formulation. A comparison isomorphism in a derived category does not itself supply such a chain map: a suitable representative must be constructed when this definition is applied. Weber, (2.3.1) and (2.4.1) and Arapura, Section 4.5, state the same axioms (GH1)/(GHM) in the vector-space and sheaf settings. The shift in the displayed weight filtration is the convention (1.1.2) used throughout this page. The definition itself uses no choice principle.

Every morphism $f\text{\hspace{0.17em}⁣}:H\to K$ of rational mixed Hodge structures is strict for $W$ and for $F$: $f(W_{n}H)=f(H)\cap W_{n}K,f(F^p{H}_{\mathbb{C}})=f(H_{\mathbb{C}})\cap F^p{K}_{\mathbb{C}}$ for all $n,p$. Its kernel, image and cokernel, with the induced or quotient filtrations, are mixed Hodge structures; the functor ${\mathrm{Gr}}_n^W$ is exact; and a morphism from a structure of weights at least $n$ (that is $W_{n-1}H=0$) to a structure of weights at most $n$ (that is $W_{n}K=K$) has pure image of weight $n$.

Deligne II (1.2.10)-(1.2.12) and Theorem (2.3.5) are the source; the proof above follows Voisin's printed deduction of Theorem 2.7 from Theorem 2.8 and uses the splitting lemma of this page in place of Deligne's three-opposed-filtrations category theorem. The statement claims no canonical splitting of the induced structures on kernel, image or cokernel, and the argument is choice-free: all decompositions used are the canonical ones supplied by the splitting lemma.