Rational mixed Hodge structures

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.