2.3 Mixed structures

A mixed Hodge structure H consists of

  1. A \mathbb{Z}-module H_\mathbb{Z} of finite type, called the “integral lattice”;
  2. A finite increasing filtration W_n on H_\mathbb{Q}=\mathbb{Q}\otimes_\mathbb{Z}H_\mathbb{Z}, called the weight filtration;
  3. A finite filtration F^p on H_\mathbb{C}=\mathbb{C}\otimes_\mathbb{Z}H_\mathbb{Z}, called the Hodge filtration.

We demand that, on H_\mathbb{C}, the filtration W_\mathbb{C} induced by extension of scalars of W, the filtration F, and the complex conjugate \overline{F} form a system (W_\mathbb{C},F,\overline{F}) of three opposite filtrations ((1.2.7) and (1.2.13)).

We also denote by W the filtration of H_\mathbb{Z} given by the inverse image of the filtration W on H_\mathbb{Q}. The axiom of mixed Hodge structures implies that, for each n, the filtration F induces on \mathbb{C}\otimes_\mathbb{Z}\operatorname{Gr}_n^W(H_\mathbb{Z}) a filtration that is n-opposite to its complex conjugate. By (2.1.9), \operatorname{Gr}_n^W(H_\mathbb{Z}) is endowed with a Hodge structure of weight n, with Hodge filtration induced by F.

If H is a Hodge structure of weight n, then we define a mixed Hodge structure with the same integral lattice and the same Hodge filtration by setting W_i(H_\mathbb{Q}) = \begin{cases} 0 &\text{for } i<n \\H_\mathbb{Q}&\text{for } i\geqslant n. \end{cases}

A morphism f\colon H\to H' of Hodge structures is a homomorphism f\colon H_\mathbb{Z}\to H'_\mathbb{Z} that is compatible with the filtrations W and F (and thus compatible with \overline{F}). We immediately deduce from (1.2.10) the following theorem.

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  1. The category of mixed Hodge structures is abelian.
  2. The integral lattice of the kernel (resp. cokernel) of a morphism f\colon H\to H' is the kernel (resp. cokernel) K of f\colon H_\mathbb{Z}\to H'_\mathbb{Z}, with K\otimes\mathbb{Q} and K\otimes\mathbb{C} being endowed with the induced filtrations (resp. quotient filtrations) of the filtrations W and F of H_\mathbb{Q} and H_\mathbb{C} (resp. of H'_\mathbb{Q} and H'_\mathbb{C}).
  3. Every morphism f\colon H\to H' is strictly compatible with the filtration W of H_\mathbb{Q} and H'_\mathbb{Q} and with the filtration F of H_\mathbb{C} and H'_\mathbb{C}. It induces morphisms of Hodge \mathbb{Q}-structures \operatorname{Gr}_n^w(f)\colon \operatorname{Gr}_n^W(H_\mathbb{Q}) \to \operatorname{Gr}_n^W(H'_\mathbb{Q}). It also induces morphisms \operatorname{Gr}_F^p(F)\colon \operatorname{Gr}_F^p(H_\mathbb{C}) \to \operatorname{Gr}_F^p(H'_\mathbb{C}) that are strictly compatible with the filtration induced by W_\mathbb{C}.
  4. The functor \operatorname{Gr}_n^W is an exact functor from the category of mixed Hodge structures to the category of Hodge \mathbb{Q}-structures of weight n.
  5. The functor \operatorname{Gr}_F^p is an exact functor.

Let H be a mixed Hodge structure. The W_n(H_\mathbb{Z}), endowed with the filtrations induced by W and F, then form mixed Hodge substructures W_n(H) of H. The quotient W_n(H)/W_{n-1}(H) can be identified with \operatorname{Gr}_n^W(H_\mathbb{Z}) endowed with its mixed Hodge structure ((2.3.2) and (2.3.3)). We denote this Hodge structure by \operatorname{Gr}_n^W(H).

We set H^{p,q} = \operatorname{Gr}_F^p\operatorname{Gr}_{\overline{F}}^q\operatorname{Gr}_{p+q}^W(H_\mathbb{C}) = (\operatorname{Gr}_{p+q}^W(H))^{p,q}. The Hodge numbers of H are the integers h^{p,q} = \dim_\mathbb{C}H^{p,q}. The Hodge number h^{pq} of H is thus the Hodge number h^{pq} of the Hodge structure \operatorname{Gr}_{p+q}^W(H)

We define a mixed Hodge \mathbb{Q}-structure H to consist of a vector space H_\mathbb{Q} of finite dimension over \mathbb{Q}, a finite increasing filtration W on H_\mathbb{Q}, and a finite decreasing filtration F on H_\mathbb{C}, with the filtrations W_\mathbb{C}, F, and \overline{F} being opposite. (2.3.5) generalises trivially to this variation.