Cohomological mixed Hodge complexes

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.