Proper cohomology and functorial mixed Hodge weights

(Actual cosimplicial Hodge complexes.) Choose the smooth projective proper hypercover $a:Y_{\bullet }\to Y$ of [F1]. For sheaves of rational or complex vector spaces, construct the Godement resolution of [F10]. Its first functor $C^0$ is exact because stalks are exact and products of surjective linear maps are surjective under AC. In a short exact sequence of sheaves, the corresponding germ embeddings into $C^0$ form a diagram with exact rows and columns; the kernel-cokernel exact sequence makes the three cokernels short exact. Iteration proves that every successive cokernel functor, every $C^t$ and every $\mathrm{\Gamma }{C}^t$ is exact, the last again being a product of stalks. In particular resolutions and global sections preserve filtration levels, their finite intersections and associated gradings. Their first-quadrant totals compute derived global sections: termwise flasque acyclicity from [F10] applies to each sheaf of the bounded de Rham complex, and each total diagonal is finite. Apply this construction on every $Y_p$ and put $K_{\mathbb{Q}}^{p,q}=\mathrm{\Gamma }(Y_p,C^q({\mathbb{Q}}_{Y_p}))$ and $K_{\mathbb{C}}^{p,q}={⨁}_{k+t=q}\mathrm{\Gamma }(Y_p,C^t({\mathrm{\Omega }}_{Y_p}^k))$, for $p,q\ge 0$, with $F^b$ given by $k\ge b$. These are the resolved global de Rham complexes, not the complex of global holomorphic forms. Godement pullback along each face is induced by pulling a germ collection at a point to the stalk over its image; the successive cokernels preserve these maps, so their composites obey the simplicial identities and preserve $F$. The constant inclusion $\mathbb{Q}\otimes \mathbb{C}\to {\mathrm{\Omega }}^0$ gives the functorial rational comparison; on cohomology it is the scalar-extension comparison of [F4]. Exactness and acyclicity of the resolution show $H^q(K_{\mathbb{C}}^p)=H^q(Y_p;\mathbb{C})$, with the holomorphic-degree image filtration, and $H^q(K_{\mathbb{Q}}^p)=H^q(Y_p;\mathbb{Q})$. The compact Hodge theorem [F4] gives pure weight $q$ and Hodge-de Rham $E_1$ degeneration on each smooth projective component. The latter makes the actual internal differential $F$-strict: if $dx\in F^b$, degeneration lifts the leading lower-filtration class of $x$ to a cycle up to a higher-filtration term and a boundary; subtracting the cycle and boundary raises the preimage filtration without changing $dx$, and finite induction supplies a preimage in $F^b$. This proves differential strictness directly. Thus every level is an actual weight-zero rational Hodge complex. Each internal total diagonal is finite, and $F^{q+1}{K}^{p,q}=0$, so [F3] applies to its first-quadrant simple complex, giving the mixed Hodge structure and $E_1^{p,q}=H^q(Y_p;\mathbb{Q})$, $E_2$ degeneration and ${\mathrm{Gr}}_q^W{H}^n(sK)=E_2^{n-q,q}$. The terms have $p=n-q\ge 0$, so weights are at most $n$, and the finitely many $E_1^{p,q}$ entries on every total diagonal are finite-dimensional by [F4]. Descent [F2] identifies the rational simple cohomology naturally with $H^n(Y;\mathbb{Q})$, carrying this structure.

Singular comparison. The point map $Y_0^{\mathrm{a}\mathrm{n}}\to Y^{\mathrm{a}\mathrm{n}}$ is a continuous surjection: a nonempty finite-type fibre over a complex point has a complex closed point, as in the residue-field argument of [F4]'s analytification supplier. Thus $Y^{\mathrm{a}\mathrm{n}}$ is compact. A finite affine cover makes it second-countable, and separatedness makes its diagonal closed in the products of these affine point loci; hence it is Hausdorff. The topology of a nonreduced scheme is that of its reduction. On each affine chart [F6] supplies a finite deleted-boundary triangulation. At a point of a retained simplex, its open star is $\mathrm{relint}\mathrm{\mid }\sigma \mathrm{\mid }\times c_{<1}(L)$. Small convex neighbourhoods of the point in the first factor and small radial cones in the second give a contractible open neighbourhood basis: compactness of the link gives a uniform radial bound at the apex. To obtain a metric, take a finite partition of unity ${\varphi }_j$ subordinate to the affine charts, with supports contained in the charts, using [F6]. If $z_j$ are the finite affine coordinate vectors, define $\mathrm{\Phi }(y)=({\varphi }_j(y),{\varphi }_j(y)z_j(y){)}_j$, extending each product by zero outside its chart. These extensions are continuous because the supports are closed and contained in their charts. If $\mathrm{\Phi }(y)=\mathrm{\Phi }(y^{\mathrm{\prime }})$, choose $j$ with ${\varphi }_j(y)>0$; equality of the scalar coordinates puts both points in that chart with equal positive ${\varphi }_j$, and equality of the product coordinates gives $z_j(y)=z_j(y^{\mathrm{\prime }})$, hence $y=y^{\mathrm{\prime }}$. A continuous injection from this compact space to Hausdorff Euclidean space is a homeomorphism onto its image: images of closed subsets are compact and therefore closed. Pulling back the Euclidean metric gives the analytic topology. The sheaf-singular comparison of [F6] now identifies the descended cohomology with $H_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^n(Y^{\mathrm{a}\mathrm{n}};\mathbb{Q})$, naturally in $Y$. The partition and comparison consume their stated AC hypotheses; no claim of a DC-only construction is made.

Independence and functoriality. If $a^{\mathrm{\prime }}:Y_{\bullet }^{\mathrm{\prime }}\to Y$ is a second split proper hypercover with smooth projective terms, [F1] supplies a third such hypercover dominating both; the two refinements give filtered comparison maps between the corresponding cosimplicial systems and hence morphisms of the mixed Hodge complexes of step 1.1, and by [F2] both simple complexes compute the constant-sheaf cohomology of $Y$, so the induced maps on $H^n(Y,\mathbb{Q})$ are the identity; strictness [F5] upgrades this to an isomorphism of mixed Hodge structures, so the structure is independent of the diagram. A morphism $g:Y\to Y^{\mathrm{\prime }}$ of proper varieties lifts, after refining the source hypercover, to a morphism of the chosen hypercovers [F1], pullback of holomorphic forms along it is a morphism of the cosimplicial mixed Hodge complexes compatible with the comparisons, and independence of the hypercovers makes the induced map $g^{\ast }:H^n(Y^{\mathrm{\prime }},\mathbb{Q})\to H^n(Y,\mathbb{Q})$ well defined and compatible with identities and composition.

(The smooth-open clause and comparison compatibility.) Let $g:Y\to U$, and use the logarithmic mixed Hodge complex realizing [F7] on an SNC compactification $X\supset U$. On its complex side, pullback along $g_0=g{a}_0:Y_0\to U$ takes logarithmic forms to regular forms: a boundary equation pulls back to a nowhere-zero holomorphic function, whose $d\log$ is regular. Resolve this map by the pullback construction of step 1.1 and place it in cosimplicial degree zero of $s{K}_{\mathbb{C}}$. Its first alternating face differential vanishes because $g_0{d}_0=g_0{d}_1$ on $Y_1$, so this is a chain map preserving $F$ by form degree. Its cohomology map is $g^{\ast }\otimes \mathbb{C}$: on $U$, ${\mathrm{\Omega }}_X^{\bullet }(\log D)$ restricts to ${\mathrm{\Omega }}_U^{\bullet }$, and the constant-sheaf de Rham comparison in the logarithmic construction restricts to the usual natural comparison; descent and the augmentation then identify this degree-zero map with pullback to $Y$. To obtain a rational chain map, split each source and target cochain group into boundaries, representatives for cohomology, and a complement to cycles, under AC. The differential identifies the last summand with the next boundary summand. Send cohomology representatives by the actual rational $g^{\ast }$ and send the other two summands to zero. This is a chain map $K_{\mathbb{Q}}(U)\to (sK{)}_{\mathbb{Q}}$ inducing $g^{\ast }$. Its scalar extension and the complex map commute with the rational comparisons up to homotopy: their difference induces zero on cohomology, and the same decompositions contract each boundary-complement pair, giving a homotopy for any such zero-cohomology map. All these maps and homotopies preserve weight, since the source pole filtration is zero for $a<0$ and the target $L_a$ is the whole simple complex for $a\ge 0$; no filtration constraint is placed on the chosen complements. Thus the two chain maps, with this weight-filtered comparison homotopy, are a morphism of the mixed Hodge complexes. By [F3] the actual rational $g^{\ast }$ is a mixed Hodge morphism.

The class map. Let $h:Y\to X$ be proper with $X$ smooth projective of pure dimension $d$. By step 2.1 and [F8], Borel-Moore homology of $Y$ equals ordinary homology. The Kronecker isomorphism $H^i(Y)=H_i(Y{)}^{\ast }$ and finite-dimensionality of $H^i(Y)$ imply that $H_i(Y)$ is finite-dimensional: any finite independent tuple extends to a basis by [F8] under AC and gives that many independent coordinate functionals, so its length is bounded by $\dim H^i(Y)$. Consequently evaluation identifies $H_i(Y)=H^i(Y{)}^{\ast }$ canonically. Give it the dual filtrations of Rational mixed Hodge structures. Their weight-$a$ graded piece is $({\mathrm{Gr}}_{-a}^W{H}^i(Y){)}^{\ast }$, with pure types $(p,q)$ changed to $(-p,-q)$; this proves the mixed Hodge axiom and the lower weight bound $-i$. Transposes of filtration-preserving maps preserve the annihilator filtrations. Naturality of the Kronecker pairing identifies $h_{\ast }$ with the transpose of $h^{\ast }$, a mixed Hodge morphism by step 3.1. For $X$, cup product preserves Hodge type by [F4], and the top cohomology on each connected component is of type $(d,d)$: forms with total degree $2d$ have only that possible type. Evaluation on the oriented fundamental classes therefore gives a Hodge morphism $H^{2d}(X)\to \mathbb{Q}(-d)$ (sum the component traces). Poincare duality and the cap/cup identity [F8] make the resulting cup/trace pairing $H^i(X)\otimes H^{2d-i}(X)\to \mathbb{Q}(-d)$ perfect. After twisting the second factor by $(d)$ it identifies $H^{2d-i}(X)(d)$ with $H^i(X{)}^{\ast }=H_i(X)$ as pure Hodge structures of weight $-i$: the complementary types become $(-p,-q)$ on the dual factor. The composite with $h_{\ast }$ is the asserted class map. By [F5] its image is pure of weight $-i$; untwisting gives weight $2d-i$. For mixed-dimensional $X$, apply this on its finitely many open-and-closed connected components and the corresponding inverse images in $Y$, obtaining the stated direct sum.

Boundaries and Choice accounting. For $Y=\mathrm{\varnothing }$ every group is zero with the trivial structure and all clauses are vacuous; the degree $n=0$ case has all weights equal to $0$, consistent with the bound; the smooth hypercover terms of differing dimensions are handled componentwise in [F4], without assuming that the singular components of $Y$ are disjoint; and the case $Y=X$ of pure dimension $d$ reduces the class map to Poincare duality itself. The Axiom of Choice [F9] is inherited from [F1], [F2] and [F7], including the sheaf-singular comparison; the exact Godement resolutions of step 1.1 and the comparison homotopy in step 3.2 use AC for products and linear complements; it also licenses the partition in step 2.1 and the duality argument in step 4.1.