Proper Hyperresolutions And Mixed Hodge Restriction Images

$X$ carries the closed immersion $i:X\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^N$ given by its projectivity, and $L_0=i^{\ast }{\mathcal{O}}_{{\mathbb{P}}_{\mathbb{C}}^N}(1)$ is invertible. The standard charts $U_0,\dots ,U_N$ are affine and cover ${\mathbb{P}}_{\mathbb{C}}^N$, so the finitely many intersections $X\cap U_j=X_{s_j}$, where $s_j=i^{\ast }(x_j)\in \mathrm{\Gamma }(X,L_0)$, are affine: each is a closed subscheme of the affine chart $U_j$, hence $\mathrm{Spec}(A_j)$ for a quotient $A_j$ of the polynomial ring over $\mathbb{C}$ [lem-closed-immersion-affine-quotient-and-base-change]. Every $A_j$ is Noetherian [lem-field-is-noetherian, cor-finite-variable-polynomial-ring-noetherian, thm-noetherian-ring-quotients-and-localisations], so this finite affine cover exhibits the scheme $X$ as locally Noetherian and quasi-compact [def-locally-noetherian-and-noetherian-scheme, def-quasi-compact-and-quasi-separated-scheme]. The coordinate section $x_j$ has nonvanishing locus $U_j$ and pullback along a closed immersion of an invertible sheaf has as nonvanishing locus the preimage of the original one [def-very-ample-invertible-sheaf-relative, def-closed-immersion-schemes], so the charts $X_{s_j}=X\cap U_j$ are affine and cover $X$; the ampleness criterion of [def-ample-invertible-sheaf] therefore makes $L_0$ ample on $X$.

By [F2] the quasi-coherent finite-type ideal sheaf $\mathcal{I}$ is coherent on the locally Noetherian $X$, and since $X$ is projective over the Noetherian ring $\mathbb{C}$ and $L_0$ is ample there is $d_0$ such that $\mathcal{I}\otimes L_0^{\otimes d}$ is globally generated for every $d\ge d_0$. Fix $d\ge \max (1,d_0)$, put $L=L_0^{\otimes d}$ and $\mathcal{F}=\mathcal{I}\otimes L\subseteq L$, and choose finitely many global sections $s_0,\dots ,s_n\in \mathrm{\Gamma }(X,\mathcal{F})$ generating $\mathcal{F}$. To justify finiteness, global generation supplies finitely many such sections generating each finite-type stalk; their coherent cokernel vanishes on a neighbourhood of that point (on an affine chart, each of its finitely many generators is killed after localizing away from that point). Quasi-compactness of $X$ supplies finitely many of these neighbourhoods, and the union of their finite section tuples generates $\mathcal{F}$ everywhere. Since $\mathcal{I}\mathrm{\ne }0$ and $L$ is invertible, $\mathcal{F}\mathrm{\ne }0$, so not all $s_i$ are zero and $\mathcal{F}={\sum }_i{\mathcal{O}}_X{s}_i$ as subsheaves of $L$.

Apply [F1] to the integral finite-type $\mathbb{C}$-scheme $X$, the invertible sheaf $L$ and the meromorphic sections $s_0,\dots ,s_n$: the fractional blowup $B={\mathrm{Proj}}_X{⨁}_{k\ge 0}(\mathcal{F}\otimes L^{-1}{)}^k$ is integral and $(\pi ,\psi ):B\to X{\times }_{\mathbb{C}}{\mathbb{P}}_{\mathbb{C}}^n$ is the schematic closure of the graph of the induced rational map, hence a closed immersion.

The canonical isomorphisms $\mathcal{F}\otimes L^{-1}=(\mathcal{I}\otimes L)\otimes L^{-1}\cong \mathcal{I}\otimes (L\otimes L^{-1})\cong \mathcal{I}$ of ${\mathcal{O}}_X$-modules [def-invertible-sheaf] identify the graded ${\mathcal{O}}_X$-algebras ${⨁}_{k\ge 0}(\mathcal{F}\otimes L^{-1}{)}^k$ and ${⨁}_{k\ge 0}{\mathcal{I}}^k$, hence induce an $X$-isomorphism $B\cong {\mathrm{Proj}}_X{⨁}_{k\ge 0}{\mathcal{I}}^k={\mathrm{Bl}}_{\mathcal{I}}X$ [def-blowup-scheme-along-ideal, def-blowup-fractional-ideal]. Composing with the closed immersion of step 3.1, the blowup ${\mathrm{Bl}}_{\mathcal{I}}X$ is a closed subscheme of $X{\times }_{\mathbb{C}}{\mathbb{P}}_{\mathbb{C}}^n$ over $X$.

Base change of the closed immersion $X\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^N$ along ${\mathbb{P}}_{\mathbb{C}}^n\to \mathrm{Spec}\mathbb{C}$ is a closed immersion $X{\times }_{\mathbb{C}}{\mathbb{P}}_{\mathbb{C}}^n\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^N{\times }_{\mathbb{C}}{\mathbb{P}}_{\mathbb{C}}^n$ [F4], and the Segre embedding of [F5] is a closed immersion ${\mathbb{P}}_{\mathbb{C}}^N{\times }_{\mathbb{C}}{\mathbb{P}}_{\mathbb{C}}^n\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^M$ with $M=(N+1)(n+1)-1$. A composite of closed immersions is again a closed immersion [F4], so ${\mathrm{Bl}}_{\mathcal{I}}X$ is a closed subscheme of ${\mathbb{P}}_{\mathbb{C}}^M$; all exhibited maps are morphisms over $\mathbb{C}$, so the structure morphism ${\mathrm{Bl}}_{\mathcal{I}}X\to \mathrm{Spec}\mathbb{C}$ factors as this closed immersion followed by the projection, and ${\mathrm{Bl}}_{\mathcal{I}}X$ is projective over $\mathbb{C}$ in the sense of [def-projective-morphism-pre-proj].

For the composition clause argue by induction on the number of factors: let $X_0$ be integral projective over $\mathbb{C}$ and $X_{k+1}={\mathrm{Bl}}_{{\mathcal{I}}_k}{X}_k$ for nonzero finite-type quasi-coherent ideal sheaves ${\mathcal{I}}_k\subseteq {\mathcal{O}}_{X_k}$. If $X_k$ is integral projective then steps 1.1-5.1 applied to $(X_k,{\mathcal{I}}_k)$ show that $X_{k+1}$ is projective over $\mathbb{C}$, and [F6] shows that it stays integral, so the induction runs through every finite composition.

Boundaries and Choice accounting. An integral scheme is nonempty and $\mathcal{I}\mathrm{\ne }0$ by hypothesis, so the empty and zero cases are excluded; the chosen twist satisfies $d\ge 1$; the endpoint $n=0$ of a single generator is allowed, the Segre factor then being ${\mathbb{P}}_{\mathbb{C}}^0$ with $M=N$; and $N=0$ is covered by the same argument with the single chart $U_0$. The Axiom of Choice [F7] is inherited exactly from [F1] and [F2]; no further selection is made.

Smooth projective covers. Let $Z$ be a reduced projective complex scheme with its closed immersion $Z\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^N$. The ambient scheme ${\mathbb{P}}_{\mathbb{C}}^N$ is smooth of finite type over the characteristic-zero field $\mathbb{C}$, so [F1] applies to the reduced closed subscheme $Z$: the embedded desingularization is a tower ${\mathbb{P}}_{\mathbb{C}}^N=X_0\leftarrow X_1\leftarrow \cdots \leftarrow X_r$ of blowups along regular centers with strict transform $\tilde{Z}=Z_r$ smooth, mapping to $Z$ properly and birationally on every irreducible component and isomorphically over the smooth locus. Each $X_i$ is integral projective over $\mathbb{C}$ by Blowups of integral projective complex schemes are projective, so the closed subscheme $\tilde{Z}\subseteq X_r$ is smooth projective, and $\tilde{Z}\to Z$ is surjective because its image is closed and contains a dense open of every irreducible component of the reduced $Z$.

Degree zero. Apply [F2] with $S=\mathrm{Spec}\mathbb{C}$ Noetherian and $T=Y$ proper, hence separated and of finite type: there are a projective $X^{\mathrm{\prime }}$ over $\mathbb{C}$ and a proper surjective $\pi :X^{\mathrm{\prime }}\to Y$. Replacing $X^{\mathrm{\prime }}$ by its reduction (The reduction of a scheme) is a closed immersion $X_{\mathrm{r}\mathrm{e}\mathrm{d}}^{\mathrm{\prime }}\hookrightarrow X^{\mathrm{\prime }}$ that is proper and surjective onto $X^{\mathrm{\prime }}$, and the reduced projective $Z=X_{\mathrm{r}\mathrm{e}\mathrm{d}}^{\mathrm{\prime }}$ still maps properly and surjectively to $Y$; step 1.1 applied to $Z$ produces a smooth projective $Y_0$ with a proper surjective $Y_0\to Y$. This is the degree-zero term of the construction, and the one-term object $\{Y_0\to Y\}$ is a split $0$-truncated proper hypercover in the sense of [F3], its only boundary map being the augmentation.

Matching objects are projective. Suppose a split $(p-1)$-truncated proper hypercover $Y_0,\dots ,Y_{p-1}$ of $Y$ with all terms smooth projective has been constructed, for some $p\ge 1$. Let $B_p$ be the finite set of order-preserving injections $\theta :[q]\hookrightarrow [p]$ with $q<p$, and write $q(\theta )=q$. The matching object $M_p$ is cut out in ${\prod }_{\theta \in B_p,Y}{Y}_{q(\theta )}$ by the boundary compatibility equations. Each equation is an equality into a separated lower term, so [F5] makes it closed. This fibre product is closed in ${\prod }_{\theta \in B_p,\mathbb{C}}{Y}_{q(\theta )}$ by the closed diagonal of $Y$. The factors are projective over $\mathbb{C}$; their finite product and its closed subscheme $M_p$ are projective by [F5]. Hence $M_p$ is projective over $\mathbb{C}$.

Extension step. Choose a smooth projective proper surjection $N_p\to M_p$ by applying step 1.1 to the reduction of the projective $M_p$; the reduction inclusion is proper surjective by [F4]. Set $Y_p={\coprod }_{\delta :[p]\twoheadrightarrow [q],q\le p}{N}_q$, using the previously chosen $N_q$ for $q<p$. Here is the explicit extension of the structure maps. Write $i_q:N_q\to Y_q$ for the summand inclusions. For any $\alpha :[m]\to [n]$ with $m,n\le p$, on the summand $N_q$ of $Y_n$ indexed by $\delta :[n]\twoheadrightarrow [q]$, factor $\delta \alpha =\iota ϵ$ with $ϵ:[m]\twoheadrightarrow [t]$ and $\iota :[t]\hookrightarrow [q]$. Its map into $Y_m$ is $Y(ϵ)Y(\iota )i_q$. The injection part is an old structure map if $q<p$, a component of the chosen boundary tuple if $q=p$ and $t<p$, and the identity if $t=q=p$. The surjection part, when $m=p$ and $t<p$, sends each old summand $N_u$ indexed by $\eta :[t]\twoheadrightarrow [u]$ into the new summand indexed by $\eta ϵ$; otherwise it is old or the identity. Boundary compatibility supplies the relations between injections; concatenation supplies the relations between surjections; refactoring their composites gives the mixed relations. Thus these prescriptions obey contravariant composition and define the $p$-truncated simplicial object. Its splitting is precisely the displayed coproduct. Each summand is smooth projective, and each map to the separated projective $M_p$ is proper by [F4]. The finite union map is proper (finite type and separatedness hold on its open-and-closed summands, and every base change has closed image as a finite union of closed images); it is surjective on the $N_p$ summand. This extends the proper hypercover through degree $p$.

Existence and finite stages. AC permits successive choices in step 4.1 for all $p$, giving one compatible augmented simplicial object; each term is smooth projective and each matching map proper surjective. In a first-quadrant cochain simple complex $K^{p,q}$ attached to this object, total degrees $n-1,n,n+1$ involve only $p\le n+1$. Thus its degree-$n$ cohomology uses only this finite skeleton. No splitness of the higher terms of ${\mathrm{cosk}}_N{\mathrm{sk}}_N{Y}_{\bullet }$ is asserted or needed.

Relative refinement construction. More generally let $T_{\bullet }\to Y$ be a proper hypercover with projective complex terms, without a smoothness or splitting assumption. Construct a split smooth projective $V_{\bullet }\to Y$ with a map to $T_{\bullet }$. At degree zero take a smooth projective proper cover of $T_0$. Once the lower degrees and their maps are constructed, form $P_p=T_p{\times }_{M_p(T)}{M}_p(V)$. This is projective by [F5], and $P_p\to M_p(V)$ is a proper surjection by base change of the matching map of $T$. Choose a smooth projective proper cover $N_p\to P_p$, and apply step 4.1 to its composite into $M_p(V)$. On $N_p$ define the map into $T_p$ by the other projection; on each degenerate summand $N_q$ define it by its existing map to $T_q$ followed by the corresponding degeneracy of $T$. The fibre-product condition and the composition check of step 4.1 give a simplicial map. Its new matching map is proper surjective on $N_p$, so the induction constructs the required $V$. This uses no inference that termwise surjections induce surjections on matching limits.

Common refinements and morphisms. For two given hypercovers $A_{\bullet },B_{\bullet }$ of $Y$, put $T_p=A_p{\times }_Y{B}_p$. Matching limits commute with these products, so the matching map of $T$ is the product over $Y$ of the matching maps of $A,B$; it is proper surjective by [F4]. Its terms are projective by [F5]. Step 6.1 gives a split smooth projective hypercover mapping to $T$ and hence to both given systems. For a morphism $g:Y\to Z$, and fixed hypercovers $A_{\bullet }\to Y$ and $B_{\bullet }\to Z$, instead put $T_p=A_p{\times }_Z{B}_p$, using $A_p\to Y\stackrel{g}{\to }Z$. This is the product over $Y$ of $A$ with the base-changed hypercover $Y{\times }_{Z}B$, so its matching maps are again proper surjective; its terms are closed in $A_p{\times }_{\mathbb{C}}{B}_p$ by separatedness of $Z$, hence projective. Apply step 6.1 over $Y$. The resulting maps to $A$ and $B$ give the required refinement of the source and lift of $g$.

Boundaries and Choice accounting. For empty $Y$ take every term empty; degree zero is step 2.1. If $Y$ is a point, the constant one-point hypercover is allowed (its $N_p$ are empty for $p>0$). Zero-dimensional and mixed-dimensional smooth terms are finite disjoint unions covered by [F5]. AC is inherited from Chow's lemma and desingularization and permits the countable sequence of cover choices, including the relative constructions; any fixed finite truncation uses only finitely many such choices.

Every ordinary singular chain is locally finite: a sum with finite support has only finitely many distinct images, so any neighbourhood of any point meets at most those finitely many images. The inclusion $C_k(X;R)\hookrightarrow C_k^{\mathrm{B}\mathrm{M}}(X;R)$ is therefore defined, commutes with the coefficientwise boundary, and is injective.

Conversely let $c={\sum }_{\sigma }{a}_{\sigma }\sigma$ be locally finite on the compact $X$. Local finiteness provides an open cover of $X$ by neighbourhoods $U$ each of which meets only finitely many images of simplices with nonzero coefficient; by compactness [F2] finitely many such neighbourhoods $U_1,\dots ,U_m$ cover $X$. If $\sigma$ has $a_{\sigma }\mathrm{\ne }0$, then its image is nonempty, so it meets some $U_j$, and for each $j$ only finitely many such $\sigma$ meet $U_j$; hence only finitely many coefficients are nonzero and $c$ lies in the image of the inclusion.

By steps 1.1 and 1.2 the inclusion is a bijection on every chain group, hence an isomorphism of chain complexes; it induces isomorphisms on homology. For a continuous map $f:X\to Y$ of compact Hausdorff spaces, for every compact $K\subseteq Y$, Hausdorffness makes $K$ closed, continuity makes $f^{-1}(K)$ closed in the compact $X$, and this preimage is therefore compact by [F2]. Thus $f$ is proper, and the pushforward $f_{\ast }$ of [F1] restricts on finite chains to the ordinary singular chain map, so the isomorphisms are natural. No choice principle was used.

(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.