Logarithmic construction of smooth-open mixed Hodge structures

(Compactification and strata.) By [F7] choose the SNC smooth projective compactification; the constructions below apply separately to its finitely many smooth connected components when dimensions differ. At a closed point of $X$, the SNC equations are a subset of a regular system of parameters; their differentials extend to a cotangent basis in a smooth chart. The parameter map has invertible complex Jacobian, so [F8] makes these parameters holomorphic coordinates, giving the coordinate divisor model of [F1]. Order the finitely many boundary components to fix residue signs. Their $a$-fold intersections and disjoint unions $D^{[a]}$ are smooth projective by [F1], so [F3] applies to each component. Empty strata contribute zero.

(Logarithmic forms and the pole filtration.) Put $e_i=d{z}_i\mathrm{/}{z}_i$ for the $s$ local boundary coordinates. The logarithmic complex $L={\mathrm{\Omega }}_X^{\bullet }(\log D)$ is locally generated by $e_1,\dots ,e_s,d{z}_{s+1},\dots ,d{z}_d$ over holomorphic functions; regular normal differentials are included because $d{z}_i=z_i{e}_i$. Define in degree $k$ the pole-count filtration by $W_a{L}^k={\sum }_{\mathrm{\mid }I\mathrm{\mid }\le a}{e}_I\wedge {\mathrm{\Omega }}_X^{k-\mathrm{\mid }I\mathrm{\mid }}$, with arbitrary regular holomorphic forms in the second factor, not merely tangential forms. Thus $W_{0}L={\mathrm{\Omega }}_X^{\bullet }$, $W_{a}L=0$ for $a<0$, and $W_{d}L=L$. Changing a boundary equation by a unit adds a regular $d\log u$, so this definition is intrinsic. Since $d{e}_i=0$ and differentiating a regular coefficient produces a regular form, $d$ preserves $W$. The degree filtration $F^{p}L={\mathrm{\Omega }}_X^{\ge p}(\log D)$ is also a finite filtration by subcomplexes.

(Residues, signs and the Hodge shift.) For each ordered set $I$ of $a$ distinct components, write a term of $W_a{L}^k$ as $e_I\wedge {\eta }_I$ with ${\eta }_I$ regular, and restrict ${\eta }_I$ to $D_I$. These are separate residue coordinates on the separate summands of $D^{[a]}$. Terms with fewer poles restrict to zero in ${\mathrm{Gr}}_a^W$; a regular normal differential restricts to zero, and a coefficient vanishing on $D_I$ lies in the ideal generated by its normal coordinates, each factor $z_i$ converting $e_i$ to the regular form $d{z}_i$ and reducing the pole count. These observations prove that the kernel of the residue map on $W_a$ is exactly $W_{a-1}$. Conversely every regular stratum form extends locally in the normal coordinates and lifts as $e_I\wedge {\eta }_I$, proving surjectivity. Changes of local equations add only lower-pole terms, so these maps glue. The differential is $d(e_I\wedge {\eta }_I)=(-1{)}^a{e}_I\wedge d{\eta }_I$, agreeing with the differential of the shifted complex $[-a]$. Hence ${\mathrm{Gr}}_a^{W}L\cong i_{a\ast }{\mathrm{\Omega }}_{D^{[a]}}^{\bullet }[-a]$, and $F^p$ corresponds to the stratum degree filtration $F^{p-a}$, the shift appropriate to a Tate twist $(-a)$.

(A pole-preserving local contraction.) Work on a smaller polydisc centered at a boundary point with $s$ normal coordinates. For tangential coordinates $w$, put ${\varphi }_t(z,w)=(z,tw)$ and $R={\sum }_j{w}_j{\mathrm{\partial }}_{w_j}$. Set $h\eta ={\int }_0^1{t}^{-1}{\varphi }_t^{\ast }({\iota }_R\eta )dt$. Differentiating coefficients and basis differentials gives $\frac{d}{dt}{\varphi }_t^{\ast }\eta =t^{-1}{\varphi }_t^{\ast }(d{\iota }_R+{\iota }_{R}d)\eta$; integration yields $dh+hd=\mathrm{i}\mathrm{d}-{\varphi }_0^{\ast }$. The integrand has a finite limit at $t=0$ because contraction and pullback supply at least one tangential factor of $t$, and the power-series bounds of [F4] justify differentiation and integration on smaller polydiscs. This preserves every pole level, since it affects only regular tangential forms. What remains is the normal logarithmic complex with holomorphic coefficients $f(z)={\sum }_{\nu \in {\mathbb{N}}^s}{c}_{\nu }{z}^{\nu }$ and differential $d(z^{\nu }{e}_I)=z^{\nu }({\sum }_i{\nu }_i{e}_i)\wedge e_I$. For a nonzero multiindex $\nu$, choose the first index $i$ with ${\nu }_i>0$ and put $h_{\nu }={\nu }_i^{-1}{\iota }_i$, contraction of the exterior algebra with the $i$-th basis vector. The identity $({\sum }_j{\nu }_j{e}_j)\wedge h_{\nu }+h_{\nu }({\sum }_j{\nu }_j{e}_j)\wedge =\mathrm{i}\mathrm{d}$ contracts that entire monomial summand. This also preserves $W$: in an $e_I$ monomial a logarithmic index with ${\nu }_i>0$ can be written as a regular differential by factoring $z_i$, so its effective pole count is precisely the number of indices $j\in I$ with ${\nu }_j=0$; $h_{\nu }$ removes only a positive-exponent index. Partitioning the series into the finitely many sectors specified by this first positive index and dividing coefficients by ${\nu }_i\ge 1$ preserves normal convergence on each smaller polydisc, so these operators define a holomorphic homotopy. The single all-zero multiindex is left unchanged: its surviving complex is the exterior algebra on the $e_i$, with zero differential and $W_a$ equal to the span of wedges of degree at most $a$. This handles every pattern of simultaneous zero exponents and proves ${\mathcal{H}}^q(W_{a}L{)}_p={\bigwedge }^q{\mathbb{C}}^s$ for $0\le q\le a$, and zero otherwise; inclusion $W_{a}L\to L$ induces the identity on these spaces.

(Every canonical truncation and the rational normalization.) Take the sheafification ${\mathcal{S}}_U^{\bullet }$ of smooth singular cochains on $U$. The flasque-resolution construction of [F6] applies to smooth simplices as well: barycentric subdivision and its prism operators preserve smoothness, local polydisc contractions are smooth, and the global small-simplex representative argument is unchanged. Thus ${\mathcal{S}}_U^{\bullet }$ resolves ${\mathbb{C}}_U$ and its terms are acyclic on every open subset. They are therefore $j_{\ast }$-acyclic, and $j_{\ast }{\mathcal{S}}_U^{\bullet }$ represents $R{j}_{\ast }{\mathbb{C}}_U$. Integration of logarithmic forms on smooth simplices in $U$, with the cochain map and normalization of [F2], gives the chain map $L\to j_{\ast }{\mathcal{S}}_U^{\bullet }$. On the polydiscs of step 2.2 the target stalk cohomology is the cohomology of $({\mathrm{\Delta }}^{\ast }{)}^s\times {\mathrm{\Delta }}^{d-s}$, the exterior algebra on meridians by [F4]. Integration gives ${\int }_{{\gamma }_i}{e}_j=2\pi i{\delta }_{ij}$, so the comparison is an isomorphism on every cohomology sheaf. Let ${\tau }_{\le a}$ denote the canonical truncation, with degree $a$ equal to the kernel of the next differential. Since every degree-$k$ logarithmic form has at most $k$ poles, the identity gives a genuine chain inclusion ${\tau }_{\le a}L\to W_{a}L$. By step 2.2 it induces an isomorphism in degrees at most $a$ and both sides have no cohomology above $a$, hence is a quasi-isomorphism for every $a$. These inclusions commute with the maps as $a$ increases. We therefore have the filtered comparison zigzag $(R{j}_{\ast }{\mathbb{Q}}_U\otimes \mathbb{C},\tau )\to (R{j}_{\ast }{\mathbb{C}}_U,\tau )\leftarrow (L,\tau )\to (L,W)$, all arrows graded quasi-isomorphisms; exact scalar extension and the local finite torus cohomology establish the first arrow. On each graded sheaf the rational meridian wedge corresponds to $(2\pi i{)}^{-a}{e}_I$, which pairs by $1$ with the degree-$a$ meridian torus, not a degree-$2a$ class. Ordered residues thus identify $R^a{j}_{\ast }{\mathbb{Q}}_U$ with $i_{a\ast }{\mathbb{Q}}_{D^{[a]}}$ and place its comparison lattice in $(2\pi i{)}^{-a}\mathbb{Q}$, exactly the rational Tate normalization $(-a)$.

(Actual filtered global-section models and a single comparison.) The sheaf comparison in step 3.1 is a zigzag, so it must not be treated as a nonexistent direct sheaf map in the opposite direction. Resolve the complexes and their filtrations by the exact Godement construction: $C^0(A)(V)={\prod }_{x\in V}{A}_x$, with the natural embedding and successive cokernels. Stalks and products are exact for vector spaces under AC. Moreover $C^0(A)={\prod }_{x\in X}{i}_{x\ast }{A}_x$ is an injective sheaf of field-vector spaces: each $A_x$ is injective over its field by extension of linear maps along a chosen complement, the exact stalk functor is left adjoint to $i_{x\ast }$, and a product of injectives is injective because the component extension maps can be chosen simultaneously under AC. The successive cokernel construction therefore gives an actual injective resolution. Exactness of $C^0$ and the short exact augmentation sequences give exactness of each successive cokernel functor by the snake lemma, so the resulting functorial resolution is exact on filtration levels, their finite intersections and both associated gradings; its global-section total computes derived global sections. Apply it to $L$ with $W,F$ and to the other weight-filtered models in the zigzag. Replace the rational and comparison models by ${\tau }_{\le d}$ first; their cohomology sheaves above $d$ vanish by the local torus calculation, so this is a quasi-isomorphism and leaves a finite weight range $0\le a\le d$. After resolution and global sections the complex-coefficient arrows in the zigzag are maps of weight-filtered vector-space complexes with graded quasi-isomorphisms. For the rational scalar-extension arrow, the global graded comparison is the isomorphism $H^{r-a}(D^{[a]};\mathbb{Q})\otimes \mathbb{C}\to H^{r-a}(D^{[a]};\mathbb{C})$ from [F3], with the normalization of step 3.1; this proves it is a graded quasi-isomorphism without interchanging tensor product with an infinite product of stalks. Such a map admits a weight-filtered homotopy inverse: its cone has acyclic weight-graded complexes; a cycle in $W_a$ has a leading graded preimage, and subtracting its differential lowers the cycle to $W_{a-1}$, so finite induction supplies a preimage in $W_a$. Thus the cone is acyclic with a $W$-strict differential. Choose filtration-compatible complements of its cycles and right inverses of its differential; the resulting degree-minus-one homotopy contracts the cone and preserves $W$, and the cone homotopy equations give a $W$-filtered homotopy inverse of the original map. AC permits these linear complements. Inverting the leftward comparison in step 3.1 and composing the remaining arrows gives an actual $W$-filtered chain map $K_{\mathbb{Q}}\otimes \mathbb{C}\to K_{\mathbb{C}}$, with $K_{\mathbb{C}}=\mathrm{\Gamma }(X,C^{\bullet }(L))$ carrying the original $W,F$. Its graded maps are quasi-isomorphisms. Different inverse choices are filtered-homotopic, so their induced comparison and image filtrations on cohomology agree. These are bounded-below degreewise-finite filtered models of the type required in [F5].

(Strictness on the weight-graded global models.) Exactness of the resolution and step 2.1 identify ${\mathrm{Gr}}_a^W{K}_{\mathbb{C}}$ with a global-section resolution of ${\mathrm{\Omega }}_{D^{[a]}}^{\bullet }[-a]$, with its degree Hodge filtration shifted by $a$. Its Hodge spectral sequence degenerates at $E_1$ by [F3]. This implies strictness of the differential of this actual filtered vector-space complex: if $dx$ belongs to $F^p$, take the leading $F$ index $b<p$ of $x$; its class in the associated-graded complex is a cycle. Degeneration means this $E_1$ class lifts to an actual cycle $y\in F^b$, up to an element of $F^{b+1}$ and a boundary in $F^b$. Subtract that cycle and boundary from $x$, preserving $dx$ and raising the filtration index of the preimage. Repeat through the finite filtration until a preimage of $dx$ lies in $F^p$. Thus $\mathrm{im}d\cap F^p=d(F^p)$. Moreover its degree-$r$ cohomology is $H^{r-a}(D^{[a]};\mathbb{Q})(-a)$ under the comparison of step 3.1, with the induced $F$, pure of weight $r-a+2a=r+a$ and finite-dimensional by [F3]. Hence the global comparison models of step 4.1 satisfy every graded axiom of a rational cohomological mixed Hodge complex, including strictness and the full induced Hodge filtration.

(The mixed structure and its spectral sequence.) Apply the full conclusion of [F5] to the proved mixed Hodge complex of step 5.1. Its cohomology is $H^n(U;\mathbb{Q})$ by steps 3.1-5.1 and [F6], and it carries the image Hodge filtration and shifted weight filtration $W_m{H}^n(U)=\mathrm{im}(H^n(W_{m-n}K)\to H^n(K))$. The weight spectral sequence is $E_1^{-a,q}=H^{q-2a}(D^{[a]};\mathbb{Q})(-a)$, pure of weight $q$, degenerates at $E_2$, and gives ${\mathrm{Gr}}_m^W{H}^n(U)\cong E_2^{n-m,m}$ with its induced Hodge filtration. In particular this step uses the supplied recurrent-to-image-filtration proof, not an assumption that higher differentials preserve the desired filtration.

(Lower weight bound and compactification image.) The pole index $a$ is nonnegative, so a term contributing to total degree $n=q-a$ has weight $q=n+a\ge n$. Hence step 6.1 gives $W_{n-1}{H}^n(U)=0$. Also $W_n{H}^n(U)$ is the image of $H^n(W_{0}K)$. On the complex side $W_{0}L={\mathrm{\Omega }}_X^{\bullet }$ by step 1.2; on the rational side ${\tau }_{\le 0}R{j}_{\ast }{\mathbb{Q}}_U={\mathbb{Q}}_X$, since a punctured-coordinate polydisc is connected. Its map to $R{j}_{\ast }{\mathbb{Q}}_U$ is restriction. Thus both sides identify $W_n{H}^n(U)$ with the actual image of $j^{\ast }:H^n(X;\mathbb{Q})\to H^n(U;\mathbb{Q})$, compatibly with the rational comparison and Hodge filtration.

(Functoriality and independence.) A compactified extension $f:(X,D)\to (Y,E)$ of an open morphism satisfies $f^{-1}(E)\subseteq D$. A target boundary equation pulls back to a nonzero regular germ on each source component, since the dense open source maps into the target open. Its zero set is contained in $D$, so [F8] factors it locally as a unit times powers of source boundary equations with nonnegative integer exponents, so $f^{\ast }d\log t=d\log u+{\sum }_i{m}_{i}d\log z_i$. Wedges preserve pole count and degree, hence $f^{\ast }$ preserves $W,F$ on the logarithmic complexes and is compatible with the rational meridian comparison. The functorial resolutions and filtered comparison homotopies of step 4.1 produce the same natural pullback on cohomology; [F5] therefore makes it a mixed Hodge morphism. The graph-extension construction in [F7] supplies such compactified extensions after modifying the source. For two SNC compactifications of one $U$, a common SNC domination gives pullback comparison maps between their logarithmic mixed Hodge structures on $H^n(U)$, whose underlying rational maps are the identity. Strictness in [F5] makes these identity maps isomorphisms for both filtrations, proving independence. For any smooth projective compactification with arbitrary boundary, the modification-image clause of [F7] identifies its restriction image with the SNC one, so the formula of step 7.1 holds for it too.

(Choice and boundary cases.) AC is inherited through the typed suppliers in [A1] and is also used exactly in step 4.1 for exact products in the Godement resolution and filtration-compatible vector-space complements to strictify the comparison zigzag. The monomial contractions, residues, rational meridian normalization, finite filtrations and weight bookkeeping are otherwise choice-free. With empty boundary $L={\mathrm{\Omega }}_X^{\bullet }$ and $W_0=L$, recovering the pure projective structure; empty $U$ may be represented by empty compactification and has zero cohomology, while the SNC compactification hypotheses already require density on every nonempty component. Components of differing dimensions are handled separately, empty strata vanish, and degrees outside the cohomological range give zero groups. This completes the retained full mixed Hodge, functoriality, independence, lower-weight and image claims.