Logarithmic Hodge Complexes And Smooth Open Weights

Closure of a quasi-compact immersion. Let $f:W\to P$ be a quasi-compact immersion from a reduced finite-type $\mathbb{C}$-scheme into a Noetherian scheme, factored as a closed immersion $W\hookrightarrow V$ followed by an open immersion $V\hookrightarrow P$. By [F2] its scheme-theoretic image $Z=V(\mathcal{K})$, $\mathcal{K}=\ker ({\mathcal{O}}_P\to f_{\ast }{\mathcal{O}}_W)$, restricts on $V$ to $W$, so $W$ is open in $Z$ with the same local rings. On an affine open $T=\mathrm{Spec}A\subseteq P$, the kernel $\mathcal{K}(T)$ is radical: if $g^r$ restricts to zero on $f^{-1}(T)$, then $g$ restricts to zero because $W$ is reduced. Thus $A\mathrm{/}\mathcal{K}(T)$ is reduced, proving that $Z$ is reduced. If $W$ were not dense in $Z$, its proper reduced closed closure in $Z$ would receive $f$: its defining sections restrict to functions zero at every point of the reduced scheme $W$, hence zero on every affine open of $W$. This contradicts minimality of the scheme-theoretic image, so $W$ is dense. Every irreducible component meets $W$: the complement of the finitely many other components is a nonempty open subset of $Z$ contained in that component, since an irreducible component cannot be contained in their finite union; density forces this open to meet $W$.

Blowups of integral projective schemes are projective. Let $Y$ be an integral finite-type $\mathbb{C}$-scheme whose structure morphism is H-projective, and let $\mathcal{I}$ be a quasi-coherent ideal sheaf of finite type. If $\mathcal{I}=0$ then ${\mathrm{Bl}}_{\mathcal{I}}Y={\mathrm{Proj}}_Y({\mathcal{O}}_Y\oplus 0\oplus \cdots )=\mathrm{\varnothing }$, which is projective as the empty closed subscheme of ${\mathbb{P}}_{\mathbb{C}}^0$; assume $\mathcal{I}\mathrm{\ne }0$. Fix a closed immersion $Y\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^N$ and let $L={\mathcal{O}}_Y(1)$ be the pullback of $\mathcal{O}(1)$, which is H-very ample by construction and ample over the affine base $\mathrm{Spec}\mathbb{C}$; $Y$ is Noetherian and $\mathcal{I}$ is coherent, so for some $m\ge 0$ the twist $\mathcal{I}(m)=\mathcal{I}\otimes L^{\otimes m}$ is globally generated. Its evaluation map $\mathrm{\Gamma }(Y,\mathcal{I}(m)){\otimes }_{\mathbb{Z}}{\mathcal{O}}_Y\to \mathcal{I}(m)$ is surjective and $Y$ is quasi-compact with $\mathcal{I}(m)$ of finite type, so finitely many global sections $s_0,\dots ,s_n$ already generate $\mathcal{I}(m)$: on a finite affine cover, a finitely generated module generated by the image of the global sections is generated by finitely many of them. Put $L^{\mathrm{\prime }}=L^{\otimes m}$, view the $s_i$ as meromorphic sections of the invertible sheaf $L^{\mathrm{\prime }}$, and set $F={\sum }_i{\mathcal{O}}_Y{s}_i=\mathcal{I}(m)$ and $I=F\otimes L^{\mathrm{\prime }-1}=\mathcal{I}$, so the ratio map of the $s_i$ is $Y⇢{\mathbb{P}}_{\mathbb{C}}^n$. The graph-closure corollary gives ${\mathrm{Bl}}_{\mathcal{I}}Y={\mathrm{Proj}}_Y({⨁}_{d\ge 0}{I}^d)=B$ with a closed immersion $(\pi ,\psi ):B\hookrightarrow Y{\times }_{\mathbb{C}}{\mathbb{P}}_{\mathbb{C}}^n$. Composing with the closed immersion $Y{\times }_{\mathbb{C}}{\mathbb{P}}_{\mathbb{C}}^n\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^N{\times }_{\mathbb{C}}{\mathbb{P}}_{\mathbb{C}}^n$ obtained by base change of $Y\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^N$ and then with the Segre embedding, itself a closed immersion, exhibits $B$ as a closed subscheme of ${\mathbb{P}}_{\mathbb{C}}^{(N+1)(n+1)-1}$; hence the structure morphism of $B$ is H-projective.

Disjoint unions of projective schemes are projective. Let $Z_1,\dots ,Z_t\subseteq {\mathbb{P}}_{\mathbb{C}}^M$ be closed subschemes with pairwise disjoint underlying sets and ideals ${\mathcal{I}}_{\alpha }$. The product ideal $\mathcal{J}={\mathcal{I}}_1\cdots {\mathcal{I}}_t$ equals the intersection ${\bigcap }_{\alpha }{\mathcal{I}}_{\alpha }$: on an affine chart $\mathrm{Spec}A$ the ideals $J_{\alpha }$ of the $Z_{\alpha }$ satisfy $V(J_{\alpha })\cap V(J_{\beta })=\mathrm{\varnothing }$ for $\alpha \mathrm{\ne }\beta$, so $J_{\alpha }+J_{\beta }=A$ and the product of pairwise comaximal ideals is their intersection. At a point $x\in Z_{\alpha }$ one has ${\mathcal{I}}_{\beta ,x}={\mathcal{O}}_x$ for $\beta \mathrm{\ne }\alpha$, so ${\mathcal{J}}_x={\mathcal{I}}_{\alpha ,x}$; therefore $V(\mathcal{J})={⨆}_{\alpha }{Z}_{\alpha }$ as subschemes of ${\mathbb{P}}^M$, and the natural morphism ${⨆}_{\alpha }{Z}_{\alpha }\to {\mathbb{P}}_{\mathbb{C}}^M$ is a closed immersion. Now let $X_1,\dots ,X_t$ be projective $\mathbb{C}$-schemes, say $X_{\alpha }\subseteq {\mathbb{P}}_{\mathbb{C}}^{M_{\alpha }}$ closed. The coordinate-block maps ${\mathbb{P}}_{\mathbb{C}}^{M_{\alpha }}\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^M$, $M={\sum }_{\alpha }(M_{\alpha }+1)-1$, identify the projective spaces with pairwise disjoint linear subspaces (vanishing of all coordinates outside one block), and are closed immersions; the images of the $X_{\alpha }$ are closed subschemes with pairwise disjoint underlying sets, so the preceding paragraph realizes ${⨆}_{\alpha }{X}_{\alpha }$ as a closed subscheme of ${\mathbb{P}}_{\mathbb{C}}^M$. Hence a finite disjoint union of projective $\mathbb{C}$-schemes is projective over $\mathbb{C}$.

Local reduction to pure dimensions and constant codimension. Let $Y$ be a smooth finite-type $\mathbb{C}$-scheme and $C\subseteq Y$ a regular closed subscheme with ideal sheaf $\mathcal{I}$, a coherent ideal. Fix $x\in C$; if $C=\mathrm{\varnothing }$ there is nothing to prove. Since smooth finite-type $\mathbb{C}$-schemes and regular finite-type $\mathbb{C}$-schemes are smooth over $\mathbb{C}$ in characteristic zero, $C$ is smooth over $\mathbb{C}$ and, being regular Noetherian and reduced, each of its points lies on exactly one irreducible component; the components of $C$ are therefore pairwise disjoint, finitely many, and each is a smooth finite-type $\mathbb{C}$-scheme of pure dimension $n$, with local rings equal to those of $C$ at its points. Let $C_0$ be the component through $x$ and let $Y^{\circ }$ be the connected component of $Y$ containing $x$; it is a single irreducible component, hence pure of dimension $m$, and open in $Y$. Choose an affine open $U\subseteq Y^{\circ }$ containing $x$ and disjoint from $C\setminus C_0$ (a finite union of closed sets missing $x$). Then $U$ is smooth of pure dimension $m$, $C\cap U=C_0\cap U$ is closed in $U$, smooth, irreducible of dimension $n$, and the closed immersion $C\cap U\hookrightarrow U$ has constant codimension $d=m-n$.

Regularity and components. A point $w\in W$ is a regular point of $Z$: the local rings of $Z$ and $W$ at $w$ coincide by step 1.1, and a smooth finite-type $\mathbb{C}$-scheme has regular local rings. Every irreducible component of $Z$ meets $W$: for a component $Z_{\alpha }$ of the reduced Noetherian $Z$, the complement of the union of the finitely many other components is a nonempty open subset of $Z$ contained in $Z_{\alpha }$, nonemptiness holding because otherwise irreducibility of $Z_{\alpha }$ would place it inside one of the other components, contradicting maximality; and the dense open $W$ meets every nonempty open subset. If $Z$ is smooth over $\mathbb{C}$, as the schemes $P$, $P_i$ and $Z^{\mathrm{\prime }}$ below are, then its irreducible components are pairwise disjoint: a common point of two components would give the local ring there two distinct minimal primes and hence a non-domain, whereas a smooth finite-type $\mathbb{C}$-scheme has regular local rings, which are domains. Consequently, for smooth $Z$, the components are finitely many pairwise disjoint integral closed subschemes, open and closed in $Z$.

Blowups preserve finite disjoint unions of integral projective schemes. Let $X={⨆}_{\alpha }{X}_{\alpha }$ with each $X_{\alpha }$ integral, finite type and H-projective over $\mathbb{C}$, and let $\mathcal{I}$ be a quasi-coherent ideal sheaf of finite type on $X$; write ${\mathcal{I}}_{\alpha }=\mathcal{I}{\mathrm{\mid }}_{X_{\alpha }}$. Since the pieces are open and closed, the blowup restricts: ${\mathrm{Bl}}_{\mathcal{I}}X\cong {⨆}_{\alpha }{\mathrm{Bl}}_{{\mathcal{I}}_{\alpha }}{X}_{\alpha }$. If ${\mathcal{I}}_{\alpha }=0$ then that piece is empty; if ${\mathcal{I}}_{\alpha }\mathrm{\ne }0$ then ${\mathrm{Bl}}_{{\mathcal{I}}_{\alpha }}{X}_{\alpha }$ is integral, finite type and H-projective by step 1.2. So the blowup is again a finite disjoint union of integral projective $\mathbb{C}$-schemes, and it is projective over $\mathbb{C}$ by step 1.3; the relevant ideals are quasi-coherent of finite type, as ideals of closed subschemes of finite-type $\mathbb{C}$-schemes (affine-locally the kernel of a surjection of Noetherian rings, hence finitely generated).

Smoothness of the blowup along a regular center. Keep the notation of step 1.4 and first blow up the coordinate subspace: for $A=\mathbb{C}[x_1,\dots ,x_m]$ and $I=(x_{n+1},\dots ,x_m)$, the chart of ${\mathrm{Bl}}_I\mathrm{Spec}A$ indexed by $x_j$ ($n<j\le m$) is $\mathrm{Spec}A[I\mathrm{/}{x}_j]$, and $A[I\mathrm{/}{x}_j]$ is generated over $A$ by the fractions $x_l\mathrm{/}{x}_j$ ($n<l\le m$, $l\mathrm{\ne }j$); substituting $x_l=y_l{x}_j$ exhibits it as the polynomial ring $\mathbb{C}[x_1,\dots ,x_n,x_j,y_l(n<l\le m,l\mathrm{\ne }j)]$, the two presentations agreeing inside $A_{x_j}$ because the $y_l$ are algebraically independent over $\mathbb{C}[x_1,\dots ,x_n,x_j]$ (clearing denominators uses only that $x_j$ is a nonzerodivisor). Hence ${\mathrm{Bl}}_{{\mathbb{A}}^n}{\mathbb{A}}^m$ is covered by affine spaces and is smooth over $\mathbb{C}$. Now by [F7] the closed immersion $C\cap U\hookrightarrow U$ of step 1.4 is, at each of its points, locally in the étale topology the inverse image of the coordinate inclusion ${\mathbb{A}}^n\subset {\mathbb{A}}^m$: there are a Zariski open $V\subseteq U$ and an étale morphism $\phi :V\to {\mathbb{A}}_{\mathbb{C}}^m$ with ${\phi }^{-1}({\mathbb{A}}^n)=C\cap V$ as closed subschemes. Since $\phi$ is flat and blowups commute with flat base change along the inverse image ideal, ${\mathrm{Bl}}_{C\cap V}V\cong {\mathrm{Bl}}_{{\mathbb{A}}^n}{\mathbb{A}}^m{\times }_{{\mathbb{A}}^m}V$. The projection of this fibre product to ${\mathrm{Bl}}_{{\mathbb{A}}^n}{\mathbb{A}}^m$ is the base change of the étale morphism $\phi$, hence étale, and étale morphisms are smooth; a composite of smooth morphisms is smooth, so the model base change, and therefore ${\mathrm{Bl}}_{C\cap V}V$, is smooth over $\mathbb{C}$.

Conclusion for one blowup and for sequences. With $\pi :{\mathrm{Bl}}_{\mathcal{I}}Y\to Y$ and $x\in C$ as in step 1.4, choose the open $V$ of step 2.3 around $x$; the open subscheme ${\pi }^{-1}(V)$ of the blowup is smooth over $\mathbb{C}$. Points of ${\mathrm{Bl}}_{\mathcal{I}}Y$ mapping outside $C$ lie in the open subscheme ${\pi }^{-1}(Y\setminus C)$, which is isomorphic to the smooth scheme $Y\setminus C$; as $x$ ranges over $C$, these opens cover ${\mathrm{Bl}}_{\mathcal{I}}Y$, and smoothness is local on the source, so ${\mathrm{Bl}}_{\mathcal{I}}Y$ is smooth over $\mathbb{C}$. The blowup is finite type over $\mathbb{C}$, so the same statement applies to it in place of $Y$: if $C_1\subseteq Y_1$ is a regular closed subscheme of a smooth finite-type $\mathbb{C}$-scheme $Y_1$, then ${\mathrm{Bl}}_{C_1}{Y}_1$ is smooth over $\mathbb{C}$. By induction, every scheme obtained from a smooth finite-type $\mathbb{C}$-scheme by a finite sequence of blowups along regular closed subschemes is smooth over $\mathbb{C}$.

The model construction: desingularizing the closure. Let $P$ be a smooth projective $\mathbb{C}$-scheme, $Z\subseteq P$ a reduced closed subscheme and $W\subseteq Z$ an open dense subscheme that is smooth over $\mathbb{C}$. Apply [F3] to $Z\subseteq P$: there is a sequence $P=P_0\leftarrow P_1\leftarrow \cdots \leftarrow P_r$ of blowups of regular centers with strict transforms $Z_i\subseteq P_i$, and $Z^{\mathrm{\prime }}=Z_r$ is smooth over $\mathbb{C}$ with the composite $P_r\to P$ an isomorphism over the smooth locus of $Z$, hence over $W$. Each $P_i$ is a finite disjoint union of integral projective $\mathbb{C}$-schemes: $P_0=P$ is smooth projective, so it is such a union by step 2.1, and each step is a blowup along the ideal sheaf of a regular closed subscheme, a coherent ideal, so step 2.2 applies inductively. Let $W^{\mathrm{\prime }}$ be the preimage of $W$ in $Z^{\mathrm{\prime }}$; it is open in $Z^{\mathrm{\prime }}$ and isomorphic to $W$. Every irreducible component of $Z^{\mathrm{\prime }}$ meets $W^{\mathrm{\prime }}$: the composite $Z^{\mathrm{\prime }}\to Z$ is proper and birational on every component, so a component of $Z^{\mathrm{\prime }}$ maps onto a component of $Z$; a component of $Z^{\mathrm{\prime }}$ disjoint from $W^{\mathrm{\prime }}$ would have image contained in $Z\setminus W$, a closed set, but every component of $Z$ meets the dense open $W$ by step 2.1, a contradiction. Consequently the ideal $\mathcal{I}\subseteq {\mathcal{O}}_{Z^{\mathrm{\prime }}}$ of the reduced closed subscheme $Z^{\mathrm{\prime }}\setminus W^{\mathrm{\prime }}$ is coherent and is not identically zero on any irreducible component of $Z^{\mathrm{\prime }}$.

The model construction: principalizing the boundary. Keep step 3.2 and decompose $Z^{\mathrm{\prime }}={⨆}_{\alpha }{Z}_{\alpha }^{\mathrm{\prime }}$ into its finitely many components, open and closed in $Z^{\mathrm{\prime }}$ by step 2.1. For each $\alpha$, apply [F4] to the smooth finite-type $\mathbb{C}$-scheme $Z_{\alpha }^{\mathrm{\prime }}$ and the restricted ideal ${\mathcal{I}}_{\alpha }$ if ${\mathcal{I}}_{\alpha }\mathrm{\ne }{\mathcal{O}}_{Z_{\alpha }^{\mathrm{\prime }}}$, and take the identity sequence if ${\mathcal{I}}_{\alpha }={\mathcal{O}}_{Z_{\alpha }^{\mathrm{\prime }}}$; the centers lie in the successive supports, which map into $\mathrm{supp}{\mathcal{I}}_{\alpha }$, so at every step the blowup is an isomorphism over the complement of the preimage of $\mathrm{supp}{\mathcal{I}}_{\alpha }$, and the composite ${\sigma }_{\alpha }:X_{\alpha }\to Z_{\alpha }^{\mathrm{\prime }}$ is an isomorphism over $Z_{\alpha }^{\mathrm{\prime }}\setminus \mathrm{supp}{\mathcal{I}}_{\alpha }$. Set $X={⨆}_{\alpha }{X}_{\alpha }$ and $\sigma ={⨆}_{\alpha }{\sigma }_{\alpha }$. Each $X_{\alpha }$ is a finite sequence of blowups of the integral projective scheme $Z_{\alpha }^{\mathrm{\prime }}$ along regular centers: by step 2.2 each intermediate scheme is a finite disjoint union of integral projective $\mathbb{C}$-schemes, and by step 3.1 it is smooth; hence each $X_{\alpha }$ is integral (or empty), smooth and projective, and $X$ is smooth and projective by step 1.3. On each $Z_{\alpha }^{\mathrm{\prime }}$ with ${\mathcal{I}}_{\alpha }\mathrm{\ne }{\mathcal{O}}_{Z_{\alpha }^{\mathrm{\prime }}}$, the total transform ${\sigma }_{\alpha }^{\ast }{\mathcal{I}}_{\alpha }$ is the ideal of an effective Cartier divisor with SNC support; on pieces with ${\mathcal{I}}_{\alpha }={\mathcal{O}}_{Z_{\alpha }^{\mathrm{\prime }}}$ it is the unit ideal, defining the zero divisor with empty support. Let $D\subseteq X$ be the union of these supports with reduced structure: its ideal is, locally, generated by the products of the components' equations, so $D$ is a reduced effective Cartier divisor with simple normal crossings support. Finally $\sigma$ is an isomorphism over $Z^{\mathrm{\prime }}\setminus \mathrm{supp}\mathcal{I}=W^{\mathrm{\prime }}$, so $X\setminus D\cong W^{\mathrm{\prime }}\cong W$, and the composite morphism $c:X\to P_r\to \cdots \to P$ restricts to the given open immersion $W\hookrightarrow Z\subseteq P$. This proves: (C) for every smooth projective $P$, reduced closed $Z\subseteq P$ and open dense smooth $W\subseteq Z$, there are a smooth projective $X$, a reduced SNC divisor $D\subseteq X$, an isomorphism $X\setminus D\cong W$ and a morphism $c:X\to P$ inducing the inclusion of $W$; moreover $X$ is a finite disjoint union of integral projective pieces.

Existence of an SNC projective compactification. By [F1] there is a quasi-compact immersion $\iota :U\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^n$; let $Z\subseteq {\mathbb{P}}_{\mathbb{C}}^n$ be its scheme-theoretic image, so that $U\cong Z\cap V$ is open and dense in the reduced closed subscheme $Z$, with $U$ contained in the regular locus, by steps 1.1 and 2.1. Apply (C) of step 4.1 with $P={\mathbb{P}}_{\mathbb{C}}^n$ and $W=U$: the output $X$ is a finite disjoint union of integral projective smooth $\mathbb{C}$-schemes with a reduced SNC boundary divisor $D$ and $X\setminus D\cong U$. The exceptional boundary divisor misses the generic point of every component, so its intersection with each irreducible component is a proper closed subset and hence nowhere dense; therefore $X\setminus D$ is dense in $X$, and $j:U\cong X\setminus D\hookrightarrow X$ is a smooth projective compactification with reduced SNC boundary. If $U=\mathrm{\varnothing }$ the same construction with $Z=\mathrm{\varnothing }$ returns $X=\mathrm{\varnothing }$ and the empty divisor, and all claims are vacuous.

Common domination. Let $U\hookrightarrow X_1,\dots ,U\hookrightarrow X_t$ be smooth projective compactifications, and put $P=X_1{\times }_{\mathbb{C}}\cdots {\times }_{\mathbb{C}}{X}_t$. Each $X_i$ is smooth over $\mathbb{C}$ and H-projective, so $P$ is smooth over $\mathbb{C}$ by stability of smoothness under base change and composition, and projective over $\mathbb{C}$: the product of closed immersions $X_1{\times }_{\mathbb{C}}\cdots {\times }_{\mathbb{C}}{X}_t\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^{N_1}{\times }_{\mathbb{C}}\cdots {\times }_{\mathbb{C}}{\mathbb{P}}_{\mathbb{C}}^{N_t}$ is a closed immersion by iterated base change, and the iterated Segre embeddings are closed immersions. The diagonal morphism $\delta :U\to P$ is an immersion: it is the composite of the diagonal $U\to U{\times }_{\mathbb{C}}\cdots {\times }_{\mathbb{C}}U$, a closed immersion because $U$ is separated over $\mathbb{C}$, with the product of the open immersions $U\hookrightarrow X_i$, which is an open immersion; it is quasi-compact because $U$ is Noetherian and $P$ is quasi-compact. Let $Z$ be its scheme-theoretic image; by step 1.1, $Z$ is reduced with $W:=\delta (U)\cong U$ open, dense and contained in its regular locus. Apply (C) of step 4.1 to $(P,Z,W)$ and compose the resulting morphism $c:X\to P$ with the projections: the composites $X\to X_i$ restrict to the identity of $U$ under the identifications $X\setminus D\cong U$, and $X$ is a smooth projective compactification of $U$ with reduced SNC boundary dominating all $X_i$. For the empty family, step 5.1 already supplies a compactification.

Extension of morphisms. Let $\phi :U\to V$ be a morphism of smooth quasiprojective complex varieties and let $U\hookrightarrow \tilde{U}$, $V\hookrightarrow \tilde{V}$ be smooth projective compactifications. Set $P=\tilde{U}{\times }_{\mathbb{C}}\tilde{V}$; as in step 6.1 it is smooth and projective. Consider the morphism $\gamma =(\iota ,\phi ):U\to P$, where $\iota$ is the open immersion $U\hookrightarrow \tilde{U}$; it factors as the graph morphism $U\to U{\times }_{\mathbb{C}}V$, a closed immersion because $V$ is separated over $\mathbb{C}$, followed by the open immersion $U{\times }_{\mathbb{C}}V\hookrightarrow \tilde{U}{\times }_{\mathbb{C}}\tilde{V}$, so $\gamma$ is an immersion, and it is quasi-compact because $U$ is Noetherian. Let $Z$ be the scheme-theoretic image of $\gamma$; by step 1.1, $Z$ is reduced and $W:=\gamma (U)\cong U$ is open, dense and smooth inside $Z$. Apply (C) of step 4.1 to $(P,Z,W)$, obtaining a smooth projective compactification $X$ of $U$ with reduced SNC boundary and a morphism $c:X\to P$ inducing $\gamma$ on $U$. Then the composite of $c$ with the second projection $P\to \tilde{V}$ is a morphism $X\to \tilde{V}$ restricting to $\phi$ on $U$, and the composite with the first projection is a morphism $X\to \tilde{U}$ restricting to the identity on $U$; so $\phi$ extends after modifying the source compactification outside $U$. Two extensions obtained from the same data coincide on $U$, and the construction of step 6.1 applied to the two source modifications makes the resulting pullback maps agree on $U$.

Choice accounting and boundaries. AC is assumed in the statement and is consumed exactly through the suppliers of [F1]--[F8]: the scheme-theoretic image theorem and the quasi-coherent ideal correspondence, the weak embedded desingularization and principalization algorithms with their canonical regular centers and finite blowup sequences, the Serre-generation theorem behind global generation, the graph-closure corollary and the Segre and ample-pullback items, the étale-local model of regular immersions together with flat base change of blowups, and the coherence identification; all of these either assume AC directly or inherit it from the relative Proj and blowup constructions. The construction uses no further selection: components, divisors and morphisms are constructed canonically or transported along given maps, and the finite disjoint unions, diagonal and graph arguments are choice-free. Boundary cases: if $U=\mathrm{\varnothing }$ then $X=\mathrm{\varnothing }$ and $D=\mathrm{\varnothing }$, the empty divisor being SNC with the empty parameter set, and all three clauses are vacuous; if $U$ is disconnected, $X$ is the finite disjoint union of the compactifications of its components and is projective by step 1.3; if a compactification already has empty boundary, the boundary ideal is the unit ideal and contributes the identity sequence and $D=\mathrm{\varnothing }$; the statements are insensitive to replacing $X$ by another compactification, as shown in step 6.1.

Structure of the spaces. For the first sentence, discard the open-and-closed components of $X$ and $X^{\mathrm{\prime }}$ that do not meet $U$. Smooth projective components are integral by the regular-local-domain argument [F9], so the remaining open $U$ is dense in each. On each retained component of $X^{\mathrm{\prime }}$, its dense part in $U$ maps to one component of $X$; continuity makes the entire component map into that closed target component. Thus $p$ restricts to the retained unions. Singular chains split over these finitely many open-and-closed components, so discarding the others does not change either restriction image in $H^n(U)$. We may therefore assume the embeddings dense. By [F1], $X$ and $X^{\mathrm{\prime }}$ are compact oriented boundaryless $2d$-manifolds, $U$ is an oriented boundaryless $2d$-manifold, and $j,j^{\mathrm{\prime }}$ are orientation-preserving open embeddings; $p$ is continuous with $p\circ j^{\mathrm{\prime }}=j$. The cohomology groups and maps below are those of these analytic spaces. When $d=0$ the spaces are finite discrete, the dense open embeddings are bijective, and the assertions hold by inspection.

The morphism is the identity over $U$. Put $V=p^{-1}(j(U))$, an algebraic open subset of $X^{\mathrm{\prime }}$ containing the dense open $j^{\mathrm{\prime }}(U)$. The maps $V\to X^{\mathrm{\prime }}$ given by inclusion and $j^{\mathrm{\prime }}\circ j^{-1}\circ p$ agree on $j^{\mathrm{\prime }}(U)$. Their equalizer is a closed subscheme by separatedness, and its defining sections vanish on the dense open $j^{\mathrm{\prime }}(U)$; reducedness and [F9] make those sections zero. Thus the two maps agree on all of $V$, and every point of $V$ belongs to $j^{\mathrm{\prime }}(U)$. Hence $p^{-1}(j(U))=j^{\mathrm{\prime }}(U)$ and $p$ restricts there to $j\circ (j^{\mathrm{\prime }}{)}^{-1}$. In particular, for every compact $K\subseteq U$, $p^{-1}(j(K))=j^{\mathrm{\prime }}(K)$ on the analytic spaces.

The augmentation evaluates the Kronecker pairing through cap. For every space $M$, $\alpha \in H^n(M;\mathbb{Q})$ and $z\in H_n(M;\mathbb{Q})$ one has $\mathrm{\Phi }(\alpha ,z)={\epsilon }_{\ast }([\alpha \cap z])$. Choose a cocycle $\phi$ and a cycle $z={\sum }_{\sigma }{c}_{\sigma }\sigma$ representing the two classes; the defining cap formula in degree $n$ gives $\phi \cap z={\sum }_{\sigma }{c}_{\sigma }\phi (\sigma )[\sigma (n)]$, a finite sum of last-vertex generators, and ${\epsilon }_{\ast }$ of its class is ${\sum }_{\sigma }{c}_{\sigma }\phi (\sigma )=\mathrm{\Phi }(\alpha ,z)$. Both sides are well defined on classes: cap descends by [F4] and $\epsilon$ descends by [F5].

The pairings. For $M=X$ or $X^{\mathrm{\prime }}$ define ${\mathrm{\Lambda }}_M(\alpha ,\beta ):=\mathrm{\Phi }(\alpha ,D_M\beta )$ for $\alpha \in H^n(M;\mathbb{Q})$ and $\beta \in H^{2d-n}(M;\mathbb{Q})$, and define ${\mathrm{\Lambda }}_U(\alpha ,c):=\mathrm{\Phi }(\alpha ,D_{U}c)$ for $\alpha \in H^n(U;\mathbb{Q})$ and $c\in H_c^{2d-n}(U;\mathbb{Q})$. Each $\mathrm{\Lambda }$ is $\mathbb{Q}$-bilinear and perfect: $D_M$ and $D_U$ are isomorphisms by [F2], so for $\alpha \mathrm{\ne }0$ there is $z$ with $\mathrm{\Phi }(\alpha ,z)\mathrm{\ne }0$ and $z=D\beta$ for a unique $\beta$, whence $\mathrm{\Lambda }(\alpha ,\beta )\mathrm{\ne }0$; conversely, if $\beta \mathrm{\ne }0$ then $D\beta \mathrm{\ne }0$ and evaluation at the nonzero class $D\beta$ is a nonzero functional on the full dual of homology, realized by some $\alpha$ because $\mathrm{\Phi }$ identifies cohomology with that full dual by [F3].

Cap, cup and the fundamental class. For a closed oriented $2d$-manifold $M$, $\alpha \in H^n(M;\mathbb{Q})$ and $\beta \in H^{2d-n}(M;\mathbb{Q})$, one has $\mathrm{\Phi }(\beta ,D_M\alpha )=⟨\alpha ⌣\beta ,[M]⟩$, where the right side abbreviates $\mathrm{\Phi }(\alpha ⌣\beta ,[M])$ for the top-degree Kronecker pairing. Indeed $D_M\alpha =\alpha \cap [M]$ by [F2], and the second cap identity of [F4] gives $(\alpha ⌣\beta )\cap [M]=\beta \cap (\alpha \cap [M])=\beta \cap D_M\alpha$; applying ${\epsilon }_{\ast }$ and step 1.3 in degrees $2d-n$ and $2d$ gives the stated equality.

Compatibility of the pairings with the open inclusion. For $\theta \in H^n(X;\mathbb{Q})$ or $\theta \in H^n(X^{\mathrm{\prime }};\mathbb{Q})$ and $c\in H_c^{2d-n}(U;\mathbb{Q})$, one has ${\mathrm{\Lambda }}_U(j^{\ast }\theta ,c)={\mathrm{\Lambda }}_X(\theta ,e_{X}c)$ and ${\mathrm{\Lambda }}_U((j^{\mathrm{\prime }}{)}^{\ast }\theta ,c)={\mathrm{\Lambda }}_{X^{\mathrm{\prime }}}(\theta ,e_{X^{\mathrm{\prime }}}c)$. Indeed ${\mathrm{\Lambda }}_U(j^{\ast }\theta ,c)=\mathrm{\Phi }(j^{\ast }\theta ,D_{U}c)=\mathrm{\Phi }(\theta ,j_{\ast }{D}_{U}c)=\mathrm{\Phi }(\theta ,D_X(e_{X}c))={\mathrm{\Lambda }}_X(\theta ,e_{X}c)$ by the naturality in [F3] and the duality naturality in [F2], and the primed identity is identical.

The identity over compact supports. For every $n$ and every $c\in H_c^{2d-n}(U;\mathbb{Q})$ one has $p^{\ast }(e_{X}c)=e_{X^{\mathrm{\prime }}}c$ in $H_c^{2d-n}(X^{\mathrm{\prime }};\mathbb{Q})$, where $p^{\ast }$ acts on the relative groups, possible by step 1.2. Choose by [F7] a compact $K\subseteq U$ and $c_K\in H^{2d-n}(U,U\setminus K;\mathbb{Q})$ representing $c$; the extension maps send $c$ to the colimit classes represented by $(j^{\ast }{)}^{-1}{c}_K$ and $((j^{\mathrm{\prime }}{)}^{\ast }{)}^{-1}{c}_K$. By step 1.2, $p^{-1}(K)=K$, so $p$ induces a map of pairs $(X^{\mathrm{\prime }},X^{\mathrm{\prime }}\setminus K)\to (X,X\setminus K)$ and $p^{\ast }$ is defined on these relative groups. Functoriality of pullback gives $(j^{\mathrm{\prime }}{)}^{\ast }\circ p^{\ast }=(p\circ j^{\mathrm{\prime }}{)}^{\ast }=j^{\ast }$, so $(j^{\mathrm{\prime }}{)}^{\ast }(p^{\ast }((j^{\ast }{)}^{-1}{c}_K))=j^{\ast }(j^{\ast }{)}^{-1}{c}_K=c_K=(j^{\mathrm{\prime }}{)}^{\ast }((j^{\mathrm{\prime }}{)}^{\ast }{)}^{-1}{c}_K$, and injectivity of the excision isomorphism $(j^{\mathrm{\prime }}{)}^{\ast }$ gives $p^{\ast }((j^{\ast }{)}^{-1}{c}_K)=((j^{\mathrm{\prime }}{)}^{\ast }{)}^{-1}{c}_K$; the two colimit classes therefore coincide.