Modification preserves open cohomology images

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.

The transfer and its adjointness. For each $n$ let $T_n:=D_X^{-1}\circ p_{\ast }\circ D_{X^{\mathrm{\prime }}}:H^n(X^{\mathrm{\prime }};\mathbb{Q})\to H^n(X;\mathbb{Q})$, a composite of the duality isomorphisms and the pushforward of [F2] and [F3]. Then, for all $a\in H^n(X^{\mathrm{\prime }};\mathbb{Q})$ and $b\in H^{2d-n}(X;\mathbb{Q})$, one has ${\mathrm{\Lambda }}_X(T_{n}a,b)={\mathrm{\Lambda }}_{X^{\mathrm{\prime }}}(a,p^{\ast }b)$. Write $\sigma =n(2d-n)$. Using step 2.1 and the definition of $T_n$, ${\mathrm{\Lambda }}_X(T_{n}a,b)=\mathrm{\Phi }(T_{n}a,D_{X}b)=⟨b⌣T_{n}a,[X]⟩=(-1{)}^{\sigma }⟨T_{n}a⌣b,[X]⟩=(-1{)}^{\sigma }\mathrm{\Phi }(b,D_X{T}_{n}a)=(-1{)}^{\sigma }\mathrm{\Phi }(b,p_{\ast }{D}_{X^{\mathrm{\prime }}}a)$, where the third equality is [F6] and the last is the defining property $D_X\circ T_n=p_{\ast }\circ D_{X^{\mathrm{\prime }}}$; naturality from [F3] turns the last term into $(-1{)}^{\sigma }\mathrm{\Phi }(p^{\ast }b,D_{X^{\mathrm{\prime }}}a)$. On the other side, ${\mathrm{\Lambda }}_{X^{\mathrm{\prime }}}(a,p^{\ast }b)=\mathrm{\Phi }(a,D_{X^{\mathrm{\prime }}}{p}^{\ast }b)=⟨p^{\ast }b⌣a,[X^{\mathrm{\prime }}]⟩=(-1{)}^{\sigma }⟨a⌣p^{\ast }b,[X^{\mathrm{\prime }}]⟩=(-1{)}^{\sigma }\mathrm{\Phi }(p^{\ast }b,D_{X^{\mathrm{\prime }}}a)$ by step 2.1 and [F6]. The two displayed values agree.

The identity on $U$. For every $a\in H^n(X^{\mathrm{\prime }};\mathbb{Q})$ one has $j^{\ast }(T_{n}a)=(j^{\mathrm{\prime }}{)}^{\ast }(a)$ in $H^n(U;\mathbb{Q})$. By perfectness of ${\mathrm{\Lambda }}_U$ in its first variable (step 1.4), it suffices to test against every $c\in H_c^{2d-n}(U;\mathbb{Q})$: ${\mathrm{\Lambda }}_U(j^{\ast }(T_{n}a),c)={\mathrm{\Lambda }}_X(T_{n}a,e_{X}c)={\mathrm{\Lambda }}_{X^{\mathrm{\prime }}}(a,p^{\ast }(e_{X}c))={\mathrm{\Lambda }}_{X^{\mathrm{\prime }}}(a,e_{X^{\mathrm{\prime }}}c)={\mathrm{\Lambda }}_U((j^{\mathrm{\prime }}{)}^{\ast }a,c)$, where the first and last equalities are step 2.2, the second is step 3.1 and the third is step 2.3.

Equality of the images. Let $b\in H^n(X;\mathbb{Q})$. Then $(j^{\mathrm{\prime }}{)}^{\ast }(p^{\ast }b)=(p\circ j^{\mathrm{\prime }}{)}^{\ast }b=j^{\ast }b$, by functoriality of pullback [F1, F3], so the image of $H^n(X;\mathbb{Q})$ in $H^n(U;\mathbb{Q})$ is contained in the image of $H^n(X^{\mathrm{\prime }};\mathbb{Q})$. Conversely, if $a\in H^n(X^{\mathrm{\prime }};\mathbb{Q})$, step 4.1 exhibits $(j^{\mathrm{\prime }}{)}^{\ast }a=j^{\ast }(T_{n}a)$ with $T_{n}a\in H^n(X;\mathbb{Q})$, so the reverse inclusion holds and the two images agree for every $n$ with $0\le n\le 2d$. For $n<0$ or $n>2d$ both cohomology groups vanish by duality [F2] and the vanishing conventions of [F3], so both images are $\{0\}$; if $U=\mathrm{\varnothing }$ then $X=X^{\mathrm{\prime }}=\mathrm{\varnothing }$ because $U$ has dense image, and all groups vanish. This proves the first sentence of the statement.

Independence of arbitrary smooth projective compactifications. Let $X$ be an arbitrary smooth projective compactification of $U$, with no SNC hypothesis on its boundary, and let $Z$ be one with reduced SNC boundary, which exists by the first clause of [F8]. Apply the extension clause of [F8] to the identity morphism $U\to U$, viewed with source compactification $Z$ and target compactification $X$: by the closing remark of [F8] the clause is available for arbitrary smooth projective compactifications of the open variety, and it yields a smooth projective compactification $Y$ of $U$ with reduced SNC boundary together with morphisms $Y\to Z$ and $Y\to X$ restricting to the identity on $U$. Since $Y$ is projective and $X$ and $Z$ are separated, the morphisms $Y\to X$ and $Y\to Z$ are proper by Morphisms from a proper scheme to a separated one are proper, and the constructions preserve the component dimensions. Apply step 5.1 on each equal-dimensional component (and take the finite direct sums if dimensions differ); this gives, for every $n$, $\mathrm{im}(H^n(X;\mathbb{Q})\to H^n(U;\mathbb{Q}))=\mathrm{im}(H^n(Y;\mathbb{Q})\to H^n(U;\mathbb{Q}))=\mathrm{im}(H^n(Z;\mathbb{Q})\to H^n(U;\mathbb{Q}))$. Hence the image subgroup attached to a smooth projective compactification of $U$ is independent of the compactification and may always be computed from one with SNC boundary: any identity expressing that subgroup for an SNC compactification, in particular the compactification-image formula of logarithmic weights, holds for every smooth projective compactification. This proves the second sentence of the statement.

Choice accounting and boundary cases. AC is assumed in the statement and is consumed exactly through the suppliers of [F1]--[F9]: the analytification with its Chern-form, Jacobian-criterion, ample-power and comparison inputs, Poincare duality with its orientation classes and exhaustions, the full-dual universal coefficient statement, the compact-support colimit and excision, the resolution, principalization and Serre-generation inputs of [F8], and the regular-local-domain supplier of [F9]; all cap, cup, pairing and sequence computations are choice-free. Boundary cases: for $d=0$ all spaces are finite discrete, the groups vanish above degree $0$, $H^0$ has dimension the number of points, $p$ is bijective over $U$ and both images agree; for $U=\mathrm{\varnothing }$ all groups vanish; for $n$ outside $0,\dots ,2d$ both groups vanish; for $p={\mathrm{id}}_X$ the transfer $T_n$ is the identity; disconnected $X,X^{\mathrm{\prime }},U$ are covered because [F2] and [F3] decompose along components and all formulas are componentwise; the compact support $K$ of step 2.3 may be enlarged and the class $c=0$ or empty $K$ give zero in the same way. Properness of $p$ is stated but not used beyond continuity and $p\circ j^{\mathrm{\prime }}=j$, both of which hold for any morphism of compactifications; no Hodge theory, strictness or mixed structure is used, and no identification of infinite-dimensional vector spaces with double duals is made.