Proof. For a diagram \(T\to Z\leftarrow T^+\), take the reduced component of \(T\times_ZT^+\) containing the graph over the common isomorphism open. Its projections are projective, and finite normalization preserves projectivity. Adjoin the remaining spaces and the carrier one at a time by the same construction. Resolving the resulting dominating component gives a common smooth model. Analytic resolution and principalization allow the stated simultaneous choices by projective modifications; see [2]. Base change, restriction to a closed analytic subspace, finite normalization, and composition preserve projectivity. For composition in the present compact setting, one can choose a uniform sufficiently large twist of a relative ample bundle by the pullback of the next one. This also proves compatibility with any finite chain of the small diagrams and divisorial contractions considered below.

A smooth space projective over a compact Kähler space is compact Kähler: a relative ample bundle supplies positivity in the fiber directions, and adding a sufficiently large multiple of a pulled-back Kähler form supplies positivity in all directions. Finally, choose Kähler classes on \(X'\) converging to \(c_1(M')\). Their pullbacks to \(W\) have semipositive representatives. Adding positive multiples tending to zero of a fixed Kähler class on \(W\) gives Kähler classes converging to \(c_1(M_W)\). Thus \(M_W\) is analytically nef. ◻

Proof. The trace definition gives \(r_*J_{W/T}=0\). On an \(r\)-contracted curve, \(-J_{W/T}=M_W-r^*M_T\) has the same degree as \(M_W\), hence has nonnegative degree by projection formula from the nef carrier. Analytic negativity gives \(J_{W/T}\geq0\). The final identity follows from functoriality of pullback and the fact that the b-divisor descends on \(W\). ◻

Proof. Choose a smooth log resolution \(r:W\to T\), projective over \(T\) and carrying \(\mathbf{M}\), with divisorial exceptional locus and simple normal crossing support for all divisors in the crepant equation \[K_W+\Delta_W+M_W=r^*(K_T+B+M_T).\] Such a simultaneous resolution is available by Lemma 8. The assumed inequalities give coefficients at most one, respectively strictly below one, for every global prime of \(\Delta_W\). The same bounds hold on their local branches. On any further local model the nef part is pulled back from \(W\), so its generalized discrepancies are the ordinary discrepancies of the simple normal crossing subpair \((W,\Delta_W)\). The simple normal crossing discrepancy criterion proves the asserted local generalized inequalities, including when some coefficients of \(\Delta_W\) are negative.

Put \(J=J_{W/T}\). The ordinary crepant boundary for \((T,0)\) is \[ \Delta_W^{(0)}=\Delta_W-r^*B-J, \qquad K_W+\Delta_W^{(0)}=r^*K_T. \tag{4}\] Both \(r^*B\) and \(J\) are effective. After resolving their joint support as above, every coefficient of \(\Delta_W^{(0)}\) is therefore strictly below one in the positive case. The same local simple normal crossing criterion gives ordinary klt singularities.

For the terminal assertion, the relative canonical divisor is exceptional, and at an exceptional prime \(R\) of \(W\) its coefficient is \[\operatorname{coeff}_R(K_W-r^*K_T) =a(R;T,B+\mathbf{M})-1+\operatorname{coeff}_R(r^*B+J)>0.\] These positive coefficients persist on every local branch. A local place exceptional over the smooth space \(W\) has ordinary log discrepancy at least two over \(W\): the Jacobian determinant of a smooth model carrying it has positive integral order along an exceptional divisor. Its discrepancy over \(T\) adds the nonnegative order of \(K_W-r^*K_T\), and hence remains greater than one. Local exceptional places already represented by divisors on \(W\) satisfy the displayed strict inequality. Thus \(T\) is locally ordinary terminal. For a normal complex fourfold with a rational canonical line bundle, local ordinary terminality implies smoothness in codimension two by [21], giving the last assertion. ◻

Proof. We first recall the fiber description for a proper bimeromorphic morphism \(v:V\to S\) of normal spaces. Its positive-dimensional-fiber locus is closed analytic. Over its complement the map is finite locally on \(S\), hence an isomorphism by normality. Stein factorization shows that every fiber is connected, because its finite bimeromorphic factor over the normal space \(S\) is an isomorphism. A reduced compact connected positive-dimensional fiber has no zero-dimensional irreducible component: such a component would be a singleton disjoint from the finitely many other closed components. Hence no point in such a fiber is a local isomorphism point. The exceptional locus of \(v\) is exactly the full inverse image of its positive-dimensional-fiber locus.

Let \(U\subset Z\) be the open set over which \(f\) is an isomorphism. On \(U\), \(D_T\) defines a \(\mathbb{Q}\)-Cartier divisor \(D_U\). The actual Weil coefficients of the two adjoints agree under the identification of prime divisors. Since \(f^+\) is small, it follows that \(D_{T^+}|_{(f^+)^{-1}U}=(f^+)^*D_U\). A positive-dimensional projective fiber over \(U\) would contain a curve on which this divisor has degree zero, contrary to \(f^+\)-ampleness. Thus \(f^+\) is an isomorphism over \(U\). Interchanging the two sides and using \(f\)-ampleness of \(-D_T\) gives the reverse inclusion of the isomorphism opens. This proves the equality of exceptional images and (6). Smallness gives \(\dim L,\dim L^+\leq2\). Over a general point of each component of \(A\) the fiber has dimension at least one, so the fiber-dimension theorem gives \(\dim A\leq1\).

The graph projections are projective and surjective, and the map from \(G\) to its component in \(T\times_ZT^+\) is finite. A prime divisor of either side maps to a prime divisor of \(Z\) by smallness. On the graph it therefore corresponds to the same prime place on the other side, with the same adjoint coefficient. Consequently \(p_*F=q_*F=0\). On a \(p\)-contracted curve, \(-F\) has the degree of \(q^*D_{T^+}\), which is nonnegative. Analytic negativity gives \(F\geq0\).

For a curve \(C\) contracted by \(h\), the projection formula and the two ample signs give \[-F\cdot C=(-D_T)\cdot p_*C+D_{T^+}\cdot q_*C>0.\] Here an image that is a point contributes zero. At least one image is a curve, because the finite map to the fiber product cannot contract \(C\) to a point. Thus at least one summand is strictly positive.

The support of \(F\) is contained in \(h^{-1}(A)\), since both projections identify the common isomorphism open and the adjoints agree there. Conversely, take \(x\in h^{-1}(A)\) and write \(z=h(x)\). The map \(h:G\to Z\) is proper bimeromorphic, so its fiber over \(z\) is connected. It is projective and positive-dimensional: its projection to the positive-dimensional fiber \(f^{-1}(z)\) is surjective. The fiber description above places \(x\) on a positive-dimensional irreducible component. Successive projective hyperplane cuts through \(x\) give an irreducible curve \(C\) in that fiber through \(x\). If \(x\) were outside \(\mathop{\mathrm{Supp}}F\), then \(C\) would not be contained in the effective \(\mathbb{Q}\)-Cartier divisor \(F\), so \(F\cdot C\geq0\). The preceding strict inequality excludes this. Hence (9) holds.

Follow a global place \(E\) on a common proper model dominating \(G\) and the nef carrier. Subtracting its two crepant equations cancels the same trace of \(\mathbf{M}\) and gives (10). The order of an effective \(\mathbb{Q}\)-Cartier divisor is positive precisely when the center of the place is contained in its support. Proper images of centers give \[\mathop{\mathrm{cent}}_Z(E)=f(\mathop{\mathrm{cent}}_T(E))=f^+(\mathop{\mathrm{cent}}_{T^+}(E))=h(\mathop{\mathrm{cent}}_G(E)).\] Together with (6) and (9), this proves both center equivalence and the stated exact strictness criterion. If the centers are not contained in these loci, their general points correspond on the common open and the order is zero. Finally, a place centered at a divisor on a normal model is that divisorial place. Smallness identifies these prime divisors on the two sides, proving equivalence of exceptionality.

Since both morphisms are nonisomorphisms, \(A\) is nonempty, so \(F\) has a prime divisor \(R\) in its support. The graph is a fourfold, hence \(\dim R=3\). Its finite map to the graph component has image in \(L\times_ZL^+\), whence \[3\leq\dim(L\times_ZL^+)\leq\dim L+\dim L^+.\] This is the remaining inequality in (7). ◻

Proof. For a divisorial step put \(E_j=D_j-v_j^*D_{j+1}\). It is exceptional over \(Y_{j+1}\) and has strictly negative degree on every \(v_j\)-contracted curve. Negativity gives \(E_j\geq0\). Every point of an exceptional fiber lies on a curve in that fiber, by the connected projective fiber argument in the proof of Lemma 11. The negative degree therefore forces that point into \(\mathop{\mathrm{Supp}}E_j\). In particular the coefficient of \(E_j\) at each contracted prime is positive. Subtracting crepant equations gives nondecreasing discrepancies and this strict increase. For a small step the same assertions, with an effective correction exceptional over its output, follow from Lemma 11.

Pull the corrections of all steps to \(W\) and add them. This gives the effective divisor in (11). Each prime in a summand is exceptional over that step’s output, and remains exceptional over every later output. Indeed, if that place became a prime divisor on a later normal model, that prime would have a strict-transform prime on each earlier model, since none of the intervening steps extracts divisors. This contradicts its earlier exceptionality. Thus the sum is exceptional over \(Y_m\). Its nonnegative coefficients also show that a strict increase from a contracted prime cannot be canceled later. For \(m=0\) the correction is zero and the assertions are immediate. ◻

Proof. Write \(r:W\to Y\) and \(t:W\to T_+\), and put \(H=P-P_+=E_+-E_Y\). Then \(r_*H=r_*E_+\geq0\). On an \(r\)-contracted curve, \(-H\) has the degree of \(P_+\), which is nonnegative because the curve lies over a point of \(S\). Negativity gives \(H\geq0\). Likewise \(t_*(-H)=t_*E_Y\geq0\), while \(H\) is \(t\)-nef because \(P\) comes from an \(S\)-nef divisor. Negativity applied to \(-H\) gives \(H\leq0\). Thus \(P=P_+\).

For a curve \(C\) in an \(r\)-fiber, equality gives \(t^*D_+\cdot C=P\cdot C=0\). The image \(t(C)\) lies in an \(S\)-fiber, where \(D_+\) is ample, so \(t(C)\) is a point. Every \(r\)-fiber is projective and connected. If an irreducible component of one such fiber had positive-dimensional image under \(t\), a curve in that projective image and projective hyperplane cuts of its inverse image would supply a curve on which \(t\) is nonconstant. Thus each component maps to a point, and connectedness makes those points equal. The map \(t\) is constant on every \(r\)-fiber.

The reduced image of \((r,t):W\to Y\times_ST_+\) is a proper analytic graph with one-point fibers over \(Y\). Its projection to \(Y\) is therefore finite and bimeromorphic, hence an isomorphism by normality of \(Y\). This gives the morphism \(\varphi\). Its graph is a closed subspace of \(Y\times_ST_+\), whose projection to \(T_+\) is projective by base change from \(Y\to S\); hence \(\varphi\) is projective. Finally, pushing the equality \(P=P_+\) to \(Y\) gives \(D_Y=\varphi^*D_+\) as actual divisors. ◻

Proof. This is the rational-coefficient version of [19]; we include its localization argument. The Borel–Moore localization sequence for \(B\subset Y\) contains \[H_{2k}(B,\mathbb{Q})\longrightarrow H_{2k}(Y,\mathbb{Q}) \longrightarrow H_{2k}^{\mathrm{BM}}(U,\mathbb{Q}) \longrightarrow H_{2k-1}(B,\mathbb{Q}).\] The outside groups vanish by \(\dim B<k\) and the real dimension bound for an analytic triangulation. Thus restriction from \(Y\) is an isomorphism in degree \(2k\). Compose restriction from \(X\) with its inverse. An irreducible compact analytic \(k\)-cycle meeting \(U\) is sent to its strict transform on \(Y\); analyticity of that transform follows by proper mapping through the given diagram. A cycle contained in \(A\) is sent to zero. Conversely, no \(k\)-cycle of \(Y\) is contained in \(B\), and its strict transform on \(X\) maps to its class. This proves the surjection on cycle spans. If \(A\) contains a \(k\)-dimensional irreducible analytic subspace, its nonzero class is in the kernel, giving strict inequality.

For a transformation that extracts no prime divisors, choose the common isomorphism open with target complement of codimension at least two. Every contracted source prime lies in the source complement. The preceding result with \(k=n-1\) gives a strict drop at each such contraction. The ranks are nonnegative integers, so there can be only finitely many drops. For a small transformation both complements have codimension at least two, and applying the result in both directions gives equality. Finally, (6) gives the common open for a small fourfold diagram. Applying the first assertion with \(k=2\), \(A=L\) and \(B=L^+\) proves the last claim. ◻

Proof. For a reduced pure projective threefold \(R\), write \(\nu_R\colon N_R\to R\) for normalization. Choose a closed reduced subset \(A_R\subset R\) of dimension at most two containing the locus where \(\nu_R\) is not an isomorphism, and put \(F_R=(\nu_R^{-1}A_R)_{\mathrm{red}}\). The square \[\begin{tikzcd} F_R \arrow[r,hook] \arrow[d] & N_R \arrow[d,"\nu_R"]\\ A_R \arrow[r,hook] & R \end{tikzcd}\] is a proper excision square. Indeed \(\nu_R\) is finite and proper and identifies the open complements. Collapsing the two closed subsets gives a continuous bijection \(N_R/F_R\to R/A_R\) of compact Hausdorff quotients, hence a homeomorphism. Comparing the two relative homology sequences gives the exact sequence \[ H_5(R)\xrightarrow{\partial_R}H_4(F_R) \longrightarrow H_4(A_R)\oplus H_4(N_R). \tag{25}\] All its maps are compatible with mixed Hodge structures. This follows from algebraic functoriality and the exact sequence of relative cohomology as a sequence of mixed Hodge structures [7], followed by duality on these compact spaces.

Take this square first for \(P_1\). For \(Q\), take \(A_Q=Q\cap A_{P_1}\). The normalization \(N_Q=\coprod_jQ_j^\nu\) is a union of entire, disjoint normalization components of \(N_{P_1}\). Thus these choices give a morphism of the two squares, with \(F_Q\) equal to the part of \(F_{P_1}\) in those selected components.

For a surface component \(V\) of \(A_Q\), let \(F_{V,1},\ldots,F_{V,r_V(Q)}\) be the surface components of \(F_Q\) mapping onto it, and let \(d_{V,\alpha}>0\) be their finite degrees over \(V\). On top homology the corresponding part of the map to \(A_Q\) is \[\bigoplus_{\alpha=1}^{r_V(Q)}\mathbb{Q}[F_{V,\alpha}] \longrightarrow\mathbb{Q}[V],\qquad \sum_\alpha t_\alpha[F_{V,\alpha}] \longmapsto\left(\sum_\alpha d_{V,\alpha}t_\alpha\right)[V].\] Its kernel has dimension \(r_V(Q)-1\). A surface not contained in \(A_Q\) has exactly one normalization divisor over it, since the normalization is an isomorphism at its general point. Conversely, finiteness and surjectivity of normalization give at least one divisor over every surface in \(Q\). It follows that \[B_Q:=\ker\bigl[H_4(F_Q)\longrightarrow H_4(A_Q)\bigr], \qquad \dim B_Q=\beta(Q).\] This also proves finiteness of the sum: the only possible nonzero terms come from the finitely many surface components of \(A_Q\).

The image of \(B_Q\) in \(H_4(N_Q)\) is spanned by classes of compact surfaces in its normalization components. Its rank is therefore at most \(\sum_jc_2(Q_j^\nu)\). Hence \[U_Q:=\ker\bigl[B_Q\longrightarrow H_4(N_Q)\bigr] \quad\text{satisfies}\quad \dim U_Q\geq\beta(Q)-\sum_jc_2(Q_j^\nu).\] By (25), \(U_Q\) is the image of \(\partial_Q\).

The map \(H_4(F_Q)\to H_4(F_{P_1})\) is injective. Both groups have surface components as their bases, and the surfaces in \(F_Q\) remain distinct surfaces on the selected normalization components of \(P_1\). Naturality of the boundary maps therefore sends \(U_Q\) injectively into the image of \(\partial_{P_1}\) on the image of \(H_5(Q)\to H_5(P_1)\). The groups \(H_4(F_Q)\) and \(H_4(F_{P_1})\) are pure of weight \(-4\). Strictness for the weight filtration [6] says that \(\partial_R\) maps \(\mathop{\mathrm{Gr}}^W_{-4}H_5(R)\) onto its image in the pure group \(H_4(F_R)\). Applying this to the naturality square, we obtain \[\dim U_Q\leq \mathop{\mathrm{rk}}\mathop{\mathrm{Gr}}^W_{-4}\bigl[H_5(Q)\longrightarrow H_5(P_1)\bigr].\] Finally, Borel–Moore localization contains \[H^{\mathrm{BM}}_6(P\setminus P_1)\longrightarrow H_5(P_1) \longrightarrow H_5(P).\] The first group vanishes because \(P\setminus P_1\) has complex dimension at most two. The second arrow is therefore injective, and remains injective on weight-graded pieces by strictness. Duality between rational homology and cohomology reverses the sign of the weight and turns the last two inequalities into (24). ◻

Proof. Write \(B\) for the target of \(f\) and \(K_f=Rf_*\mathbb{Q}_{\widetilde Y}[4]\). Fix an \(f\)-relatively ample line bundle and let \(\ell\) be its rational first Chern class. Cup product defines a global morphism \(\ell\colon K_f\to K_f[2]\).

We first obtain relative hard Lefschetz locally on \(B\). Around a point of \(B\), choose a small chart \(V\) embedded as a closed analytic subspace of a complex manifold. After making the chart smaller, relative ampleness embeds \(f^{-1}V\) in a projective space over \(V\); composing with the closed embedding of \(V\) gives a projective map from the smooth manifold \(f^{-1}V\) to the ambient manifold. On each connected component of the smooth source, the constant shifted sheaf underlies the constant polarizable Hodge module with that strict support. Saito’s projective direct-image theorem [23] applies to precisely this situation: its morphisms are projective morphisms of smooth complex analytic manifolds, its Lefschetz class is the Chern class of a relatively ample line bundle, and its conclusion includes relative hard Lefschetz and polarizable perverse direct images. Passing through the closed embedding, which is fully faithful and exact for perverse sheaves, gives on \(V\) \[\ell^a\colon {}^p\!H^{-a}(K_f)|_V \xrightarrow{\ \sim\ }{}^p\!H^a(K_f)|_V \qquad(a\geq0).\] After forgetting Hodge structures, the Tate twist has the same underlying rational perverse sheaf. The operator is the restriction of the single global \(\ell\), and a morphism of perverse sheaves is an isomorphism if it is so on an open cover. Thus these are global relative hard Lefschetz isomorphisms on \(B\).

The formal Lefschetz splitting criterion [5] now gives \[ K_f\simeq\bigoplus_i{}^p\!H^i(K_f)[-i]. \tag{27}\] The criterion uses truncation triangles and the induced Lefschetz isomorphisms; the same argument applies to the bounded perverse \(t\)-structure. This explains why the local analytic theorem gives a global derived splitting here.

We also need to separate the supports of the perverse terms in (27). Locally, Saito’s pure direct images decompose by strict support [23]; their underlying perverse terms are middle extensions from smooth opens of those supports. The support decomposition glues without requiring semisimplicity of local systems on arbitrary analytic opens. Indeed, a perverse morphism between middle extensions with different irreducible supports must be zero: its image would be a subobject or a quotient with smaller support in at least one of them. Restriction to an open preserves the middle-extension property; every resulting strict support is a local branch of the original support. The local system may decompose further without changing that support. For each perverse term and each dimension \(d\), the sum of the local projectors onto all \(d\)-dimensional strict-support summands is intrinsic. Restriction preserves dimensions of local branches, so these sums agree on overlaps and glue as perverse morphisms. The support of each glued dimension summand is closed analytic in every chart, hence globally. For each irreducible component \(Z\) of this support, sum all local projectors whose strict supports are local branches of \(Z\). These sums agree by the same zero-morphism argument and glue to a global projector whose image has strict support \(Z\); the middle-extension property is local. Local finiteness and compactness make the number of these summands finite.

Finally \(f\) is an isomorphism over a dense open of its target. On that open \(K_f\) is \(\mathbb{Q}[4]\) in perverse degree zero. The only full-support term is consequently the middle extension of this constant local system, namely \(\mathop{\mathrm{IC}}_B\), in degree zero. This proves the lemma. ◻

Proof. Apply Lemma 19 to \(f=pg\). Besides \(\mathop{\mathrm{IC}}_X\), write a strict-support summand as \(\mathcal K_{Z,i}[-i]\), where \(\mathcal K_{Z,i}\) is perverse with proper irreducible support \(Z\subset X\) of dimension \(z\). Over a general point \(x\) of \(Z\), the fiber of \(f\) has dimension at most \(3-z\): \(f^{-1}(Z)\) is a proper analytic subset of the irreducible fourfold \(\widetilde Y\) and therefore has dimension at most three. Proper base change and the topological dimension of a compact analytic fiber give \[\mathcal H^k(K_f)_x=H^{k+4}(f^{-1}(x))=0 \quad\text{if }k>2(3-z)-4.\] The generic nonzero stalk of \(\mathcal K_{Z,i}[-i]\) is in degree \(i-z\), because a perverse middle extension on \(Z\) has its generic local system in degree \(-z\). It is a direct summand of this stalk, so \[ i-z\leq2(3-z)-4,\qquad\text{or equivalently}\qquad i+z\leq2. \tag{28}\]

We now restrict the decomposition to \(C\). Suppose first that \(Z\not\subset C\), and stratify \(Z\cap C\) compatibly with \(\mathcal K_{Z,i}\). A stratum \(S\) has dimension \(s\leq z-1\) and \(s\leq1\). Its stalk degrees for \(\mathcal K_{Z,i}\) are at most \(-s-1\) by (26). This remains valid under the refinement by \(C\): a stratum cut out of an old boundary stratum inherits the stronger bound from that stratum, while a stratum cut out of the local-system open has only degree \(-z\leq-s-1\). Compactly supported cohomology on the smooth complex \(s\)-fold \(S\) vanishes above degree \(2s\). Thus its contribution to the hypercohomology of \(\mathcal K_{Z,i}[-i]|_C\) vanishes in degrees greater than \[i-s-1+2s=i+s-1\leq i+z-2\leq0.\] Filtering \(C\) by its finitely many strata proves that this summand has no degree-one hypercohomology on \(C\). The full-support summand \(\mathop{\mathrm{IC}}_X\) has the same property: for each stratum of \(C\), the upper bound is \(-s-1+2s=s-1\leq0\); on the smooth open its only stalk degree is \(-4\), which is smaller still.

Only summands with \(Z\subset C\) can therefore contribute in degree one. Such a complex is supported on the closed subset \(C\), so its restriction induces an isomorphism on hypercohomology from \(X\) to \(C\). The chosen global splitting and proper base change now show that \[\mathbb H^1(X,K_f)=H^5(\widetilde Y) \longrightarrow \mathbb H^1(C,K_f|_C)=H^5(\widetilde P)\] is surjective.

It remains to justify what this surjection says about weights. Compact Kähler Hodge theory gives \(H^5(\widetilde Y,\mathbb{Q})\) its pure Hodge structure of weight \(5\). On the projective \(\widetilde P\), Deligne’s mixed Hodge structure has weights at most \(5\) [7]. We verify directly that the restriction map preserves its Hodge filtration, because the ambient \(\widetilde Y\) may be nonprojective.

Choose a proper smooth projective hyperresolution \(\epsilon_\bullet\colon Z_\bullet\to\widetilde P\). Deligne’s construction computes the Hodge filtration on \(H^5(\widetilde P)\) from the total simplicial de Rham complex of the \(Z_n\), filtered by holomorphic form degree [7]. The terms can be chosen projective because \(\widetilde P\) is projective and the construction can use projective resolutions and modifications. We may use smooth forms in this complex, with the same filtration: the Dolbeault resolutions give filtered quasi-isomorphisms from the holomorphic de Rham complexes on the smooth \(Z_n\). A class in \(F^aH^5(\widetilde Y,\mathbb{C})\) has a closed smooth representative \(\eta\) whose components have holomorphic degree at least \(a\), by the Kähler Hodge decomposition. Pullback along the holomorphic map \(Z_0\to\widetilde P\hookrightarrow\widetilde Y\) preserves these bidegrees. Put that pullback in simplicial degree zero of the total complex. Its de Rham differential vanishes, and its simplicial differential is zero because the two face pullbacks to \(Z_1\) have the same composite with the augmentation. It is therefore a cocycle in the filtered total de Rham complex. It represents the ordinary topological restriction by de Rham naturality and cohomological descent. Thus restriction preserves \(F^a\) for every \(a\).

The restriction is rational on underlying cohomology. It also preserves \(W\): the source is pure of weight \(5\), and \(W_5\) is all of the target. It is consequently a morphism of mixed Hodge structures. Strictness for \(W\) [6], applied to the surjection just proved, yields \[W_4H^5(\widetilde P) =\mathop{\mathrm{im}}\bigl[W_4H^5(\widetilde Y)\to H^5(\widetilde P)\bigr]=0.\] Together with the upper weight bound, this proves purity of weight \(5\). Strictness here concerns mixed Hodge structures themselves and does not require a rational polarization on the compact Kähler source. ◻

Proof. Use the full-support projection in the chosen decomposition for \(g\) to form the morphism \[\alpha\colon\mathbb{Q}_Y[4]\longrightarrow Rg_*\mathbb{Q}_{\widetilde Y}[4] \longrightarrow\mathop{\mathrm{IC}}_Y.\] The first arrow is the natural pullback map. Both arrows restrict to the identity on \(Y_{\mathrm{reg}}\). Let \(\mathcal L\) be the cone of \(\alpha\), so that \[ \mathbb{Q}_Y[4]\xrightarrow{\alpha}\mathop{\mathrm{IC}}_Y\longrightarrow\mathcal L \longrightarrow\mathbb{Q}_Y[5]. \tag{31}\] It is supported on \(\mathop{\mathrm{Sing}}Y\). Enlarge the finite puncture sets, if necessary, so that the cohomology sheaves of \(\mathcal L\) are local systems on each \(T^\circ\). Analytic constructibility permits this enlargement, and the preceding coinvariant description shows that it does not change \(b_T\).

By proper base change, the map induced by \(\alpha\) on \(P\) factors as \[H^5(P)\longrightarrow H^5(\widetilde P) \longrightarrow\mathbb H^1(P,\mathop{\mathrm{IC}}_Y|_P).\] The first arrow comes from a holomorphic morphism between the projective spaces \(\widetilde P\) and \(P\), and is algebraic by GAGA. It therefore preserves weights. Lemma 20 shows that it kills \(W_4H^5(P)\). Hence the displayed composite kills that subspace as well; we need no weight statement about the chosen perverse projection. The long exact sequence of (31) on \(P\) gives \[W_4H^5(P)\subset \mathop{\mathrm{im}}\bigl[\mathbb H^0(P,\mathcal L|_P)\longrightarrow H^5(P)\bigr].\] Restricting the triangle from \(P\) to \(Q\) shows that the image of this subspace in \(H^5(Q)\) is contained in the image of \(\mathbb H^0(Q,\mathcal L|_Q)\). Passing to a weight-graded quotient can only lower rank, and therefore \[ \mathop{\mathrm{rk}}\mathop{\mathrm{Gr}}^W_4\bigl[H^5(P)\longrightarrow H^5(Q)\bigr] \leq\dim\mathbb H^0(Q,\mathcal L|_Q). \tag{32}\]

We estimate the last group by its strata. On a stratum of \(\mathop{\mathrm{Sing}}Y\) of dimension \(s\in\{0,1\}\), the bound (26) and the triangle give \[ \mathcal H^q(\mathcal L)|_S=0\qquad(q>-s-1). \tag{33}\] The constant complex in the triangle has its only stalk cohomology in degree \(-4\); its shifted contribution to the cone is in degree \(-5\), so it cannot alter this upper bound. On a curve \(T^\circ\), the same triangle identifies \(\mathcal H^{-2}(\mathcal L)\) with \(\mathcal H^{-2}(\mathop{\mathrm{IC}}_Y)\). The curve stratum has complex codimension three in \(Y\). In the middle-extension construction, attaching a codimension-\(c\) stratum retains the unshifted cohomology of \(Rj_*\mathbb{Q}\) through degree \(c-1\). For \(c=3\) this gives \[ \mathcal H^{-2}(\mathcal L)|_{T^\circ} =\mathcal H^{-2}(\mathop{\mathrm{IC}}_Y)|_{T^\circ} =(R^2j_*\mathbb{Q})|_{T^\circ}=\mathcal V_T. \tag{34}\] This is the attaching truncation of [11], with the shift by four from our perverse normalization.

The restriction \(\mathcal L|_Q\) is supported on the closed set \(\mathop{\mathrm{Sing}}Y\cap Q\), so its hypercohomology may be computed on that set. The complement there of the disjoint opens \(T^\circ\) for the curves contained in \(Q\) is finite. A point stratum contributes nothing to degree-zero hypercohomology by (33). The open–closed cohomology sequence therefore gives a surjection \[\bigoplus_{T\subset\mathop{\mathrm{Sing}}Y\cap Q} \mathbb H_c^0(T^\circ,\mathcal L|_{T^\circ}) \longrightarrow\mathbb H^0(Q,\mathcal L|_Q).\] On a curve, compactly supported cohomology of a local system vanishes above degree two. In the spectral sequence \[H_c^a(T^\circ,\mathcal H^b(\mathcal L)|_{T^\circ}) \ \Longrightarrow\ \mathbb H_c^{a+b}(T^\circ,\mathcal L|_{T^\circ}),\] the stalk bound gives \(b\leq-2\). The only term in total degree zero is consequently \(a=2\), \(b=-2\), whose dimension is \(b_T\) by (34). The dimension of each degree-zero abutment is at most \(b_T\). This bound and (32) prove (30). ◻

Proof. The isomorphism of \(g\) over the regular locus and adjunction give a natural morphism \[\rho\colon Rg_*\mathbb{Q}_{\widetilde Y}[4] \longrightarrow Rj_*\mathbb{Q}_{Y_{\mathrm{reg}}}[4].\] By the definition of a higher direct-image stalk and proper base change for \(g\), its degree-\(-2\) map at \(t\) is precisely (37). These are the usual sheaf cohomology groups of constant sheaves, equal to singular cohomology on the locally contractible analytic spaces in question.

Let \(\kappa\colon\mathop{\mathrm{IC}}_Y\to Rg_*\mathbb{Q}_{\widetilde Y}[4]\) be the full-support inclusion in the chosen decomposition. Its restriction to \(Y_{\mathrm{reg}}\) is the identity. The composite \(\rho\kappa\colon\mathop{\mathrm{IC}}_Y\to Rj_*\mathbb{Q}[4]\) is therefore the canonical comparison morphism: by adjunction a morphism to \(Rj_*\mathbb{Q}[4]\) is determined by its restriction to \(Y_{\mathrm{reg}}\). The codimension-three attaching construction used in (34) says that this comparison is an isomorphism on ordinary stalk cohomology in degree \(-2\) along \(T^\circ\). Its degree-\(-2\) map factors through \[\mathcal H^{-2}(Rg_*\mathbb{Q}[4])|_{T^\circ}=\mathcal W_T.\] Thus the degree-\(-2\) map induced by \(\rho\), namely (36), is surjective. It is a morphism of the local systems in (29), which gives the asserted monodromy equivariance. ◻

Proof. Fix \(T\), and write \[q_T:\mathcal W_T\longrightarrow\mathcal V_T\] for the natural quotient of Lemma 22. If \(b_T=0\) there is nothing to prove, so assume \(b_T>0\). We may remove finitely many further points of \(T^\circ\) whenever needed. The local systems already defined restrict to the smaller curve, and the induced map on fundamental groups is surjective. Thus their monodromy images and the coinvariant dimension \(b_T\) do not change.

We first construct sections \[\sigma_1,\ldots,\sigma_{b_T}\in H^0(T^\circ,\mathcal W_T)\] whose images under \(q_T\) are linearly independent in every fiber of \(\mathcal V_T\), after the permitted further finite puncturing.

Step 1. Invariant lifts from finite monodromy. First we show that \(\mathcal W_T\), and therefore its quotient \(\mathcal V_T\), has finite monodromy. Let \(\mathcal F=\widetilde Y\times_Y T\), with its full complex-space structure. It is projective over the compact projective curve \(T\). A relative embedding and GAGA identify this morphism with an algebraic projective family, including its possibly nonreduced scheme fibers. After deleting finitely many points we may assume that \(\mathcal F\to T^\circ\) is flat and that all \((R^kg_*\mathbb{Q})|_{T^\circ}\) are local systems.

For \(t\in T^\circ\), put \(F_t=\mathcal F_t\). Every integral class in \(H^2(F_t,\mathbb{Z})\) lifts, by proper base change, to \(H^2(g^{-1}U_t,\mathbb{Z})\) for a sufficiently small Stein neighborhood \(U_t\) of \(t\) in \(Y\). Terminal singularities are rational [10]. Hence \(g_*\mathcal O_{\widetilde Y}=\mathcal O_Y\) and \(R^kg_*\mathcal O_{\widetilde Y}=0\) for \(k>0\); Leray and Cartan’s Theorem B give \[H^2(g^{-1}U_t,\mathcal O_{\widetilde Y})=0.\] The exponential sequence therefore realizes the lifted integral class as the first Chern class of a holomorphic line bundle on \(g^{-1}U_t\). Its restriction to the projective scheme \(F_t\) is algebraic by the scheme form of GAGA [22]. In particular, \[ \mathop{\mathrm{Pic}}(F_t)\otimes_{\mathbb{Z}}\mathbb{Q} \xrightarrow{\ c_1\ } H^2(F_t,\mathbb{Q})=(\mathcal W_T)_t \quad\text{is surjective}. \tag{38}\] Here and below the cohomology of a scheme fiber means that of its analytic topological space; nilpotents do not change it.

We use the parameter argument of [20] to show that a spanning set of fiber classes has finite monodromy orbits. Fix a relatively very ample \(\mathcal O_{\mathcal F}(1)\). Every line bundle on a fiber is a quotient of some \(\mathcal O_{F_t}(-n)^{\oplus r}\). Letting \(n,r\), and the Hilbert polynomial vary gives countably many relative Quot schemes of finite type over \(T^\circ\) [13]. In each take the open set on which the universal quotient is invertible on the whole pulled-back family. These open sets still contain a point representing every fiber line bundle. Indeed the family and universal quotient are flat over the Quot scheme, so the local criterion for flatness makes a fiberwise locally free quotient of rank one locally free near that fiber; properness removes the image of the complementary closed locus.

There are countably many irreducible components of these parameter spaces. Choose \(t\) outside the closures of the images of all components which do not dominate \(T^\circ\). Such a point exists, since each of those closures is a finite set and a complex curve is uncountable. Put \(\Gamma=\pi_1(T^\circ,t)\). By (38), choose finitely many line bundles on \(F_t\) whose Chern classes span \((\mathcal W_T)_t\). Each is represented on a dominating reduced irreducible parameter component \(H\). The Chern class of the universal bundle gives a section of the pullback of \(\mathcal W_T\) to \(H^{\mathrm{an}}\): restrict its class on the total family to fibers and use proper base change.

Choose a closed point of the generic fiber of \(H\to T^\circ\). Its residue field is a finite extension of the function field of \(T\). Spreading this point, normalizing its closure, and deleting finitely many base points gives a connected finite étale cover of a dense open of \(T^\circ\), together with a morphism from that cover to \(H\). The analytification of an irreducible complex algebraic variety is path connected. A path in \(H^{\mathrm{an}}\) from the original parameter point to a point of this cover identifies the transported Chern class with the class there. The subgroup fixing that sheet has finite index and fixes this class. The inclusion of the further punctured curve, with the projected path for change of basepoint, still surjects on \(\Gamma\). The image of the finite-index sheet stabilizer is therefore finite index in \(\Gamma\) and fixes the original class. Thus each of our spanning classes has a finite orbit under \(\Gamma\). The intersection of their stabilizers has finite index and acts trivially on \((\mathcal W_T)_t\). The monodromy group \(G\) of \(\mathcal W_T\) is therefore finite, as is that of \(\mathcal V_T\).

Poincaré duality with local coefficients on the oriented curve gives \[b_T=\dim(\mathcal V_T)_t{}_{\Gamma}.\] For a representation of the finite group \(G\) over \(\mathbb{Q}\), averaging identifies coinvariants with invariants and makes invariants an exact functor. The quotient \(q_T\) therefore supplies the asserted sections \(\sigma_i\) with independent images.

Step 2. Line bundles near a compact curve core. Take \(T^\circ\) noncompact, removing one more point if needed. Recall the fixed choice made after (29): for every prime component \(E\) of \(\mathop{\mathrm{Exc}}(g)\) dominating \(T\), its Stein factor restricts to \[ E|_{T^\circ}\longrightarrow \widehat T_E^\circ\longrightarrow T^\circ, \tag{39}\] where the first map is a smooth proper family with connected fibers and the second is a connected finite étale cover. Further finite puncturing preserves these properties. Choose an ambient open \(Y^\circ\subset Y\) in which \(T^\circ\) is closed: remove the discarded points of \(T\), the other singular components, and their intersection points. These choices will be used when we identify the local divisors globally.

Let \(h:g^{-1}T^\circ\to T^\circ\) be the restricted resolution. The curve \(T^\circ\) has the homotopy type of a finite graph, so cohomology of its local systems vanishes in degrees at least two. The Leray spectral sequence for \(h\) consequently makes the edge map \[H^2(g^{-1}T^\circ,\mathbb{Q}) \longrightarrow H^0(T^\circ,\mathcal W_T)\] surjective. Choose lifts of the \(\sigma_i\). The algebraic space \(g^{-1}T^\circ\) has finite triangulation type; after multiplying each lift by a positive integer, it comes from a class \(\alpha_i\in H^2(g^{-1}T^\circ,\mathbb{Z})\). Replace \(\sigma_i\) by the same multiple. Their images are still independent.

To extend the \(\alpha_i\) simultaneously while retaining every loop of \(T^\circ\), choose a compact curve core as follows. In the smooth compactification of \(T^\circ\), remove small open coordinate discs about its punctures, with disjoint closures, and denote the complement by \(K_*\). This is a connected compact semianalytic subset of \(T^\circ\). The inclusion of its interior into \(T^\circ\) induces a surjection on fundamental groups. The core \(K_*\) is also Runge: every component of \(T^\circ\setminus K_*\) runs to a missing puncture and is therefore not relatively compact. The compact set \(g^{-1}K_*\) is semianalytic. Continuity of cohomology on neighborhoods of this compact set extends the restrictions of the finitely many \(\alpha_i\) to integral classes on one open neighborhood \(\widetilde U_0\) of \(g^{-1}K_*\) in \(\widetilde Y\). One may use Čech cohomology for this continuity statement; semianalytic triangulation and local contractibility identify it with singular cohomology here. Properness of \(g\) makes \[U_0=Y\setminus g(\widetilde Y\setminus\widetilde U_0)\] an open neighborhood of \(K_*\) with \(g^{-1}U_0\subset\widetilde U_0\).

The closed curve \(T^\circ\subset Y^\circ\) is Stein. Siu’s Stein-neighborhood theorem, in the form [8] with empty compact set, gives a Stein neighborhood \(\Omega\) of this curve in \(Y^\circ\). The Runge property makes \(K_*\) holomorphically convex in \(T^\circ\), hence also in \(\Omega\) by the paragraph preceding that theorem. The usual holomorphic-polyhedron construction for a convex compact in a Stein space gives a Stein neighborhood \[K_*\subset U\Subset\Omega\cap U_0.\] Take its connected component containing \(K_*\). It is normal, being open in \(Y\), and therefore irreducible. For the local \(\mathbb{Q}\)-factorialization, Fujino’s compact construction [10] supplies an ambient semianalytic Stein compact \(K'\Subset U\), with \(K_*\subset\operatorname{int}_U(K')\), such that \[K'\cap Z\ \text{has finitely many connected components}\] for every analytic subset \(Z\) defined on a neighborhood of \(K'\). This is the compact condition (P4) in Fujino’s condition (P); normality of the source and Steinness of the base supply its other ambient conditions.

The classes just extended to \(g^{-1}U\) are Chern classes of line bundles \(L_1,\ldots,L_{b_T}\) there, by the same rational-singularity, Cartan, and exponential-sequence argument used above. For every \(t\) in the interior of \(K_*\), their fiber classes are the \(\sigma_i(t)\); hence their images under \(q_T\) remain independent. We also need divisor representatives before making any further modification. The sheaf \(g_*L_i\) is coherent and has rank one on the regular open of \(U\). Cartan’s Theorem A supplies a global section nonzero at a point of that open. Its pullback is a nonzero section of \(L_i\), whose zero divisor is a Cartier divisor \(A_i\) on \(g^{-1}U\) representing \(L_i\).

Step 3. Fiber degrees and surface families. We will obtain at least \(b_T\) surface families by proving that the \(b_T\) line bundles constructed below have linearly independent vectors of fiber degrees. Apply Fujino’s small \(\mathbb{Q}\)-factorialization theorem [10] to the ordinary pair \((U,0)\), the identity \(U\to U\), and the full compact \(K'\). The identity is projective, \(U\) is terminal and therefore klt, and \(K_U\) is \(\mathbb{Q}\)-Cartier. The preceding construction supplies condition (P). After restricting the base to a neighborhood of \(K'\), the theorem gives a projective small bimeromorphic morphism \[\ell:Y'\longrightarrow U\] with \(Y'\) normal and \(\mathbb{Q}\)-factorial over \(K'\). We retain the notation \(U\) after this restriction and restrict \(g,L_i,A_i\) accordingly. Compactness and local terminality of \(Y\) give a common \(m_{\mathrm{can}}>0\) for which \(\omega_Y^{[m_{\mathrm{can}}]}\) is invertible; its restriction to \(U\) is \(\omega_U^{[m_{\mathrm{can}}]}\). Since \(\ell\) is small, the natural identification on the common codimension-one open extends by reflexivity to \[\omega_{Y'}^{[m_{\mathrm{can}}]} \simeq\ell^*\omega_U^{[m_{\mathrm{can}}]}.\] In particular, the left side is invertible and the natural relative canonical divisor \(K_{Y'/U}\) is zero. This is canonical crepancy, expressed using compatible local canonical generators as \(K_{Y'}=\ell^*K_U\). A place exceptional over \(Y'\) is exceptional over \(U\), because \(\ell\) is small. Crepancy therefore preserves the terminal discrepancy inequalities, so \(Y'\) is terminal. Moreover \(\ell\) is an isomorphism over \(U_{\mathrm{reg}}\). Indeed, for a relatively ample line bundle \(\mathcal A\) on \(Y'\), the rank-one reflexive sheaf \((\ell_*\mathcal A)^{**}\) is locally invertible on the smooth base. Its pullback agrees with \(\mathcal A\) off codimension two and hence everywhere there, by reflexivity on the normal source. A positive-dimensional fiber over a smooth point would then have degree zero against \(\mathcal A\), contradicting relative ampleness.

Push \(A_i\) forward to a Weil divisor on \(U\), and let \(D_i'\) be its strict transform on \(Y'\). These are actual locally finite analytic divisors defined on all of the current \(Y'\). We apply \(\mathbb{Q}\)-factoriality precisely in the sense of [10]: every prime divisor defined on a neighborhood of the whole set \(\ell^{-1}(K')\) is \(\mathbb{Q}\)-Cartier at each point of that set. Properness makes \(\ell^{-1}(K')\) compact, and local finiteness leaves only finitely many prime components of the \(D_i'\) near it. A finite cover of this full compact and a common multiple therefore give an integer \(m_{\mathrm{div}}>0\) and an open neighborhood of \(\ell^{-1}(K')\) on which every \(m_{\mathrm{div}}D_i'\) is Cartier. By properness, choose an open set \(U_1\) with \(K'\subset U_1\subset U\) whose full inverse image is contained in that neighborhood, and put \[\mathcal N_i=\mathcal O_{Y'}(m_{\mathrm{div}}D_i')|_{\ell^{-1}U_1}.\] This completes the use of \(\mathbb{Q}\)-factoriality on the specified full compact; all later restrictions use these fixed line bundles. In particular, the change-of-compact issue in [10] does not arise. On the regular open, where both \(g\) and \(\ell\) are isomorphisms, \[ \mathcal N_i|_{U_{1,\mathrm{reg}}} \simeq L_i^{\otimes m_{\mathrm{div}}}|_{U_{1,\mathrm{reg}}}. \tag{40}\]

Choose connected curve domains \[B_0\Subset B_1\Subset\operatorname{int}_{T^\circ}(K_*)\] such that \(\pi_1(B_0)\to\pi_1(T^\circ)\) is surjective; nested smaller cores obtained by enlarging the omitted discs have this property. The larger domain lets us control finitely many components near \(\overline B_0\) and make their discrete exceptional sets finite on that compact. Work in the ambient open of \(U_1\) obtained by removing \(T^\circ\setminus B_1\), so that \(B_1\) is closed there. The inverse image of \(B_1\) is proper over \(B_1\). Local finiteness of its analytic components, compactness of \(\ell^{-1}(\overline B_0)\), and properness allow us first to replace \(B_1\) by a neighborhood of \(\overline B_0\) on whose inverse image only finitely many components occur.

We claim that deleting finitely many points from \(B_0\) gives a connected curve domain \(B\) with \(\pi_1(B)\to\pi_1(T^\circ)\) surjective and the following fiber description. All fibers of \(\ell\) over \(B\) have dimension at most one. The finitely many irreducible surface components \(S_1,\ldots,S_s\) of the reduced inverse image of \(B\), counted after this restriction, all dominate \(B\). Their normalizations factor as \[\widehat S_\alpha\xrightarrow{\,h_\alpha\,}C_\alpha \longrightarrow B \qquad (1\le\alpha\le s),\] where \(C_\alpha\to B\) is a connected finite étale cover and \(h_\alpha\) is a smooth proper family of connected curves. A cover that splits on restriction contributes separate components to this list. For each fixed \(t\in B\), the pairs \((\alpha,c)\) with \(c\in C_\alpha\) over \(t\) supply distinct irreducible curves and exhaust the irreducible curves of \(\ell^{-1}(t)\). Each smooth connected normalization fiber maps birationally to the curve it supplies.

We verify this description by spelling out the finite exclusions. Smallness gives \(\dim\mathop{\mathrm{Exc}}(\ell)\le2\), so a general fiber above \(B_1\) has dimension at most one. The locus of larger fibers is a proper analytic subset of \(B_1\), by the fiber-dimension theorem for proper maps. For each surface component of the reduced inverse image that dominates \(B_1\), take its normalization and then its Stein factor over the curve. The normalization is proper and is a normal surface. Its singularities are isolated. Away from the branch values of the finite Stein factor and the critical values of the map from the normalization, it is a smooth proper family of connected curves over a finite unramified cover. The values at which an intersection of two components, or the nonisomorphism locus of a normalization, contains a whole fiber curve also form a discrete analytic subset: these loci have dimension at most one, and their general fibers over the base are zero-dimensional. Finally remove the images of components lying over a point. Each of these excluded sets is discrete on the larger curve domain \(B_1\); there are only finitely many components, so their union meets \(\overline B_0\) in a finite set.

Delete those finitely many points from \(B_0\) and call the result \(B\). It is connected, and \(\pi_1(B)\to\pi_1(T^\circ)\) is still surjective: loops in \(B_0\) may be perturbed away from a finite set. The exclusions give the asserted fiber description and families.

For each \(\alpha,i\), let \[d_{\alpha i} =\deg\bigl(\mathcal N_i|_{h_\alpha^{-1}(c)}\bigr), \qquad c\in C_\alpha,\] where pullback to \(\widehat S_\alpha\) is understood. This integer is independent of \(c\), by constancy of degree in a smooth proper family over the connected curve \(C_\alpha\). Because its fiber maps birationally to the curve it supplies, this is also the degree of \(\mathcal N_i\) on that fiber curve of \(\ell\). We claim that the map \[ \mathbb{Q}^{b_T}\longrightarrow\mathbb{Q}^s,\qquad (c_i)_i\longmapsto \left(\sum_i c_i d_{\alpha i}\right)_\alpha \tag{41}\] is injective.

Suppose its value is zero, and fix \(t\in B\). Every irreducible curve in \(F'_t=\ell^{-1}(t)\) then has degree zero against \(\sum_i c_i c_1(\mathcal N_i)\). The fiber has dimension at most one. Its rational \(H_2\) is spanned by the fundamental classes of its irreducible curves, also when the fiber is nonreduced, and \(H^2(F'_t,\mathbb{Q})\) is the rational dual of \(H_2(F'_t,\mathbb{Q})\). The restricted Chern class is therefore zero. Proper base change identifies it with the germ in \((R^2\ell_*\mathbb{Q})_t\). Restriction through the isomorphism over the regular open gives the natural map \[(R^2\ell_*\mathbb{Q})_t\longrightarrow (R^2j_*\mathbb{Q})_t =(\mathcal V_T)_t,\] so the corresponding punctured class is zero as well. By (40), the restrictions of \(\sum_i c_i c_1(\mathcal N_i)\) and \(m_{\mathrm{div}}\sum_i c_i c_1(L_i)\) agree on their common regular open. Taking their germ in \((R^2j_*\mathbb{Q})_t\) and applying (37) to the latter identifies that zero class with \[m_{\mathrm{div}}\sum_i c_i\,q_T(\sigma_i(t)).\] These vectors are independent; hence every \(c_i=0\). This proves (41), and in particular \[ s\ge b_T. \tag{42}\]

Step 4. Distinct global discrepancy-two divisors. Take an ambient open \(U_B\subset U_1\) whose intersection with \(T^\circ\) is the closed subspace \(B\), obtained as above by removing \(T^\circ\setminus B\). Each \(S_\alpha\) is a closed analytic surface in \(\ell^{-1}U_B\). Since \(Y'\) is terminal, it is smooth in codimension two by [21]. Thus at a general point of \(S_\alpha\) both the ambient space and the surface are smooth. Blow up its coherent ideal and normalize. The unique divisor dominating that dense smooth part defines a local place \(v_\alpha\) of ordinary log discrepancy \(2\) over \(Y'\), the value for a smooth codimension-two blowup. Canonical crepancy of \(\ell\) gives \[a(v_\alpha;U_B,0)=2,\qquad \mathop{\mathrm{cent}}_{U_B}(v_\alpha)=B.\] The \(v_\alpha\) are distinct because their centers on \(Y'\) are the distinct \(S_\alpha\).

It remains to represent each \(v_\alpha\) by a prime divisor on \(g^{-1}U_B\) and show that these local divisors arise from distinct global components of \(\mathop{\mathrm{Exc}}(g)\).

On the fixed global resolution, the relative canonical divisor is intrinsic even if the canonical bundle has no global divisor representative. Using the fixed \(m_{\mathrm{can}}\), divide by \(m_{\mathrm{can}}\) the divisor of the natural meromorphic comparison \(g^*\omega_Y^{[m_{\mathrm{can}}]}\dashrightarrow \omega_{\widetilde Y}^{\otimes m_{\mathrm{can}}}\), and denote the result by \(K_{\widetilde Y/Y}\). It is exceptional; write \[K_{\widetilde Y/Y}=\sum_E\kappa_EE.\] The natural canonical comparisons commute with restriction and composition, so this divisor supplies the relative discrepancy term on \(U_B\) as well. Every \(\kappa_E\) is strictly positive by terminality. The principalization of the ideal of \(\mathop{\mathrm{Sing}}Y\) says that \(g^{-1}(\mathop{\mathrm{Sing}}Y)\) is supported on these exceptional divisors. Put \(\widetilde U_B=g^{-1}U_B\), and consider \(v_\alpha\) on a common proper model with \(\widetilde U_B\). If it were exceptional over this smooth resolution, then \(a(v_\alpha;\widetilde U_B,0)\ge2\). Its center on the resolution maps into \(B\subset\mathop{\mathrm{Sing}}Y\), so it is contained in at least one of the \(E\), with \(\mathop{\mathrm{ord}}_{v_\alpha}(E)>0\). The discrepancy formula would give \[a(v_\alpha;U_B,0) =a(v_\alpha;\widetilde U_B,0) +\sum_E\kappa_E\mathop{\mathrm{ord}}_{v_\alpha}(E)>2,\] a contradiction. Thus \(v_\alpha\) is represented by a prime divisor on \(g^{-1}U_B\). Since \(U_B\cap T=B\), this local divisor is an irreducible component of \(E|_B\) for a global exceptional divisor \(E\) with \(g(E)=T\).

Such a global divisor cannot split into two components over \(B\). Indeed the connected finite cover \(\widehat T_E^\circ\to T^\circ\) in (39) remains connected over \(B\): the action of \(\pi_1(T^\circ)\) on a fiber is transitive, and \(\pi_1(B)\) has the same image. Over this connected cover the restriction of \(E\) is a smooth proper family with connected fibers, so its total space is connected and smooth, hence irreducible. Therefore the distinct local divisors representing the \(v_\alpha\) come from distinct global components \(E\) of \(\mathop{\mathrm{Exc}}(g)\). Their global centers are \(T\), and their ordinary log discrepancies are \(2\), since this equality holds on the open over \(B\). Together with (42), this proves the proposition. ◻

Proof of Theorem 17. Every global place of ordinary log discrepancy \(2\) whose center is a singular curve is represented by a component of \(\mathop{\mathrm{Exc}}(g)\). Indeed the same discrepancy computation in the last part of Proposition 23 would otherwise make its discrepancy strictly greater than \(2\). There are only finitely many such components on the compact resolution, so \(\tau_Y(Q)\) is finite. Proposition 23, summed over the singular curves contained in \(Q\), gives \[\sum_{T\subset\mathop{\mathrm{Sing}}Y\cap Q} b_T\le\tau_Y(Q).\] Combining this with (35) yields \[\beta(Q)\le \sum_j c_2(Q_j^\nu)+\tau_Y(Q),\] as required. ◻

Proof. Take a projective resolution \(\pi:W\to Y\) with \(W\) smooth and carrying \(\mathbf{M}\), with divisorial exceptional locus, and resolving all labels. Further blowups of intersections in the simple normal crossing divisor make the strict transforms of the labels smooth and pairwise disjoint, while keeping the full divisor simple normal crossing. Write \[K_W+\Theta_W+M_W=\pi^*(K_Y+\Theta+M_Y).\] Every \(\pi\)-exceptional component of \(\Theta_W\) has coefficient \(1-a(E;Y,\Theta+\mathbf{M})<0\), by generalized terminality. The only possible positive components of \(\Theta_W\) are therefore the strict transforms of the positive labels. On higher models the nef datum pulls back from \(W\), so the discrepancy calculation is the ordinary one for the signed boundary \(\Theta_W\).

Consider a place \(E\) that is also exceptional over \(W\), and let its center on \(W\) have codimension \(c\geq2\). At a general smooth point of this center choose normal parameters \(x_1,\ldots,x_c\). The Jacobian formula on a smooth model carrying \(E\) gives \[a(E;W,0)\geq\sum_{i=1}^c\mathop{\mathrm{ord}}_E(x_i), \qquad \mathop{\mathrm{ord}}_E(x_i)\geq1.\] Indeed, in the pullback of a local volume form at most one differential factor can lose one order along \(E\); adding one to the Jacobian order gives the displayed bound. If the center lies in no positive component of \(\Theta_W\), the signed boundary can only raise this discrepancy, which is at least two. Otherwise it lies in a unique positive smooth label of coefficient \(b<1\). Taking \(x_1\) as its local equation and discarding the negative boundary terms gives \[ a(E;Y,\Theta+\mathbf{M}) \geq (1-b)\mathop{\mathrm{ord}}_E(x_1)+\sum_{i=2}^c\mathop{\mathrm{ord}}_E(x_i). \tag{46}\] A discrepancy less than two thus forces \(c=2\) and containment in a positive label. A label of coefficient zero cannot give such a place.

If this codimension-two center is contained in \(\mathop{\mathrm{Exc}}(\pi)\), it is a component of the intersection of its label with one of the finitely many exceptional divisors. There are only finitely many such centers. If it meets the isomorphism open, its general point maps isomorphically to a surface downstairs, and the center is the strict transform of that surface. Thus a center mapping to codimension at least three belongs to the first finite list, and for any fixed surface downstairs there are only finitely many possible centers on \(W\).

There are only finitely many places of discrepancy less than two whose center on \(W\) equals a fixed one of these codimension-two centers. To see this, discard the negative components of \(\Theta_W\); this lowers discrepancies, so it suffices to work with the single smooth label of coefficient \(b\). Blow up the center generically. The new exceptional coefficient is \(-(1-b)\). A further place with discrepancy less than two has its center in that exceptional divisor; the estimate (46) also forces a codimension-two center in the strict label. It must therefore follow their intersection above the general point of the original center. At stage \(j\) the most recent exceptional coefficient is \(-j(1-b)\), and the older exceptional divisors do not meet this generic intersection. A place above it that is still exceptional has discrepancy at least \[(1-b)+1+j(1-b)=1+(j+1)(1-b).\] This reaches two after finitely many stages. The same argument shows that, where no other boundary or nef data enter, the places below two are exactly the echoes in (44).

This generic computation identifies global places. At each global blowup there is a unique irreducible component of its exceptional divisor that dominates the dense smooth part of the center. The global analytic intersection of that prime with the strict label then has a unique irreducible component dominating the same part. These global components have coherent ideals, which define the next global blowups. Conversely, one can follow any given global place on common higher models through these blowups; when its center becomes a divisor on a normal model, that divisor represents the place. Thus the computation does not count different local pieces of one place separately.

Finally include the finitely many divisors on \(W\) that are exceptional over \(Y\). The complement of the isomorphism open, and the loci where a label fails to be smooth and alone or the nef datum fails to descend, are contained in a proper analytic subset of \(Y\) of codimension at least two. Such a subset has only finitely many surface components on the compact fourfold. Take these components for \(\mathcal V_Y\), enlarging the set by the finitely many surface images already encountered if necessary. Outside them the preceding one-label calculation applies, or there is no positive label and no place of positive weight. This proves all assertions. ◻

Proof. Write \(Y=Y_i\) and let \(Q=\bigcup_{h\text{ varying}}S_h\) with its reduced structure. If \(Q\) is empty, (50) makes every surface bracket nonnegative, and the other terms of (49) are nonnegative. Suppose \(Q\ne\varnothing\). At this junction the map \(h_i\) is projective bimeromorphic from an ordinarily terminal compact Kähler fourfold, and \(\dim h_i(Q)\leq1\) by Proposition 16. Thus Theorem 17 applies. For a surface \(V\subset Q\), its branch count in that theorem is \(r_V(Q)=r_V^{\mathrm{var}}\geq1\); the last inequality follows from the finite surjective normalizations. Surfaces outside \(Q\) have \(r_V^{\mathrm{var}}=0\). The total possible deficit in (50) is consequently \(\beta(Q)\).

Every varying label has \(d_h\geq1\). The places counted by \(\tau_Y(Q)\) in Theorem 17 also each supply a weight of at least one in the first sum of (49). In fact such a place \(E\) has ordinary log discrepancy two and center a singular curve in \(Q\). It is exceptional over \(Y\). Its center is contained in a varying label of positive coefficient, and that effective label is \(\mathbb{Q}\)-Cartier on the strong model. Its order at \(E\) is therefore strictly positive. On a common carrier the discrepancy formula reads \[a(E;Y,\Theta+\mathbf{M}) =2-\mathop{\mathrm{ord}}_E(\Theta)-\operatorname{coeff}_E(J)<2, \qquad J\geq0.\] The count \(\tau_Y(Q)\) uses distinct global places. They occur as distinct terms of the codimension-at-least-three sum, and none is a place in a surface bracket. Theorem 17 now gives \[\beta(Q) \leq\sum_{h\text{ varying}}c_2(S_h^\nu)+\tau_Y(Q) \leq\sum_{h\in\mathcal H}d_hc_2(S_h^\nu) +\sum_{\mathop{\mathrm{codim}}_Y\mathop{\mathrm{cent}}_Y(E)\geq3}w(E).\] Summing (50) and substituting this inequality in (49) proves the assertion. The branch inequality and this lower bound have been used only at the junctions. ◻

Proof. Let \(T_h\) be the normalization of the common image of the two labels in \(Z\). Restricting the projective maps, composing with finite normalization, and factoring through \(T_h\) gives projective bimeromorphic maps of normal compact threefolds \[S_h^\nu\longrightarrow T_h\longleftarrow (S_h^+)^\nu.\] They identify over the common isomorphism open. By Lemma 11, the common exceptional image \(A=f(L)=f^+(L^+)\) has dimension at most one. A normalization divisor lying over \(L\) has a surface as its image in \(S_h\), because normalization is finite, and its image in \(T_h\) lies over \(A\). It is therefore contracted by the displayed map. Conversely, the map is an isomorphism off the preimage of \(A\), so every contracted prime divisor lies over \(L\). These are exactly the primes counted in the first sum in (53). The same reasoning holds on the positive side. The exceptional loci have finitely many surface components, which also proves finiteness.

We use the following independence fact. For a projective bimeromorphic map \(g:S\to T\) of normal compact threefolds, the classes in \(H_4(S,\mathbb{Q})\) of the prime divisors contracted by \(g\) are linearly independent. Take a resolution \(r:\widetilde S\to S\) that is projective, has smooth source, and is an isomorphism over \(U=S_{\mathrm{reg}}\). Put \(R=r^{-1}(S\setminus U)\). Normality gives \(\dim(S\setminus U)\leq1\), and \(\dim R\leq2\). If a rational combination of the contracted divisor classes vanished on \(S\), the corresponding combination of strict transforms would restrict to zero in \(H_4^{\mathrm{BM}}(U,\mathbb{Q})\). The localization sequence \[H_4(R,\mathbb{Q})\longrightarrow H_4(\widetilde S,\mathbb{Q}) \longrightarrow H_4^{\mathrm{BM}}(U,\mathbb{Q})\] then shows that its class is a rational combination of the classes of the surface components of \(R\); those classes span \(H_4(R,\mathbb{Q})\). Subtract that combination. We obtain a homologically trivial rational Cartier divisor on \(\widetilde S\), all of whose components are exceptional over \(T\). For the components of \(R\), this follows since their images in \(S\), hence in \(T\), have dimension at most one. On the smooth compact resolution, homological triviality makes this divisor numerically trivial on every curve by the ordinary class pairing. Analytic negativity in both signs, in the form of [21], makes the divisor zero. The strict transforms of the original divisors are distinct from the resolution-exceptional ones, so all original coefficients vanish. This proves independence without any factoriality assumption on \(S\).

For either label normalization, restrict its span of compact analytic surface classes to the common open over \(T_h\setminus\nu^{-1}(A)\), where \(\nu:T_h\to f(S_h)\) is normalization. The complement of this open in either normalization has dimension at most two. Localization shows that the kernel of this restriction is spanned by its surface components, precisely the contracted prime divisors just counted. Their independence makes the kernel dimension \(e_h\) on one side and \(e_h^+\) on the other. The two images agree: any compact analytic prime surface not removed has a strict transform on the other normalization through their proper graph over \(T_h\), and the two classes restrict to the same class on the common open. The same holds in the reverse direction. Subtracting the dimensions of these two cycle spans proves the last equality in (53). ◻

Proof. For a coefficient change, the removed boundary \(\Theta-\Theta'\) is effective and \(\mathbb{Q}\)-Cartier on the strong model. For every place, \[a(E;Y,\Theta'+\mathbf{M})-a(E;Y,\Theta+\mathbf{M}) =\mathop{\mathrm{ord}}_E(\Theta-\Theta')\geq0.\] Thus \(w(E)\geq w'(E)\). The model, centers, branch counts, and normalization ranks are unchanged, and the defaults \(d_h\) are constant by the choice of the tail. At a fixed surface the two raw sums are finite, and therefore \[q_Y(V)-q'_Y(V)=\sum_{\mathop{\mathrm{cent}}_Y(E)=V}\bigl(w(E)-w'(E)\bigr).\] Let \(\Sigma\) be the finite union of the supports of the two functions \(q_Y\) and \(q'_Y\) on surfaces. Outside \(\Sigma\) the displayed sum is zero. Every summand is nonnegative, so no exceptional place with such a surface center can lose weight. At each surface of \(\Sigma\) only finitely many places have positive old or new weight, and there are only finitely many positive weights at centers of codimension at least three. This proves that \(\mathcal C\) is finite. Summing the displayed identity over \(\Sigma\), adding the finite comparison in codimension at least three, and canceling the unchanged rank term gives (54).

For a small step, Lemma 11 says that exceptionality of a global place is the same on the two sides, that the two center containment conditions in the statement are equivalent, and that \(w(E)\geq w^+(E)\). If a center is not contained in the exceptional locus, its general point corresponds on the common isomorphism open and its discrepancy is unchanged. The set \(\mathcal P\) is finite: each exceptional locus has dimension at most two and has finitely many surface components; a surface center contained in it is one of these components. Proposition 24 gives finiteness above each such surface and finiteness of all positive-weight places with lower-dimensional centers, on both sides.

Surface centers not contained in \(L\) correspond by proper strict transform to those not contained in \(L^+\). At their general points the places, discrepancies, and normalization primes over the surfaces correspond, so the two local brackets agree. The analogous correspondence holds for the raw terms at lower-dimensional centers outside the exceptional loci. In the exceptional loci, every contributing place is counted exactly once in the raw part on each side of (49). This remains true if its center changes from a surface to a curve or point, or conversely. The finite comparison of those raw parts is therefore \(\sum_{E\in\mathcal P}(w(E)-w^+(E))\). The subtracted defaults on the exceptional surfaces have difference \(-\sum_h d_h(e_h-e_h^+)\). Hence \[\begin{align*} &\mathfrak D(Y,\Theta+\mathbf{M})-\mathfrak D(Y^+,\Theta^++\mathbf{M})\\ &\quad=\sum_{E\in\mathcal P}\bigl(w(E)-w^+(E)\bigr) -\sum_h d_h(e_h-e_h^+) +\sum_h d_h\bigl(c_2(S_h^\nu)-c_2((S_h^+)^\nu)\bigr)\\ &\quad=\sum_{E\in\mathcal P}\bigl(w(E)-w^+(E)\bigr), \end{align*}\] where the last equality is Lemma 26. This is (55). Every summand is nonnegative. A strictly decreasing exceptional weight must have its center in the exceptional loci, since weights agree off them, and is in \(\mathcal P\). A positive weight that becomes zero is included by the definition of \(\mathcal P\). The same positivity is immediate in (54); this proves strictness in both cases. ◻