Proof. The strict boundary coefficients belong to \([0,1]\), so positive coefficients of \(H\) occur only on exceptional primes. Coefficient by coefficient, \[H-R=-\left\lfloor G\right\rfloor, \qquad (H-R)-(K_U+\{G\})=-p^*(K_T+B_T).\] The fractional boundary \(\{G\}=G-\left\lfloor G\right\rfloor\) is effective SNC with coefficients below one. The displayed adjoint difference is relatively nef and big for the bimeromorphic \(p\): it is relatively trivial, and the generic relative dimension is zero. Analytic relative Kawamata–Viehweg vanishing (Das et al. 2024, Theorem 2.41) gives \(R^ip_*\mathcal O_U(H-R)=0\) for \(i>0\). Exceptional Hartogs extension gives \(p_*\mathcal O_U(H)=\mathcal O_T\). The sequence for the Cartier divisor \(R\) therefore yields a surjection \[\mathcal O_T\longrightarrow p_*\mathcal O_R(H).\] It factors through \(p_*\mathcal O_R\). Multiplication by the canonical section of \(H\) embeds \(\mathcal O_R\) into \(\mathcal O_R(H)\), because \(R\) is reduced SNC and shares no component with \(H\). The surjection consequently makes that embedding an isomorphism after pushforward and makes \(\mathcal O_T\to p_*\mathcal O_R\) surjective. Its kernel is exactly the reduced ideal of the proper image \(p(R)\): a holomorphic function vanishes after restriction to \(R\) precisely when it vanishes on that image. This proves (4); Stein factorization gives connected fibers.

For the last assertion, take a klt resolution, so \(R=0\). The same vanishing and Hartogs calculation identify \(Rp_*\mathcal O_U(H)\simeq\mathcal O_T\). The factorization \[\mathcal O_T\longrightarrow Rp_*\mathcal O_U\longrightarrow Rp_*\mathcal O_U(H)\simeq\mathcal O_T\] is the identity, as can be checked on degree zero. It splits the first map. Apply proper coherent duality. Canonical higher direct images of the resolution vanish: the canonical torsion-freeness theorem (Fujino 2023, Theorem 2.9 and Proposition 2.11) applies locally on the base, and these higher images vanish generically for a bimeromorphic map. The required local Kählerness follows by restricting to a small Stein base neighborhood and combining a relative ample metric with a pulled-back strictly plurisubharmonic potential. Thus the dual split surjection is the trace \[p_*\omega_U[\dim U]\longrightarrow\omega_T^\bullet,\] where \(\omega_T^\bullet\) is the dualizing complex. It follows that this complex is concentrated in degree \(-\dim U\). On that cohomology sheaf the trace is also injective: its source is torsion-free and the map is generically an isomorphism. Hence it is an isomorphism. Biduality gives \(Rp_*\mathcal O_U=\mathcal O_T\). This is rational singularity, and the concentration of the dualizing complex is the Cohen–Macaulay property. ◻

Proof. The assertion is the identity map if \(C\) is empty. Otherwise give \(C\) its reduced structure and write \(C_1,\ldots,C_s\) for its irreducible components.

We will construct a positive line on a resolution and use it to give the conormal of the full analytic inverse image a positive presentation. Grauert’s contraction criterion will then apply to that conormal.

Proof. Use a projective resolution \(\pi:\widehat X\to X\) with smooth compact Kähler source as above. The class \(\beta=\pi^*\alpha\) is nef with positive cube by (6). Collins–Tosatti makes \(\operatorname{Null}(\beta)\) a proper analytic set.

Each irreducible component \(Z\) of this analytic set is itself null. It has positive dimension, since each of its points lies on a positive-dimensional null subspace. Suppose instead that \(\beta^{\dim Z}\cdot Z>0\). Resolve \(Z\) projectively. The restricted pulled class on the smooth compact Kähler resolution is nef with positive top intersection, so Collins–Tosatti gives it a proper null set. Choose a point of \(Z\) outside the image of that set, the target nonisomorphism locus of the resolution, the singular locus of \(Z\), and the other components of \(\operatorname{Null}(\beta)\). Such a point exists because these are proper analytic subsets of \(Z\). By the definition of \(\operatorname{Null}(\beta)\), a null irreducible subspace passes through it. That subspace lies in \(Z\), since the point is on no other component. Its strict transform is null for the restricted class, contradicting the choice of the point.

There are finitely many components of the compact analytic null set. By (7), each of its surface components is exceptional, since a nonexceptional one would map to a null surface on \(X\). Thus their images, as well as those of its curve components, have dimension at most one. For any downstairs null curve \(C_0\), choose an irreducible component \(V\) of \(\pi^{-1}C_0\) dominating \(C_0\). Proper surjectivity supplies such a component. If \(\dim V=1\), cycle projection gives \(\beta\cdot V=\deg(V/C_0)\,\alpha\cdot C_0=0\); if \(\dim V>1\), it gives zero by dimension drop. Thus \(V\) lies in an upstairs null component whose image contains \(C_0\). Every downstairs null curve is therefore contained in the finite union of images. An irreducible curve in a finite union of curves and points is one of the curve components. There are therefore only finitely many downstairs null curves, as asserted. ◻

Proof. Canonical rings and their successive kernels.

All direct images defining \(\mathscr B_k\) first use the continuous map \(|\phi|:|S|\to|W|\). The equality \(\mathscr B_1=\mathcal O_W\) is the canonical equality for the holomorphic map \(\phi\). Apply derived direct image for abelian sheaves to the conormal sequence. Its derived image on an \(\mathcal O_S\)-module is the underlying sheaf of the analytic higher direct image. To see this formally, an injective \(\mathcal O_S\)-module is flasque: for open sets \(V\subset U\), the extension-by-zero modules satisfy \(j_{V!}\mathcal O_V\hookrightarrow j_{U!}\mathcal O_U\). Adjunction identifies \(\operatorname{Hom}_{\mathcal O_S}(j_{U!}\mathcal O_U,J)\) with \(\Gamma(U,J)\), so injectivity of \(J\) makes the restriction of sections surjective. Flasque abelian sheaves are acyclic for continuous direct image. An injective module resolution therefore computes both the analytic module higher images and their underlying abelian-sheaf higher images. Thus (19) gives exact sequences \[ 0\longrightarrow \mathscr F_k:=\phi_*\mathscr E_k \longrightarrow\mathscr B_{k+1} \longrightarrow\mathscr B_k\longrightarrow0\qquad(k\geq1). \tag{29}\] The displayed quotients are ring maps; exactness here concerns their underlying abelian sheaves. Proper direct-image coherence for the already holomorphic \(\phi\) (Grothendieck 1960-1961c, Theorem 1.1, p. 15-01) makes \(\mathscr F_k\) a coherent \(\mathcal O_W\)-module.

In \(\mathcal O_{S_{k+1}}\), the ideal \(\mathscr E_k\) is square-zero and is annihilated by \(\mathscr I_1\): use \(\mathscr I_k^2\subset\mathscr I_{2k}\subset\mathscr I_{k+1}\) and \(\mathscr I_1\mathscr I_k\subset\mathscr I_{k+1}\). Consequently the action of \(\mathscr B_{k+1}\) on the ideal \(\mathscr F_k\) factors through \(\mathscr B_1=\mathcal O_W\). This is the \(\mathcal O_W\)-module structure on the square-zero ideal used in the following construction.

The composites of (29) are epimorphisms \(\mathscr B_k\to\mathcal O_W\). Their kernels are \[\mathscr N_k=|\phi|_*(\mathscr I_1/\mathscr I_k), \qquad \mathscr N_k^k=0,\] by left exactness and \(\mathscr I_1^k\subset\mathscr I_k\). Since \(W\) is reduced, \(\mathscr N_k\) is exactly the nilradical. Each stalk of \(\mathscr B_k\) is local: an element lifting a unit in \(\mathcal O_{W,w}\) has an inverse by a finite geometric series in the nilpotent kernel, whereas an element mapping to the maximal ideal is not a unit. Its residue field is \(\mathbb C\). Thus \(W_k\) is already a locally ringed space, and it remains to construct local analytic presentations for it.

A local analytic presentation from source functions.

Fix \(k\) and \(w\in W\). Choose a local closed analytic embedding of a neighborhood of \(w\) into an open polydisc. Lift the finitely many germs of its coordinate restrictions through \(\mathscr B_k\to\mathcal O_W\), represent the lifts on a common neighborhood, and shrink the polydisc. This gives a closed embedding \(i:U\hookrightarrow D\subset\mathbb C^d\) and sections \(b_1,\ldots,b_d\in\mathscr B_k(U)\) reducing to the coordinate restrictions. Put \(Z=S_{k,U}\), the open subspace of \(S_k\) with underlying set \(\phi^{-1}(U)\). By definition, \(\mathscr B_k(U)=H^0(Z,\mathcal O_Z)\).

For a possibly nonreduced complex space, a tuple of holomorphic functions defines a holomorphic map to affine space, functorially on structure sheaves (Grothendieck 1960-1961b, Theorem 1.1, p. 10-02). Apply this theorem to the \(b_j\). Their point values lie in \(D\), so the map factors as \[g:Z\longrightarrow D.\] The same tuple theorem identifies its actual analytic restriction to the reduction with \(i\phi\), because the reduced coordinate functions are those of \(i\phi\). In particular \(|g|=i|\phi|\). The restriction \(\phi^{-1}(U)\to U\) is proper and \(i\) is closed, so \(g\) is proper; nilpotents do not change properness of the underlying continuous map.

The holomorphic map \(g\) makes \(\mathscr A=g_*\mathcal O_Z\) a genuine \(\mathcal O_D\)-algebra. Its underlying module is coherent by the proper direct-image theorem (Grothendieck 1960-1961c, Theorem 1.1). Directly from the underlying maps, as sheaves of \(\mathbb C\)-algebras, \[\mathscr A=i_*(\mathscr B_k|_U).\] Reduction gives an epimorphism \(\mathscr A\to i_*\mathcal O_U\). It is \(\mathcal O_D\)-linear because the actual analytic restriction \(g|_S=i\phi\) identifies the reduced action of every ambient holomorphic function with its restriction to \(U\). The target is the coherent quotient of \(\mathcal O_D\) by the ideal of \(U\). Hence its kernel \(\mathscr N\) is a coherent \(\mathcal O_D\)-module. It is also a nilpotent ideal. Shrink \(D\) once more and choose sections \(n_1,\ldots,n_r\) generating \(\mathscr N\) as a module. They are actual sections of \(\mathcal O_Z\) whose reductions vanish. Since \(\mathcal O_D\to i_*\mathcal O_U\) is an epimorphism, subtraction of an ambient lift of a reduction gives the stalkwise module equality \[ \mathscr A=\mathcal O_D\cdot1+ \sum_{\nu=1}^r\mathcal O_D\cdot n_\nu. \tag{30}\] No splitting of the reduction map is used here.

Apply the tuple theorem again to the \(b_j\) and \(n_\nu\). It gives \[H=(g,n_1,\ldots,n_r):Z\longrightarrow P=D\times\mathbb C^r.\] If \(j:U\hookrightarrow P\) is \(u\mapsto(i(u),0)\), then \(|H|=j|\phi|\), since the \(n_\nu\) are nilpotent. Thus \(H\) is proper. The sheaf \(\mathscr A'=H_*\mathcal O_Z\) is a coherent \(\mathcal O_P\)-module and the unit is an algebra map \[\theta:\mathcal O_P\longrightarrow\mathscr A'.\] As sheaves of rings \(\mathscr A'=j_*(\mathscr B_k|_U)\), so its stalk at \(j(u)\) is \(\mathscr B_{k,u}=\mathscr A_{i(u)}\), and its stalk outside \(j(U)\) is zero. The stalk identification uses the cofinality of restrictions of ambient neighborhoods among neighborhoods in the closed subspace \(U\).

The map \(\theta\) is surjective on every stalk. At \(j(u)\), any element of the target has, by (30), the form \(a_0+\sum a_\nu n_\nu\) with \(a_\nu\in\mathcal O_{D,i(u)}\). The projection of \(H\) to \(D\) is \(g\), and the extra coordinate \(t_\nu\) pulls back to \(n_\nu\). The element is therefore the image of the convergent germ \[a_0(z)+\sum_{\nu=1}^r a_\nu(z)t_\nu\in\mathcal O_{P,j(u)}.\] At all other stalks the target is zero. The kernel \(\mathscr Q=\ker\theta\) is a coherent analytic ideal (Grothendieck 1960-1961a, 9-07–9-08). It defines a closed complex subspace \(Y=V(\mathscr Q)\subset P\) by (Grothendieck 1960-1961a, Definition 2.2 and adjacent construction, p. 9-10). Its support is exactly \(j(U)\), since the target stalk there surjects onto the nonzero ring \(\mathcal O_{U,u}\). Under the homeomorphism with \(U\), its structure sheaf is \(\mathscr B_k|_U\), and its reduction is \(U\).

Every germ of \(\mathscr Q\) pulls back to zero by the definition of \(\theta\). The closed-subspace factorization criterion (Grothendieck 1960-1961a, 9-04–9-05) therefore factors \(H\) holomorphically through \(Y\). Its sheaf map, under the identified structure sheaf, is the canonical evaluation \[|\phi|^{-1}(\mathscr B_k|_U)\longrightarrow\mathcal O_Z.\] Indeed every germ of \(\mathscr B_k|_U\) lifts through the surjective \(\theta\) to an ambient germ, whose pullback is precisely its value as a section on the inverse image. This proves the assertion for every germ, not only the chosen coordinates.

Gluing and closed transition maps.

The ringed space \((|W|,\mathscr B_k)\) was fixed before any of the local choices. The presentations just constructed make each of its restrictions an analytic space by the local definition and adjacent construction in (Grothendieck 1960-1961a, Definitions 2.1–2.2 and adjacent prose). On overlaps the identifications are the identity on the same sheaf of \(\mathbb C\)-algebras, and hence are analytic isomorphisms satisfying the cocycle condition (Grothendieck 1960-1961a, Definition 2.4 and Proposition 2.5). This proves that \(W_k\) is a complex space. It retains the Hausdorff topology of \(W\), and its reduction is canonically \(W\). Write \(\varepsilon_k:|\phi|^{-1}\mathscr B_k\to\mathcal O_{S_k}\) for the global sheaf evaluation. It is locally the holomorphic factorization above, so it defines \(\phi_k\). Its reduction is \(\phi\), and its underlying map is \(|\phi|\); thus it is proper and surjective. The direct-image equality is the definition of its target structure sheaf, with the canonical unit as the identification.

For the closed transition, use the construction with \(H\) at level \(k+1\) and restrict its coordinate functions to the closed subspace \(S_{k,U}\). They give another proper holomorphic map to the same \(P\), with underlying map \(j|\phi|\). Its unit is the composite \[\mathcal O_P\twoheadrightarrow j_*(\mathscr B_{k+1}|_U) \twoheadrightarrow j_*(\mathscr B_k|_U).\] The second quotient stays surjective under the closed embedding \(j\), as seen on its stalks. The last sheaf with this ambient action is coherent by proper direct image. The two units therefore have nested coherent ideal kernels, so their analytic subspaces give the required closed inclusion. These local inclusions glue because their ring maps are the fixed quotients. Naturality of the evaluation maps gives \[\varepsilon_k\circ|\phi|^{-1}(\mathscr B_{k+1}\to\mathscr B_k) =(\mathcal O_{S_{k+1}}\to\mathcal O_{S_k})\circ\varepsilon_{k+1},\] which proves commutativity with the source inclusions. No retraction \(W_k\to W\), Cartesian square, or realization of the infinite tower as one formal completion is asserted or needed. ◻

Proof. Choose a positive global index \(\ell\) such that \(\ell S\) is Cartier and \(\mathcal O_S(-\ell S)\) is \(\phi\)-ample. Put \[A=S_\ell=\ell S,\qquad A'=W_\ell, \qquad f=\phi_\ell:A\longrightarrow A'.\] Here \(A\) is the full effective Cartier divisor, including its nilpotents; its local equation on normal \(X\) is a nonzerodivisor. Lemma 8 makes \(f\) a proper surjective holomorphic map of complex spaces and gives the natural equality \(f_*\mathcal O_A=\mathcal O_{A'}\).

Cartier multiplication gives \(\mathscr I_\ell\mathscr I_j=\mathscr I_{j+\ell}\). Hence, for every integer \(n>0\), on the full space \(A\) one has \[ \mathcal O_A(-nA) =\mathscr I_{n\ell}\otimes\mathcal O_A =\mathscr I_{n\ell}/\mathscr I_{(n+1)\ell}. \tag{32}\] Its finite filtration by \(\mathscr I_j/\mathscr I_{(n+1)\ell}\), for \(n\ell\leq j\leq(n+1)\ell\), has successive quotients \(\mathscr E_j\) for \(n\ell\leq j<(n+1)\ell\). All indices are at least one. These filtration terms are \(\mathcal O_A\)-modules, because \(\mathscr I_\ell\mathscr I_j\subset\mathscr I_{(n+1)\ell}\).

Let \(a:S\hookrightarrow A\) and \(i:W\hookrightarrow A'\) be the closed reduction embeddings. The compatibility in Lemma 8 gives \(fa=i\phi\). Pushforward by a closed embedding is exact, so the composition identity for derived direct images of abelian sheaves gives \[R^q f_*(a_*\mathscr E_j)=i_*R^q\phi_*\mathscr E_j=0\quad(q>0).\] The long exact sequences of the finite filtration in (32) therefore give \[ R^q f_*\mathcal O_A(-nA)=0\qquad(q>0,\ n>0). \tag{33}\] This uses no vanishing for \(\mathscr E_0\) and keeps the full Cartier thickening throughout.

The actual line \(\mathcal O_A(-A)\) restricts to \(\mathcal O_S(-\ell S)\). The underlying map of \(f\) is \(\phi\), so the reductions of their fibers, as closed subspaces of \(X\), coincide: a reduced closed analytic subspace is determined by its support. Thus \(\mathcal O_A(-A)\) is ample on the reduction of every \(f\)-fiber. For a line on a compact nonreduced complex space, ampleness on the reduction implies ampleness on the whole space. Indeed the positive metric from (Fujino 2026, Corollary 1.12) has local strictly plurisubharmonic weights in common ambient embeddings, by its Lemma 2.4. The same weights define a metric on the original line, since the pointwise absolute values of transition units are unchanged by nilpotents. The converse positive-metric criterion gives ampleness on the full space. The proper fiberwise criterion (Fujino 2026, Definition 3.1 and Remark 3.2) now makes \(\mathcal O_A(-A)\) \(f\)-ample.

We apply the analytic blowdown criterion stated in (Das and C. Hacon 2024, Theorem 4.2, p. 14), attributed there to (Fujiki 1975, Theorem 2, p. 495). For a reduced complex space, an effective Cartier divisor \(A\) with its full possibly nonreduced structure, and a proper surjective holomorphic map \(f:A\to A'\), this criterion assumes \(f\)-ampleness of \(\mathcal O_A(-A)\) and \[R^1f_*\mathcal O_A(-nA)=0\qquad\hbox{for every }n>0.\] It supplies a blowing down: a complex target \(Z\) containing \(A'\) as an embedded subspace, a proper surjective holomorphic map \(F:X\to Z\) whose restriction to \(A\) is \(f\), and an analytic isomorphism of the complements. In its universal form it also supplies the natural equality of sheaves of rings \[ F_*\mathscr S_{X,A,f}=\mathcal O_Z. \tag{34}\] Here \(\mathscr S_{X,A,f}\) consists along \(A\) of the germs in \(\mathcal O_X\) whose restrictions lie locally in \(\operatorname{im}(f^{-1}\mathcal O_{A'}\to\mathcal O_A)\), and is \(\mathcal O_X\) away from \(A\). This is a subsheaf of rings; no module coherence of this particular sheaf is used. The hypotheses of the stated criterion follow from (33) and the preceding ampleness argument. No flatness or reducedness of \(A\) or \(A'\) is required.

The embedded \(A'\) is closed because it is compact. Its reduction embeds the original \(W\) as a closed subspace of \(Z\). The compatible reductions give \(F|_S=\phi\) followed by this embedding. Removing a subspace depends only on its support, so the complement isomorphism is \(X\setminus S\simeq Z\setminus W\). It follows that \(F^{-1}(|W|)=|S|\), and the prescribed restriction to \(A\) gives the fiber-set equalities in (31). Outside \(W\) the fibers are single reduced points. These assertions concern underlying fiber sets over \(W\); they require no Cartesian square of the closed thickenings. The fibers are connected because the floor fibers are connected. The target is compact as the image of compact \(X\).

We next prove normality; it is not an extra conclusion assumed from the blowdown criterion. For every open \(V\subset Z\), a section \(s\in\mathcal O_X(F^{-1}V)\) restricts to a section on the open space \[A\cap F^{-1}V=f^{-1}(V\cap A').\] The equality \(f_*\mathcal O_A=\mathcal O_{A'}\) makes this restriction the pullback of a unique section on \(V\cap A'\). Thus every germ of \(s\) along \(A\) satisfies the defining condition of \(\mathscr S_{X,A,f}\). The reverse inclusion follows from \(\mathscr S_{X,A,f}\subset\mathcal O_X\). Consequently \[F_*\mathscr S_{X,A,f}=F_*\mathcal O_X, \qquad F_*\mathcal O_X=\mathcal O_Z\] as sheaves of rings, the second equality by (34). This reasoning uses equality of the displayed open subspaces, not of any closed fiber products.

The equality first makes \(Z\) reduced. The complement is dense: the inverse image of a nonempty open of \(Z\) is a nonempty open of \(X\), and therefore meets \(X\setminus S\). The complement isomorphism thus makes \(F\) bimeromorphic. Let \(\nu:Z^\nu\to Z\) be the finite analytic normalization. The finite normalization exists by (Houzel 1960--1961, Part B, Section 4, Corollaries 2–3). The dominant bimeromorphic map from normal \(X\) lifts to \(Z^\nu\). Pullback along this lift gives inclusions, injective on the dense common complement, \[\mathcal O_Z\subset\nu_*\mathcal O_{Z^\nu}\subset F_*\mathcal O_X=\mathcal O_Z.\] The normalization is consequently an isomorphism and \(Z\) is normal. The natural direct-image equality also gives connected fibers by analytic Stein factorization.

Finally consider the actual global line \(\mathcal O_X(-\ell S)\). Its restriction is ample on every reduced \(F\)-fiber: this is clear off \(W\), and over \(W\) the reduction is the same embedded reduced space as that of the corresponding \(\phi\)-fiber. The nonreduced upgrade and the proper fiberwise criterion used above therefore make this one line \(F\)-ample. The same proof works for any actual ambient rational line with \(\phi\)-ample restriction after clearing its global index. In particular it works for the negative full adjoint in Proposition 7. Thus \(F\) is projective. A projective resolution with smooth source of the compact Kähler \(X\) is again compact Kähler, and its composite with \(F\) is bimeromorphic. Hence \(Z\) lies in Fujiki’s class \(\mathcal C\). ◻

Proof. We make the potential in the class equality explicit. Use the normal-space potential description of Boucksom–Guedj (Boucksom and Guedj 2013, sec. 4.6.1), specifically Lemma 4.6.1 and the paragraph following Definition 4.6.2. The soft-sheaf descriptions of Bott–Chern cohomology by smooth functions and distributions first give a global real distribution \(u_Y\) for which \[T=p^*\eta+i\partial\bar\partial u_Y.\] Locally write \(T=i\partial\bar\partial v\) with \(v\) plurisubharmonic and \(p^*\eta=i\partial\bar\partial h\) with \(h\) smooth. Then \(u_Y+h-v\) is pluriharmonic as a distribution. By the cited lemma it is a smooth real part of a holomorphic germ. Thus \(u_Y\) is locally represented by a quasi-plurisubharmonic function. These representatives agree as distributions on overlaps and hence, after taking their upper-semicontinuous representatives, agree pointwise. They give one global quasi-plurisubharmonic \(u_Y\), which is locally integrable and locally bounded above.

In local ambient embeddings, \(p^*g\) is represented by a smooth semipositive form and \(\kappa\) by a smooth positive definite form. A finite relatively compact cover of \(Y\) therefore gives a constant \(C>0\) with \(p^*g\leq C\kappa\). Consequently \[p^*\eta+i\partial\bar\partial u_Y\geq C^{-1}p^*g.\] The map \(p\) is a modification between normal spaces, and \(u_Y\) has the local upper bound needed in the current descent argument. (Das et al. 2024, Corollary 2.32 and its proof, p. 18) gives a global quasi-plurisubharmonic \(u_Z\) with (35) for \(c=C^{-1}\). It is an \(L^1\) potential, and the formula keeps the descended current in the chosen Bott–Chern class. The local potentials of \(\eta\) give local potentials for that current. No Kähler form on \(Z\) has been used. ◻

Proof. By (Das et al. 2024, Lemma 2.3 and Proposition 2.4), the natural pairing is perfect, \(\operatorname{Nef}(Z)=C^*\), and the Kähler cone is open with closure the nef cone. In particular \(\kappa\) lies in the interior of \(C^*\). A sequence in \(\Sigma\) with unbounded norm would, after division by its norm and passage to a subsequence, give a nonzero class \(z\in C\) with \(\kappa\cdot z=0\), contradicting that interior property. The slice is closed and hence compact.

If it is nonempty, let \(\mu=\min_\Sigma\ell>0\). Homogeneity gives \(\ell-\frac\mu2\kappa\in C^*=\operatorname{Nef}(Z)\). A smooth local-potential nef approximation for this last class, bounded below by \(-\mu\kappa/4\), becomes a positive representative after adding \(\mu\kappa/2\). Thus \(\ell\) is Kähler. If \(\Sigma\) is empty, then \(C=\{0\}\) by the interior property; the same conclusion follows because every class is nef and a positive small multiple of \(\kappa\) can be subtracted first. All cones here are the local-potential cones in the cited perfect pairing. ◻

Proof of Proposition 4. The proof of (Das et al. 2024, Theorem 5.5) first uses a projective relative small \(\mathbb Q\)-factorialization and then (Das and C. D. Hacon 2024, Theorem 1.7). The latter proof passes to a small strongly \(\mathbb Q\)-factorial model and treats a nef nonbig class by its Theorem 5.5 and a nef big class by its Theorem 6.4. Here strong \(\mathbb Q\)-factoriality means that every global coherent reflexive rank-one sheaf has an invertible reflexive power. It implies ordinary global \(\mathbb Q\)-factoriality; it does not assert the analogous property for all analytic open subsets. The small models use the projective relative MMP of (Das and C. D. Hacon 2024, Proposition 2.26 and Lemma 2.27), whose maps are already projective. Compactness of the final bases is retained when projective maps are composed, as required by (Das et al. 2024, Remark 2.11).

Proof. Das–Hacon–Păun’s dlt modification theorem (Das et al. 2024, Theorem 6.1) applies to a compact Kähler lc fourfold with effective rational boundary and \(\mathbb Q\)-Cartier adjoint. It gives a projective bimeromorphic morphism \[g:(X',\Delta')\longrightarrow(X,\Delta)\] with \(X'\) compact Kähler and ordinary \(\mathbb Q\)-factorial, the pair \((X',\Delta')\) dlt, and the adjoint crepant. The input is klt, so equality of discrepancies makes the output klt as well. The crepant equality is an equality of actual rational lines: pull a local pluriadjoint frame to a common resolution with smooth source as in (2), compare the crepant coefficients, and extend the identical meromorphic map across codimension two on the normal \(X'\). Thus for a common index \[\mathcal O_{X'}(m(K_{X'}+\Delta'))\simeq g^*\mathcal O_X(mD).\] There is no possible undetected flat-line difference in this comparison.

The pulled-back line is analytically nef by the metric criterion. Also, normality and projection formula identify all its sections with those of \(\mathcal O_X(mD)\). The meromorphic maps agree on the common dense open, so their Iitaka dimensions agree. Theorem 4.1 of Höring–Lazić–Lehn (Höring et al. 2025) states that a nef adjoint of a compact ordinary \(\mathbb Q\)-factorial Kähler klt pair of positive Iitaka dimension is semiample, unconditionally in dimension at most four. Its analytic positivity and canonical-sheaf conventions are the ones in use here. Apply it to \((X',\Delta')\), then enlarge the generated degree to a multiple of the original index. All sections are pullbacks. A section nonzero at a chosen point above \(x\in X\) descends to a section nonzero at \(x\); Nakayama’s lemma gives the evaluation surjectivity on \(X\). ◻

Proof. Principalize the reduced coherent ideal of \(\operatorname{Supp}(\Delta+M)\), using (Das et al. 2024, Theorem 2.13 and Remark 2.14). This use of an ideal is important: the original boundary need not be \(\mathbb Q\)-Cartier. The resulting modification has smooth source, is projective, and has locally normal crossing strict and exceptional support. Mark each of its finitely many global prime components separately as an ordered Cartier boundary and apply the ordered-boundary resolution of (Temkin 2017, Theorems 1.1.3 and 1.1.13). Its final indexed components and their intersections are smooth, and its complete support consists of strict transforms and new exceptional components (Temkin 2017, Lemmas 2.1.10 and 2.2.9(iv)). Denote the composite by \(p:Y\to X\). It is projective over the compact base. A relative ample metric plus a sufficiently large pullback of a Kähler form makes \(Y\) compact Kähler.

Let \(G_Y\) be the crepant subboundary for \(D\), defined by the actual meromorphic pluriadjoint pullback. Then \(p_*G_Y=\Delta\), and klt gives \(\operatorname{coeff}_E(G_Y)<1\) at every prime. Define \(B_Y\) only after the preceding resolution: assign coefficient one to the actual \(p\)-exceptional primes and the strict transforms of primes in \(M\), and retain the coefficients of \(\Delta\) on the other strict boundary primes. An exceptional label in the resolution bookkeeping that is not an actual exceptional divisor does not change this rule. This is an effective rational SNC boundary, hence dlt.

Set \[ P_Y=p^*M+(B_Y-G_Y). \tag{43}\] Pullback of the holomorphic equation of a Cartier multiple of \(M\) makes \(p^*M\) effective. On a strict prime of \(M\), the second summand has coefficient \(1-\operatorname{coeff}(\Delta)>0\); on an exceptional prime it has coefficient \(1-\operatorname{coeff}(G_Y)>0\); on any other strict boundary prime it is zero. Thus \(P_Y\) is effective and its reduced support is exactly the floor of \(B_Y\). The actual identity \(p^*D\sim_\mathbb Qp^*M\), together with (2), gives \(K_Y+B_Y\sim_\mathbb QP_Y\).

Choose an integer \(t>0\) at least the ratios of the coefficients of \(P_Y\) to those of \(p^*M\) on every nonexceptional prime of \(M\). The positive part of \(P_Y-tp^*M\) is then an effective rational exceptional divisor \(E\), and \[P_Y\leq tp^*M+E.\] For every sufficiently divisible \(n\), exceptional Hartogs extension and projection formula give \[H^0\bigl(Y,\mathcal O_Y(n(tp^*M+E))\bigr) =H^0\bigl(X,\mathcal O_X(ntM)\bigr).\] The space on the right has dimension one: it is nonzero and \(\kappa(X,D)=0\). The displayed divisor inequality bounds \(h^0(Y,\mathcal O_Y(nP_Y))\) by one, while effectivity makes it nonzero. This controls every sufficiently divisible degree. If two independent sections existed in another Cartier degree, their powers \(s_0^v\) and \(s_0^{v-1}s_1\) would remain independent in a degree divisible by the fixed common index, a contradiction. Therefore \(\kappa(Y,P_Y)=0\). ◻

Proof. Bott–Chern descent first supplies a target class whose pullback is \(\alpha_i\). We prove that this class is nef, contains a current dominating a positive Hermitian form, and has positive top intersection on every positive-dimensional compact reduced subspace. These are the three conditions in (Das et al. 2024, Theorem 2.29) that make the target class Kähler.

Proof. Clear the index and pass to the reduction of \(F\). If an irreducible component \(F_\alpha\) is not contained in \(\operatorname{Supp}E\) but meets it, the restricted canonical section cuts a nonempty effective Cartier divisor on the integral projective \(F_\alpha\). This component has positive dimension: a point that is an irreducible component cannot meet another component, and a connected zero-dimensional fiber is a point. Cutting by sufficiently general hyperplanes of a very ample line on \(F_\alpha\) gives a curve on which \(E\) has strictly positive degree, a contradiction. If some components are contained in the support and others are not, connectedness gives a noncontained component meeting a contained one and the same contradiction. These alternatives prove the claim, with no reducedness assumption on the original fiber. ◻

Proof. Write \(b:\Gamma\to Y\) for the normal common graph. Choose a relatively very ample and generated line \(\mathscr L\) for \(b\); compactness permits one global power. Its evaluation embeds \(\Gamma\) in \(\mathbb P_Y(\mathscr E)\), where \(\mathscr E=b_*\mathscr L\). This coherent sheaf is torsion-free of rank one: a section killed by a nonzero function vanishes on the dense isomorphism locus and hence everywhere. Since \(Y\) is smooth, \(\mathscr E^{**}\) is a line, and \[\mathscr J=\mathscr E\otimes(\mathscr E^{**})^{-1}\subset\mathcal O_Y\] is a coherent ideal equal to \(\mathcal O_Y\) on \(U_Y\).

Apply the ideal principalization functor of (Temkin 2017, Theorems 1.1.11 and 1.1.13) to \(\mathscr J\). For smooth input it is supported on the cosupport of the ideal, so it avoids \(U_Y\); its total transform is invertible on a smooth space \(Y_1\). The resulting invertible quotient of the pulled-back \(\mathscr E\) defines a map \(Y_1\to\mathbb P_Y(\mathscr E)\). It lands in the closed subspace \(\Gamma\) on a dense open, and hence everywhere because \(Y_1\) is reduced. The map \(Y_1\to\Gamma\) is projective: its graph is a closed immersion into the base change of the projective map \(Y_1\to Y\). Its composite to \(V\) is projective and is an isomorphism over \(U_V\).

This principalization is not required to have transverse intermediate centers. Instead, after it is complete, mark as separate ordered Cartier boundary components all strict initial boundary primes, all primes in the principalized support, and all divisorial exceptional primes. On \(U_Y\) this is the original globally simple SNC list. Apply the ordered-boundary functor of (Temkin 2017, Theorems 1.1.3 and 1.1.13). Its centers lie over the original labeled bad locus and therefore avoid \(U_Y\). The final indexed components and intersections are smooth, and the total support consists of the strict support and new exceptional support. The already invertible ideal remains invertible, so the map to \(\Gamma\) persists.

Finally apply the same two stages to the pullback of the reduced coherent ideal of \(V\setminus U_V\). In the second stage mark, each distinct prime once, every component of the prior resolved support, every component of the newly principalized support, and every divisorial exceptional prime. Its total transform has divisorial support equal to the entire inverse image of \(V\setminus U_V\); every such divisor maps to a subset of codimension at least two by (67). Outside this support the resulting map \(\pi:\widehat V\to V\) is an isomorphism. Conversely a point on this support cannot be a local isomorphism point, because the divisor germ through it would map to a hypersurface contained in \(V\setminus U_V\). Thus this is exactly the nonisomorphism locus. The complete support is globally simple SNC and includes every strict boundary component of \(V\).

Each \(\pi\)-exceptional prime has center in \(V\setminus U_V\). The preceding discrepancy argument shows that its log discrepancy is positive, proving property (2). Let \(\widehat Z\) be an irreducible intersection of \(k\) distinct transformed floor components. At its general point exactly those components occur, by SNC. If \(k=1\), its prime valuation has log discrepancy zero. If \(k\geq2\), the blowup of the smooth stratum has log discrepancy \(\operatorname{codim}(\widehat Z)-k=k-k=0\). This is a prime on a projective resolution, so its image meets \(U_V\). Since \(\pi\) is an isomorphism there, it is generically an isomorphism on that image. This proves property (3). ◻

Proof of Proposition 12. All statements except \(P\ne0\) have been established. Suppose that \(P=0\). Every component of \(P_Y\), and hence every component of \(p^*M\), has disappeared. On a common projective resolution with smooth source \(r:W\to Y\), \(s:W\to V\), set \(Q=r^*p^*M\). It is effective and nonzero: the pullback of a nonzero effective Cartier multiple has a nonzero strict transform. It is \(s\)-exceptional. Indeed, a prime of \(W\) mapping onto a prime of \(V\) corresponds, by no extraction, to a surviving prime of \(Y\); its coefficient in \(Q\) is zero under the supposition that every component of \(p^*M\) disappeared.

The actual rational line of \(Q\) is \((pr)^*D\). It is analytically nef on the compact Kähler \(W\), by pullback of the metric inequalities for the original \(D\). In particular it is relatively nef for the projective \(s\). Exceptional negativity gives \(Q\leq0\), contradicting its effectivity and nonvanishing. Thus \(P\ne0\). ◻

Proof. Work in a relatively compact local embedding in \(\mathbb C^N\), and refine the regular loci of \(Z\), its singular locus, \(D\), \(\Gamma\), the boundary primes, and their intersections into a locally finite collection of smooth strata. After a neighborhood shrink only finitely many relevant strata meet the chosen compact closure. Choose a linear map \(\ell:\mathbb C^N\to\mathbb C^{n-2}\) whose differential is an isomorphism on the tangent space of \(\Gamma\) at a general smooth point where \(D\) is smooth. Such points exist because \(D\) is normal. For the restriction of \(\ell\) to each smooth stratum of dimension at least \(n-2\), Sard’s theorem makes its critical values a measure-zero subset of the parameter space; strata of smaller dimension have measure-zero image (Sard 1942). The image of \(\Gamma\) contains a neighborhood of the chosen value. We may therefore choose a nearby value regular for all these restrictions and still meeting \(\Gamma\) at a general point. The corresponding level is transverse to every stratum it meets.

At regular points of \(Z\) the level is a smooth surface. The singular locus of a normal \(n\)-fold has dimension at most \(n-2\); its strata therefore meet this level only discretely. Every irreducible component of the level has dimension at least two by the principal ideal theorem. It cannot be contained in that discrete singular intersection, and hence has dimension exactly two by its smooth dense part. The \(n-2\) level equations have height \(n-2\) at every point. In the Cohen–Macaulay local rings of \(Z\) they are a regular sequence. The quotient is consequently Cohen–Macaulay of dimension two. It is generically reduced and regular in codimension one, since its possible singular points are discrete; Serre’s criterion makes it normal. Transversality on \(D\) gives a smooth curve at the cut point, and the fact that \(d\ell\) is an isomorphism on \(T\Gamma\) makes this curve transverse to \(\Gamma\).

Each boundary prime cuts an effective curve with its original rational coefficient, counted with its positive intersection multiplicity. At a generic codimension-one point of the normal surface, the ambient space and the slice are smooth and transverse to these primes. Dividing the ambient canonical form by the parameter volume identifies the restriction of the frame with the pluriadjoint frame of this sliced effective boundary. Both are rank-one reflexive sheaves on the normal surface, so the identification extends from codimension one and is an actual line identity.

To compare the orders, on the smooth \(D\) near a general point of \(\Gamma\) complete the parameters to coordinates \((z_1,\ldots,z_{n-2},u)\) with \(\Gamma=(u=0)\). Write the meromorphic residue as \[u^c h(z,u)(du\wedge dz_1\wedge\cdots\wedge dz_{n-2})^{\otimes m},\] where \(h\) is a unit at a general point of \(\Gamma\). Choose the regular level above also outside the proper zero set of its leading coefficient. Division by the base volume and restriction to that level then has order exactly \(c\) on the cut curve. On the ambient smooth locus the two operations commute: in coordinates with \(D=(x=0)\), both take the residue of \(h(x,u,z)(dx/x\wedge du\wedge dz)^{\otimes m}\) to \(h(0,u,z)(du)^{\otimes m}\) on the level, up to the ordering sign. The preceding reflexive identification makes the surface term the residue of the same actual frame. The restricted original residue is already a meromorphic form on the smooth curve \(C\), so equality on its dense smooth-ambient part extends across the cut point. This proves the order assertion. In even degree the ordering sign is one. ◻

Proof. We induct on the number of indices. The assertions for the empty index set are the given normality, effective boundary, and crepant identity on \(V\). Assume them for a stratum \(Z\) and all allowed weights there. Lower all unused floor weights. The resulting crepant SNC boundary on \(\widehat Z\) has every coefficient below one. The effective pair \((Z,B_Z^{\mathbf e})\) is therefore klt. Lemma 3 shows that \(Z\) has rational singularities and is Cohen–Macaulay.

To add an index \(j\), retain only that unused weight at one and lower the others. The coefficient-one locus on \(\widehat Z\) is \(R=\widehat Z\cap\widehat S_j\), a disjoint union of smooth components. Apply Lemma 3 to \(\pi_Z\). Its connected fibers prevent the images of two such components from meeting. On each image \(D\), the equality \(\pi_{Z,*}\mathcal O_R=\mathcal O_{\pi_Z(R),\mathrm{red}}\) identifies the proper bimeromorphic pushforward of the structure sheaf of its smooth resolving component \(\widehat D\) with \(\mathcal O_D\). Factoring through the finite normalization shows that \(D\) is normal. It is a compact analytic subspace of \(V\), so it inherits a Kähler form by restricting local strictly plurisubharmonic potentials.

For arbitrary allowed weights retaining \(j\), define \(B_D^{\mathbf e}=\pi_{D,*}\widehat B_{\widehat D}^{\mathbf e}\). At this point it is a rational divisor whose effectivity remains to be shown. In a divisible even degree, a local frame of the ambient adjoint line pulls back and takes SNC residue to a meromorphic pluriform on \(\widehat D\). The proper bimeromorphic map \(\widehat D\to D\) is an isomorphism at the generic points of divisors of the normal \(D\). There the same map identifies the divisorial adjoint for \(B_D^{\mathbf e}\) with the restriction of the existing invertible line. Normal reflexive extension gives (70) on all of \(D\). Pulling the line back to \(\widehat D\) recovers the original meromorphic map, since the maps agree on a dense open. This proves (69), including its meromorphic interpretation.

We verify that \(B_D^{\mathbf e}\) is effective. It is enough to test the coefficient at a general point of each prime in \(D\), a codimension-two locus in \(Z\). Apply Lemma 19, using the Cohen–Macaulay property already proved for \(Z\). It reduces exactly that coefficient, with the actual residue order preserved, to the smooth cut curve in a normal surface germ with effective boundary. On a minimal resolution of this surface germ, write its crepant boundary as the effective strict transform plus an exceptional divisor \(E\). For every exceptional curve \(C\), \[E\cdot C=-(K_{\widetilde T}+B^{\mathrm{str}})\cdot C\leq0.\] Indeed \(B^{\mathrm{str}}\cdot C\geq0\), while surface adjunction gives \(K_{\widetilde T}\cdot C=2p_a(C)-2-C^2\geq0\): exceptional self-intersections are negative and a smooth rational exceptional \((-1)\)-curve is excluded by minimality. The negative-definite exceptional intersection matrix, with nonnegative off-diagonal entries, then implies \(E\geq0\). Explicitly, if \(E=P-N\) has disjoint nonnegative parts and \(N\ne0\), then \(E\cdot N=P\cdot N-N^2>0\), contradicting \(E\cdot C\leq0\) on each component of \(N\). The strict transform of the smooth cut curve is finite birational over it and hence isomorphic. Adjunction to that smooth curve in a smooth surface with effective remaining boundary has a nonnegative coefficient. This is the coefficient originally tested. There is no coefficient to test when the new stratum is a point. Effectivity follows and completes this part of the induction.

For clarity, the full-weight pair on each stratum is dlt and has exactly the claimed floor. The SNC subboundary has coefficients at most one, so it is lc, and crepancy makes the pair on \(Z\) lc. On the smooth \(\widehat Z\), let a divisorial valuation have center of codimension \(c\), and choose local coordinates \(x_1,\ldots,x_c,\ldots\) at a general point of that center, with the SNC boundary components among the coordinate divisors. Give an unmarked coordinate weight zero and write the boundary weights as \(b_i\leq1\). The reduced coordinate arrangement is lc, so \[a(v;\widehat Z,\widehat B_{\widehat Z}) \geq\sum_{i=1}^c(1-b_i)v(x_i).\] All \(v(x_i)\) here are positive. A zero discrepancy therefore forces every \(b_i=1\); its center is an intersection of coefficient-one components. Each such intersection has image meeting the isomorphism locus of the original special resolution. In particular each coefficient-one divisor is nonexceptional for \(\pi_Z\); the others have coefficients below one. Let \(F_Z\subset Z\) be the complement of the largest open over which \(\pi_Z\) is an isomorphism. It has codimension at least two on the normal target \(Z\), and contains no entire image of a zero-discrepancy center. Its inverse image on \(\widehat Z\) may already be divisorial. If necessary, principalize the pulled reduced ideal \(\mathscr I(F_Z)\mathcal O_{\widehat Z}\), and then apply the ordered-boundary resolution as in Lemma 17, marking each distinct prime in the prior SNC support, the newly principalized support, and every divisorial exceptional prime. All new exceptional centers lie over \(F_Z\). The displayed inequality makes their discrepancies positive. This is a dlt resolution in the standard sense. It also proves that the coefficient-one primes on \(Z\) are exactly the images of the one-index intersections.

It remains to check the asserted descent of sections. With full weights, let \(R=\widehat B_{\widehat Z}^{=1}\). The preceding floor identification gives \((\pi_Z(R))_{\mathrm{red}}=C_Z\); write \(\pi_R:R\to C_Z\) for the induced restriction of \(\pi_Z\). Sections on the incident strata pull back to sections on its smooth components. Residue compatibility means equality on the entire scheme-theoretic pairwise intersections, which are reduced smooth SNC strata. The local equalizer for an SNC union therefore glues them to a section on \(R\); it is the elementary equalizer for the coordinate ideals of its components. Even-degree iterated residues make all paths to deeper intersections identical. By Lemma 3 and projection formula, \[(\pi_R)_*\pi_R^*(L|_{C_Z})=(L|_{C_Z})\] for any ambient Cartier line \(L\) under consideration. The section descends uniquely to all of \(C_Z\). This reasoning applies also to \(Z=V\) and \(C_Z=S\). In particular it retains the conductor coefficient at a general double crossing on each normalized component; it does not posit an additional conductor boundary on the nonnormal union itself. ◻

Proof. The equality with the ideal holds because on the normal \(T\) both sheaves consist of holomorphic functions vanishing at every height-one prime in \(C\). Work on a small connected base open trivializing the line whose pullback is \(m(K_T+G)\). Its nonvanishing pulled-back frame is a local meromorphic \(m\)-pluricanonical form \(\theta\).

Adjoin the full set of \(m\)th roots of this form. This construction is local even without a canonical Cartier frame. At a normal local domain \(A\), choose a meromorphic canonical generator \(\eta\) and write \(\theta/\eta^m=a/b\). The monic algebra \[A[w]/(w^m-a b^{m-1})\ \subset\ \operatorname{Frac}(A)[t]/(t^m-a/b),\qquad w=bt,\] is finite and reduced; its analytic normalization is finite (Houzel 1960--1961, Part B, Section 4, Corollaries 2–3). Changes of meromorphic canonical generator multiply the root coefficient by an \(m\)th power, and identify the total algebras and their integral closures. They therefore glue to the normalized cover. Keep every component and the full \(\mu_m\)-action. The tautological form \(\tau=t\eta\) is a meromorphic top form on a projective resolution with smooth source; take this resolution functorially and equivariantly, after shrinking near the compact fibers. Write \(l:H\to T\) for the composite.

At a general prime of \(T\) with boundary coefficient \(b\), the order of \(\tau\) upstairs is \[ e(1-b)-1, \tag{72}\] where \(e\) is the ramification index. This follows from the pole order \(-eb\) of the root coefficient and the differential ramification order \(e-1\). It is \(-1\) when \(b=1\) and is an integer between zero and \(e-1\) when \(0\leq b<1\). A meromorphic form in the \(\tau\)-character has the form \(h\tau\), where \(h\) is an invariant meromorphic function and hence descends to \(T\); this uses the entire root algebra, whose invariants are the normal base. Regularity in codimension one forces \(h\) to be holomorphic and to vanish on \(C\).

Conversely, suppose \(h\in\mathcal O_T(-C)\). On a log resolution of the pair, every crepant coefficient is at most one. A coefficient-one exceptional prime has center in \(C\), because the pair is klt away from \(C\); the pullback of \(h\) has positive order there. Thus in SNC coordinates the density \(|h|^2|\theta|^{2/m}\) has every exponent strictly above the integrability threshold. It is locally integrable. Change of variables makes \(l^*h\tau\) locally square integrable on the smooth resolution, and a square-integrable meromorphic top form is holomorphic. This proves the exact character equality \[ (l_*\omega_H)_{\tau}=\mathcal O_T(-C)\tau. \tag{73}\]

Each component of the full normalized cover dominates the normal base, and each resolved component does so as well. The inverse image in \(T\) of the connected base open is connected, because \(f\) is proper with connected fibers; normality makes it irreducible. Finite maps followed by projective resolutions have Kähler sources near the whole inverse image of a sufficiently small base neighborhood: patch relative ample metrics and add a large pulled-back Kähler form. Canonical torsion-freeness (Fujino 2023, Theorem 2.9 and Proposition 2.11) applies componentwise. For the generically finite map to \(T\), its higher canonical images vanish generically and hence vanish. For the map to \(P\), its first canonical image is torsion-free. Leray and the character projector in (73) make \(R^1f_*\mathcal O_T(-C)\) a direct summand of that torsion-free sheaf. This proves the assertion locally, and hence on \(P\). ◻

Proof. Choose a small rational \(\varepsilon>0\) for which \(B_0-\varepsilon C_0\) is effective klt. This uses dlt and the fact that all its lc centers are contained in the floor. Let \(H\) be an integral very ample divisor on \(Y\) and choose a rational effective \(H'\sim_\mathbb QdH\) with \(d>2\dim Z\) such that \[B_0-\varepsilon C_0+f_0^*H'\] is still klt. A sum of sufficiently many general members of \(|H|\) with small rational weights does this by Bertini on a log resolution. If \(Y\) is a point take \(H'=0\).

The adjoint of this perturbed pair is not pseudo-effective. On a smooth general fiber of a resolution of \(f_0\), its class is the negative of the nonzero effective divisor \(\varepsilon C_0\) restricted to that fiber: the full adjoint and \(H'\) are pulled back from \(Y\), and the floor is horizontal. Its integral against a Kähler power is strictly negative. If the ambient class were pseudo-effective, the potential of a positive current representing its pullback would restrict to a positive current on almost every such smooth fiber, contradicting this integral. The smooth horizontal-fiber conditions hold on a nonempty open, so the almost-everywhere restriction is sufficient.

This same integral also produces a negative class in the cone used by the Kähler cone theorem. If the smooth fiber has dimension \(v>0\), push its positive closed bidimension-\((1,1)\) current \(\omega_F^{v-1}\) through the fiber inclusion and the resolution to \(Z_0\). It represents a class in \(\overline{\operatorname{NA}}(Z_0)\), and its pairing with the perturbed adjoint is exactly that strictly negative integral. The cone decomposition therefore supplies a negative extremal ray.

Run the klt Kähler program for this perturbed adjoint. In dimension three, the negative extremal-ray and contraction theorems (Das et al. 2024, Corollary 5.3 and Theorem 5.5) are used through Subsection 4.4, which incorporates Proposition 9 and the specified external results. They give connected projective contractions to normal compact Kähler targets. The supporting class differs from the klt adjoint by a Kähler class, which is the projectivity condition in that theorem. The supporting class need not be big, and the perturbed adjoint here is not pseudo-effective. Flips and termination of flip sequences are provided by (Das and C. D. Hacon 2024, Theorems 2.24–2.25). In dimensions at most two the non-pseudo-effective pair is projective and the ordinary projective program applies. For a surface, otherwise a non-Moishezon Kähler resolution has a nonzero holomorphic two-form: if \(H^{2,0}=0\), rational approximation of a Kähler class would make it projective. That form descends reflexively and, with the effective boundary, gives a section of a positive adjoint multiple, contradicting non-pseudo-effectivity. A Moishezon Kähler klt surface is projective by Namikawa’s criterion (Namikawa 2001, Corollary 6’). Curves are projective.

We specify the ordinary \(\mathbb Q\)-factorial and finiteness details used by this program. For a projective negative extremal contraction, a global rational line of degree zero on all contracted curves descends in a multiple by (Das et al. 2024, Theorem 2.44(3c)). Here we apply its projective relative theorem to the effective klt pair, with the compact set equal to the entire compact target. For this projective surjection, property Q of (Das et al. 2024, Definition 2.42) requires a normal source and compact source and target, as here. All fiber curves span the one relative ray, so the theorem’s contraction is the given contraction. Indeed its map is constant on each connected projective fiber of the given contraction, and rigidity factors it through that contraction. Its structure map back to the relative base supplies the reverse factorization; surjectivity makes the two factorizations inverse. This use of Theorem 2.44 does not assert a factoriality hypothesis for that theorem. At a divisorial step, adjust the transform of a destination Weil divisor by a \(\mathbb Q\)-Cartier exceptional prime of nonzero degree on the ray, and descend the resulting degree-zero line. Comparison in codimension one makes the destination divisor \(\mathbb Q\)-Cartier. At a flip, adjust on the negative side by the negative adjoint, descend, and pull to the positive side; the positive adjoint is \(\mathbb Q\)-Cartier and the same comparison applies. Boundary components and the adjoint then also give the canonical reflexive power. This preserves ordinary \(\mathbb Q\)-factoriality.

The rank of global rational divisor classes modulo curve numerical equivalence is finite. Each running model is a connected compact complex analytic space. Its reduced underlying real analytic space has bounded Zariski tangent dimension by a finite analytic chart cover. Acquistapace–Broglia–Tognoli (Acquistapace et al. 1979, Theorem 1) embed it closedly in Euclidean space, and Łojasiewicz (Łojasiewicz 1964, Theorem 1) gives a compatible locally finite triangulation; the subcomplex corresponding to the compact model is finite. Hence its rational \(H^2\) is finite-dimensional. The first Chern class sends global rational Cartier divisors to that group. Every divisor in its kernel has degree zero on each compact curve, by restriction to the curve’s normalization. The rational divisor space modulo curve numerical equivalence is consequently a quotient of the Chern-class image, and has finite rank. The rank drops at a divisorial contraction, by the nonzero exceptional ray degree, and does not increase at a flip. For the latter assertion a numerically trivial line on the negative side descends as above; the descended line is curve-numerically trivial because every curve downstairs is covered by a curve upstairs, using projectivity over that curve. Its pullback is trivial on curves on the positive side, and every divisor there is a transform by smallness. There can therefore be only finitely many divisorial steps. Together with termination of flip sequences this gives a finite program to a nef model or a Mori contraction. All steps extract no divisors.

Every step factors over \(Y\). Inductively write the running perturbed adjoint as \[A_i^{\varepsilon}=K_{X_i}+B_i-\varepsilon C_i+f_i^*H'.\] For a negative contraction apply the relative cone and length theorem (Das et al. 2024, Theorem 2.45) to \(K_{X_i}+B_i-\varepsilon C_i\) over its entire compact contraction target. The source is ordinary \(\mathbb Q\)-factorial and the pair is klt. Property Q holds as above; because the compact set is the whole target, Theorem 2.45’s factoriality over that set is precisely the global factoriality of the source, including its canonical reflexive power. A negative generator \(\Gamma\) has \[-(K_{X_i}+B_i-\varepsilon C_i)\cdot\Gamma\leq2\dim Z.\] If \(f_i(\Gamma)\) were a curve, then \(f_i^*H'\cdot\Gamma\geq d\), contradicting the negativity of \(A_i^{\varepsilon}\). On the nonnegative part of that relative cone, both \(K_{X_i}+B_i-\varepsilon C_i\) and the nef \(f_i^*H'\) are nonnegative, while their sum is nonpositive on the contracted cone; its \(H'\)-degree is zero there as well. All contracted curves are vertical over \(Y\). A holomorphic map from a connected projective fiber that is nonconstant would map a curve nontrivially, so \(f_i\) is constant on each fiber. Rigidity and normality of the target give the factorization, and the positive side of a flip factors through the same base.

We check the boundary properties needed to repeat the step. No prime is extracted, so the transformed full boundary is effective and its floor is exactly the transform \(C_{i+1}\) of \(C_i\). The running MMP preserves the klt pair \(B_i-\varepsilon C_i+f_i^*H'\). Removing the effective divisor \(f_{i+1}^*H'\) shows that \(B_{i+1}-\varepsilon C_{i+1}\) is klt. In particular the full pair is klt away from its floor.

The full adjoints remain actually crepant pullbacks from \(Y\). Indeed the chosen meromorphic pluriadjoint identity extends to the new model in codimension one, since no prime is extracted, and then reflexively. On a common graph both meromorphic maps start from the same line pulled back from \(Y\) and agree on the dense isomorphism open. They agree everywhere as meromorphic maps. Equality of discrepancies therefore preserves log canonicity of the full pair. If \(r,s\) are the graph projections, negativity for the perturbed step gives \[ 0\leq r^*A_i^{\varepsilon}-s^*A_{i+1}^{\varepsilon} =\varepsilon(s^*C_{i+1}-r^*C_i). \tag{75}\] A horizontal component of \(C_i\) cannot disappear into an entirely vertical \(C_{i+1}\), since the coefficient of the right side along its strict valuation would then be negative. Thus the floor remains horizontal. On every subsequent model the full adjoint and \(H'\) still come from \(Y\). Repeating the resolved smooth-fiber integral and pushing its Kähler-power current as above gives a negative class in its \(\overline{\operatorname{NA}}\) cone and hence a negative extremal ray. In particular a nef endpoint is impossible.

We have obtained a Mori contraction \[u:T\longrightarrow W,\qquad g:W\longrightarrow Y,\] with \(T\) normal compact Kähler, \(u\) projective of fiber type, and both maps having connected fibers. For \(g\), this follows also from \((g u)_*\mathcal O_T=\mathcal O_Y\) and \(u_*\mathcal O_T=\mathcal O_W\). The full boundary \(G\) is effective lc, its adjoint is an actual pullback from \(Y\), and it is klt away from \(C^+=\left\lfloor G\right\rfloor\): subtracting \(\varepsilon C^+\) is klt. The running perturbed adjoint satisfies \[A^{\varepsilon}\equiv_u-\varepsilon C^+, \qquad -A^{\varepsilon}\equiv_u+\varepsilon C^+.\] The left side of the second equivalence is relatively ample for the Mori contraction. Thus \(C^+\) is relatively ample for \(u\). The graphs of the divisorial steps and flips are projective over both sides. A main component of their iterated fiber product, followed by normalization and projective resolution, gives the common projective resolution in the statement. Its smooth source is compact Kähler. ◻

Proof. If \(C_Z\) is vertical over \(Y_Z\), the consequence of (74) gives extension in a uniform large-degree tail without a comparison. Suppose therefore that it is horizontal, and use the small model \(q_Z\) and the Mori model \(u:T\to W\), \(g:W\to Y\) of Lemma 22. Extending the pulled-back section on \(C_0\) suffices: a section on \(Z_0\) descends to \(Z\), and equality of its restriction after pullback implies equality on the reduced \(C_Z\), since every component is covered birationally.

Proof. Let \(i:Z\hookrightarrow V\) be the parent stratum, and choose a common projective resolution with smooth compact Kähler source \(U\), with maps \(a:U\to Z\) and \(b:U\to T\) to its original and Mori models. Use the single class \[\alpha=[(i\circ a)^*\omega_V]\in H^{1,1}(U,\mathbb R).\] Resolve the marked-branch comparison graph and its maps to \(U\) and the minimal surfaces. We obtain a smooth compact Kähler surface \(D\) with maps \(r_i:D\to U\), bimeromorphic maps \(d_i:D\to P_i\), and bimeromorphic maps to the original \(Z_i\), such that \(a r_i\) is the inclusion of that original branch after its resolution and \(d_2=\tau d_1\). These identities hold on the common marked open and hence everywhere. This construction includes a self-link: its two maps to the same original prime may differ by the normalized exchange. Resolution independence and projection formula give \[ c_{Z_1}-\tau^*c_{Z_2} =d_{1,*}(r_1^*-r_2^*)\alpha. \tag{79}\]

The exceptional image of the modification \(b:U\to T\) has codimension at least two in the normal threefold, and hence dimension at most one. The Mori base in this two-marking case has dimension two. After removing its image, the discriminant, and the singular base locus, \(U\) is therefore a smooth proper \(\mathbb P^1\)-fibration over a dense open of the base. For such a fibration \(\pi:U^\circ\to W^\circ\), the differential sequence and \(H^0(\mathbb P^1,\Omega^1)=0\) give \[\pi_*\Omega^2_{U^\circ}=\Omega^2_{W^\circ}.\] Indeed the quotient of \(\Omega^2_{U^\circ}\) by \(\pi^*\Omega^2_{W^\circ}\) is \(\pi^*\Omega^1_{W^\circ}\otimes\Omega^1_{U^\circ/W^\circ}\), whose direct image is zero, while \(\pi_*\mathcal O_{U^\circ}=\mathcal O_{W^\circ}\). Thus the two restrictions of a holomorphic two-form on \(U\) agree on the dense paired-branch open of \(D\), even for the double branch after an étale local ordering. They agree on all of \(D\).

Holomorphic pullback and proper pushforward on compact Kähler manifolds preserve Hodge type. The map \[d_{1,*}(r_1^*-r_2^*):H^2(U,\mathbb Q)\longrightarrow H^2(P_1,\mathbb Q)\] is therefore a rational Hodge morphism; pushforward here is between equal-dimensional surfaces. The preceding equality kills its \((2,0)\) part and, by conjugation, its \((0,2)\) part. Its rational image is of type \((1,1)\), hence belongs to \(\operatorname{NS}(P_1)_\mathbb Q\) by Lefschetz \((1,1)\). Extend scalars to \(\mathbb R\) in (79) to obtain (78). ◻

Proof. The intersection form on \(H^{1,1}(P,\mathbb R)\) has Lorentz signature \((1,h^{1,1}-1)\). Its restriction to \(N\) is nonpositive. To see this, a rational positive-square class in \(N\), after scaling and choosing its sign, is \(c_1(L)\) with positive Kähler degree. Riemann–Roch and Serre duality give \(h^0(P,L^v)\gg v^2\): the Euler characteristic has positive quadratic leading term and \(H^0(P,K_P\otimes L^{-v})=0\) for large \(v\) by its negative Kähler degree. This would make \(P\) Moishezon and hence projective. Rational approximation in \(N\) rules out a positive real class as well.

If \(N\) is negative definite, orthogonally project \(c\) to \(N^\perp\). The projection \(c_\perp\) has positive square and is fixed by every \(\sigma^*\): the congruence kills its projected difference, and each automorphism preserves \(N\) and the intersection form. The perpendicular of \(c_\perp\) is negative definite, so all eigenvalues on \(H^{1,1}(P,\mathbb R)\) have modulus one.

If \(N\) is degenerate, its radical is a rational isotropic line \(\ell\); nonpositivity in a Lorentz space permits no larger radical. An integral automorphism preserves a primitive integral generator of \(\ell\) up to sign. The invariant flag \[\ell\subset\ell^\perp\subset H^{1,1}(P,\mathbb R)\] has eigenvalues \(\pm1\) on the first and last quotients, which pair dually, and a negative-definite middle quotient \(\ell^\perp/\ell\). All \((1,1)\) eigenvalues again have modulus one. This argument permits unipotent action and makes no finiteness assertion for all cohomology.

Write \(\sigma^*\eta=\delta_\sigma\eta\). Integration of \(\eta\wedge\bar\eta\) gives \(|\delta_\sigma|=1\), and the conjugate eigenvalue has the same modulus. Since a nowhere-vanishing canonical form spans \(H^{2,0}(P)\), all eigenvalues on \(H^2(P,\mathbb C)\) now have modulus one. The action on \(H^2(P,\mathbb Z)/\mathrm{torsion}\) is integral. Kronecker’s theorem makes each eigenvalue a root of unity; if its order is \(n\), its cyclotomic polynomial has degree \(\varphi(n)\leq b_2(P)\). There are only finitely many such \(n\). The scalars of every element of \(G\) therefore lie in one finite set of roots of unity, which proves finite image on \(\mathbb C\eta^{\otimes m}\). ◻

Proof. The analytic Kummer sequence \[1\longrightarrow\mu_r\longrightarrow\mathcal O_X^\times \xrightarrow{(\cdot)^r}\mathcal O_X^\times\longrightarrow1\] is exact also for singular or nonreduced analytic spaces. A unit germ has a local holomorphic root; the kernel is the locally constant sheaf of \(r\)th roots of unity, since \(u^r-1\) has distinct roots, including in a local ring with nilpotents. Line bundles are classified by \(H^1(\mathcal O^\times)\), and abelian torsors by \(H^1\) of the corresponding sheaf (The Stacks Project Authors, n.d., Tags 09NU and 02FQ).

For an abelian sheaf \(F\) and a compact subset whose distinct points have disjoint neighborhoods, SGA 4 continuity (SGA 4, n.d., Exposé Vbis, Lemma 4.1.3) gives \[ \varinjlim_{U\supset A}H^i(U,F)=H^i(A,F|_A) \quad\text{for every }i. \tag{82}\] The separation hypothesis holds in the Hausdorff analytic space. Although the intrinsic structure sheaf of \(A\) need not be the inverse image of \(\mathcal O_X\), the inverse image of \(\mu_r\) is exactly \(\mu_r\) on \(A\). Naturality of Kummer therefore makes the obstruction to a root of \(L\) restrict to zero in \(H^2(A,\mu_r)\), because \(Q\) is such a root there. Continuity in degree two kills that obstruction on a smaller neighborhood, giving a root pair \((P,\alpha)\).

The difference between \((P|_A,\alpha|_A)\) and \((Q,\beta)\) is a \(\mu_r\)-torsor. By continuity in degree one it extends to a torsor on a smaller neighborhood. The associated line with its trivialized \(r\)th power twists \(P\) to give the prescribed pair on \(A\). Finally, the sheaf of compatible isomorphisms of two root pairs is itself a locally trivial finite \(\mu_r\)-torsor. A prescribed section on \(A\) trivializes its restriction there. Continuity in degree one therefore trivializes this torsor on a smaller neighborhood; choose a section there. The ratio of its boundary restriction to the prescribed section is a section of \(\mu_r\) on \(A\). Continuity in degree zero extends that ratio uniquely as a germ, and correcting the chosen section gives the prescribed extension. The same degree-zero injectivity gives uniqueness as a germ. These sections are holomorphic, being locally solutions of \(z^r=u\) for a holomorphic unit. ◻

Proof. Local finiteness and compactness give an open neighborhood \(\mathcal N_1\) of the central fiber which meets only finitely many labels. Remove from the base the closed image of \(H\setminus\mathcal N_1\); properness then puts the entire restricted support in \(\mathcal N_1\). There are now finitely many closed indexed intersections to consider. The connected components of each form a locally finite family in \(H\): the intersection is closed and smooth, so near one of its points a small smooth neighborhood meets only its own component, and near a point outside it a neighborhood avoids it. Compactness, followed by the same properness argument, gives a second neighborhood \(\mathcal N_2\) meeting only finitely many such components and a base restriction on which the whole support lies in \(\mathcal N_2\).

Let \(C_1,\ldots,C_N\) be the whole connected components of the pre-restriction closed intersections which meet \(\mathcal N_2\). We retain the whole components, not just their intersections with \(\mathcal N_2\). Each \(C_a\) is closed in a closed stratum and is a connected complex manifold. Thus \(f_a=h|_{C_a}\) is proper over the preceding base and \(C_a\) is irreducible. Their restrictions cover every closed intersection after the current restriction, including the singleton intersections.

Put \(S_a=f_a(C_a)\) with its reduced structure. Remmert’s proper mapping theorem makes \(S_a\) closed analytic (Grauert 1960, sec. 7, no. 1, Satz 1, p. 60), and it is irreducible because \(C_a\) is. Inside any prescribed open coordinate neighborhood \(V\ni t_0\) contained in the preceding restrictions, take the locally finite irreducible decompositions of all \(S_a\cap V\). Let \(B\) be the union of those components which do not contain \(t_0\). A union of a subfamily of the locally finite irreducible components of an analytic set is closed analytic. Since the list of \(a\)’s is finite, \(B\) is closed analytic in \(V\) and misses \(t_0\). Set \(U=V\setminus B\).

This removal does not split any retained irreducible component \(A\) of an \(S_a\cap V\). Its regular locus is connected, and its intersection with \(B\) is a proper analytic subset: otherwise closedness and density would imply \(A\subset B\), contrary to \(t_0\in A\setminus B\). The complement of a proper analytic subset in a connected complex manifold is connected. One may see this by perturbing a path, in a finite chain of coordinate balls, off a subset of positive complex codimension. Hence \(\operatorname{Reg}(A)\setminus B\) is connected and \(A\cap U\) is irreducible. Only finitely many components of \(S_a\cap V\) contain \(t_0\), by local finiteness. Their restrictions give a finite cover of \(S_a\cap U\) by irreducible closed analytic subsets, all containing \(t_0\). Therefore every irreducible component of \(S_a\cap U\) contains \(t_0\); if \(S_a\) misses \(t_0\), its restriction is empty.

Let \(W\) be a connected component of \(f_a^{-1}(U)\). It is open in the connected smooth \(C_a\), is closed in \(f_a^{-1}(U)\), and is smooth and irreducible. Its map to \(U\) is proper. Put \(r_a=\dim S_a\). The maximal-rank locus of \(f_a\) is dense in \(C_a\): the generic rank theorem gives maximal rank \(r_a\), and the lower-rank locus is a proper analytic subset defined by the maximal minors. The nonempty open \(W\) meets the maximal-rank locus. Its proper image is consequently an irreducible closed analytic subset of \(S_a\cap U\) of dimension \(r_a\). Every component of \(S_a\cap U\) has this dimension because \(S_a\) is irreducible and hence pure-dimensional. The image is therefore an irreducible component, since a proper analytic subset of such a component has smaller dimension. It therefore contains \(t_0\), and \(W\) meets \(f_a^{-1}(t_0)\).

The compact analytic fiber has finitely many connected components: on its reduction, irreducible components are locally finite and compactness makes their number finite. Each \(W\cap f_a^{-1}(t_0)\) is a nonempty open-and-closed subset of this fiber. Different \(W\)’s give disjoint such subsets, so \[ \#\pi_0(f_a^{-1}(U)) \leq \#\pi_0\bigl((f_a^{-1}(t_0))_{\mathrm{red}}\bigr)<\infty. \tag{90}\] This includes zero-dimensional and other vertical images and permits nonreduced scheme fibers.

Reindex the singleton strata by these finitely many components \(W\). An intersection of the new labels is an open-and-closed part of the restriction of the corresponding old closed intersection, hence is a union of some of its finitely many connected components. It remains smooth and proper, and the relative form restricts. Distinct new pieces from one old label are disjoint, so the local SNC branches and their ordering are unchanged. This proves the assertion. ◻

Proof. Let \(v\) be a local meromorphic frame of \(\mathcal O_Z(aE)\). By (84) and \(aq=r\), the \(q\)-pluriform \(v^q\pi^*s\) is, up to a holomorphic unit, the pullback of a local frame \(\ell_V\) of \(L\). On a log resolution of \((V,B,G)\), write \(b_F\) for a crepant boundary coefficient and \(m_F=\operatorname{ord}_F(s/\ell_V)\). Then \(m_F\geq0\), \(b_F\leq1\), and \(b_F=1\) implies \(m_F>0\): the dlt pair is klt off \(S\), and the section vanishes exactly on \(S\). Thus the local density \[|\ell_V|^{2/q}|s/\ell_V|^{2\epsilon}\] has exponent \(-b_F+\epsilon m_F>-1\) at each prime for every \(\epsilon>0\), and is locally integrable. Change of variables under the proper generically finite map \(\pi\rho\) preserves this integrability. For \(\alpha=(\rho^*v)\tau\), let \(k_F\) be its integral order at a prime upstairs and \(n_F\) the order of the pulled section ratio. Integrability says \(k_F+\epsilon n_F>-1\) for all \(\epsilon>0\). The order \(n_F\) is positive exactly on \(H\). Hence \(k_F\geq-1\) on \(H\) and \(k_F\geq0\) elsewhere. This proves the logarithmic map. Cartier adjunction on the reduced SNC divisor gives its residue map on \(H\), with \(\rho_H^*\mathcal A_{-a}=\rho^*\mathcal O_Z(aE)|_H\).

Let \(\chi\) be the \(\tau\)-character of the second cover. At a general point of \(E\) over \(S_i\), differential pullback gives \[\operatorname{ord}_E(\pi^*s) =\frac r{d_i}(d_i-q)+q\left(\frac r{d_i}-1\right) =r-q=q(a-1).\] The second root is unramified there and \(\operatorname{ord}_E\tau=a-1\). At any other codimension-one prime the first cover is étale and the original boundary coefficient is \(\beta\in[0,1)\). For \(e=q/\gcd(q,q\beta)\), the second-cover order is \(e(1-\beta)-1\in\{0,\ldots,e-1\}\), by (72). A meromorphic form in character \(\chi\) is \(f\tau\) for an invariant meromorphic \(f\) descended to \(Z\). The orders force \(\operatorname{ord}_E f\geq-(a-1)\) and nonnegative orders at all other primes. Normality extends it as a section of the indicated invertible line. Conversely, the logarithmic map multiplied by the equation of \(E\) has no pole because \(D_E\geq H\). We obtain the exact sheaf identities \[\begin{align*} (\rho_*\omega_{\widehat Z})_\chi &=\mathcal O_Z((a-1)E)\tau,\tag{92}\\ (\rho_*\omega_{\widehat Z}(D_E))_\chi &=\mathcal O_Z(aE)\tau. \tag{93}\end{align*}\] The second is projection formula for the invertible line \(\mathcal O_Z(E)\).

Canonical torsion-freeness (Fujino 2023, Theorem 2.9 and Proposition 2.11) applies locally near \(E\) to each dominating resolved component. More precisely, around a compact fiber choose the Kähler neighborhood just constructed. Properness of \(\rho\) supplies a smaller normal target neighborhood whose entire inverse image lies in it, by removing the closed image of its complement. After a base shrink this neighborhood contains all of \(E\) in the root chart. The restricted morphism is proper with Kähler smooth source, so the cited theorem applies there. These neighborhoods cover \(E\). On each, the higher direct images for the generically finite resolution vanish because they vanish generically. Finite pushforward is exact. Thus on their union \(Z^\circ\subset Z\), an open neighborhood of \(E\), \[(R^i\rho_*\omega_{\widehat Z})|_{Z^\circ}=0\quad(i>0).\] Projection formula gives the same restricted vanishing for \(\omega_{\widehat Z}(D_E)\).

Put \(A_E=\mathcal A_{-a}=\mathcal O_E(aE)\). Cartier adjunction for the possibly nonreduced divisor \(D_E\) is the exact sequence \[0\longrightarrow\omega_{\widehat Z}\longrightarrow \omega_{\widehat Z}(D_E)\longrightarrow\omega_{D_E}\longrightarrow0.\] Since \(D_E\) is the scheme inverse image of \(E\), it has a morphism \(\rho_D:D_E\to E\). The preceding vanishings and character identities identify \[ \bigl(R(\rho_D)_*\omega_{D_E}\bigr)_\chi\simeq A_E\tau[0] \quad\text{in }D(\mathcal O_E). \tag{94}\] One can see this first after exact closed pushforward from \(E\) to \(Z^\circ\), where it is the quotient of (93) by (92). Closed pushforward detects its cohomology sheaves, and the canonical truncation of a complex concentrated in degree zero gives (94) on \(E\). No derived full-faithfulness assertion is needed.

Because \(A_E\) is invertible, its pullback followed by the residue map defines \(A_E[0]\to R(\rho_H)_*\omega_H\). The inclusion \(\omega_H\hookrightarrow\omega_{D_E}\), followed by the character projection and (94), gives a map back to \(A_E\tau[0]\). On degree zero their composition sends \(f\) to \(f\tau\) in the quotient of the two character lines. Under the chosen identification it is the identity. An endomorphism of the sheaf \(A_E[0]\) in degree zero is determined by its sheaf map, so this is a derived retraction. Applying \(Rg_*\) proves the split injections (91). All maps were defined by the tautological form, the actual divisor ideal, and residue, so they respect root comparisons. Under an ordinary product the tautological form is relative to the new factor; wedging with a local frame of its canonical line gives the absolute map and the stated \(\omega_{B_0}\) factor. ◻

Proof. Choose finitely many pairs of parameter opens \(U_i'\Subset U_i\), with the \(U_i'\) covering \(f(S)\), such that a local root chart and its proper map \(h_i:H_i\to U_i\) are defined on \(U_i\). Each \(h_i^{-1}(\overline{U_i'})\) is compact. Local finiteness therefore leaves only finitely many smooth component intersections meeting these compact sets. For each such stratum \(C\) and each subset of variable hyperplanes in \(T\), consider the incidence \[\{(z,H_1,\ldots,H_v):h(z)\in H_1\cap\cdots\cap H_v\}.\] It is smooth, since varying each hyperplane independently supplies its normal direction. Sard’s theorem for the projection to the hyperplane parameter space gives transversality for a general tuple; a countable atlas handles any noncompact stratum. Choose a tuple satisfying all the finitely many conditions and also the open conditions that it forms a coordinate basis and that none of its hyperplanes contains \(t_*\). Use those coordinates for \[\psi([w_0:\cdots:w_b])=[w_0^r:\cdots:w_b^r].\] Its derivative at a point has image the tangent space to the intersection of exactly the coordinate hyperplanes corresponding to vanishing coordinates, or is invertible when none vanish. The chosen transversality therefore gives, at \(h(z)=\psi(w)\), \[ dh(T_zC)+d\psi(T_wT')=T_{h(z)}T \tag{96}\] for every smooth SNC stratum. The map is unbranched over \(t_*\).

In local coordinates \(t_i\) on \(U\), extend the coordinate functions of \(h\) holomorphically off \(H\) to functions \(G_i\) on local opens of \(\widehat Z\). In \(\widehat Z\times U'\), the equations \(\psi_i(w)-G_i=0\) have surjective differential on each \(H\)-stratum by (96). They cut a smooth submanifold of codimension \(b\) transverse to the SNC divisor. The cut \(H^\flat=H\times_U U'\) is therefore reduced SNC of pure dimension \(\dim V-1\), and has codimension \(b+1\) in the product ambient. Complete-intersection adjunction gives (95). In coordinates its last factor has the wedge frame \(dw/dt\); this means a frame of \(\omega_{U'}\otimes\psi^*\omega_U^{-1}\), not an inverse Jacobian.

The graph \(\Xi\) is a compact closed analytic subspace. Its actual line identity is \[\operatorname{pr}_V^*L|_\Xi\simeq \operatorname{pr}_{T'}^*R^r|_\Xi.\] Lemma 29 gives a root \(P_*\) of \(\operatorname{pr}_V^*L\) on a neighborhood of \(\Xi\), identified with \(\operatorname{pr}_{T'}^*R\) along \(\Xi\). Form the same two full normalized covers there. For the second cover use coefficients of \(s\) in meromorphic canonical generators from the first factor \(V\). This defines its relative tautological form without requiring a Cartier relative canonical line on the singular cover; changing to a generator after differential pullback changes the coefficient by a \(q\)th power and identifies the total root algebra and its normalization. Resolve and principalize the full tautological ideal functorially on this neighborhood, before cutting by the graph.

Near \(w_0\), choose a frame \(\bar p\) of \(R\) whose \(r\)th power is \(\psi^*\ell\) for a downstairs frame \(\ell\); a local root of a unit gives such a choice. Along the compact graph slice, compare the pullback of \(P_1\) with \(P_*\) by sending the boundary frame \(p_1\) to \(\bar p\). Lemma 29 extends this to an ambient root isomorphism as a germ. On the graph, its ratio with the prescribed comparison is a \(\mu_r\)-section equal to one on the compact slice, hence equal to one on an open graph neighborhood of that slice. Properness of \(\Xi\to T'\) permits a base shrink on which the ambient comparison is defined and agrees with the prescribed one along the entire graph: remove the closed image of the complement of that neighborhood. This argument also applies to a nonreduced graph. The comparison identifies the first cover with the local ordinary product cover, and product normality identifies the second cover as well. Apply the same fixed smooth-functorial principalization to the same full ideal and ordered boundary data on both sides; functoriality identifies the results. These are isomorphisms of resolved ambient germs over the product base preserving the whole divisor ideal, not only isomorphisms of their supports.

Let \(H'\) be the graph cut inside the reduced inverse section divisor on this global resolution. Locally it is exactly \(H^\flat\) above, so it is reduced SNC with the asserted pure dimension. It is compact, being over the compact \(\Xi\) by proper maps. The ambient projection is a submersion near it by the local product comparison. The transverse cuts of the globally smooth inverse-divisor components are smooth; split them into their disjoint connected components. Compactness and local finiteness leave only finitely many. Their closed intersections are smooth and compact, hence proper over \(T'\). Finally, the relative metric construction above, applied over the Kähler product neighborhood of \(\Xi\), gives one Kähler form near \(H'\). Its restrictions supply all the required relative Kähler forms. ◻

Proof. We use Saito’s analytic category of pure real Hodge modules (Saito 1990, sec. 1.4 and 1.8), which permits discrete real indices for the \(V\)-filtration. The pure properties supplied above are in that category. If the constant-source theorem is read with the stronger rationally indexed pure conditions, those imply the broad real specializability conditions, and induction on support dimension gives the pure clauses of Definition 1.8 as well. On a smooth dense analytic Zariski open \(Z^\circ\) of \(Z\), the piece \(Q\) gives a polarized real variation \(\mathbb H\) of weight \(w-\dim Z\), by Lemmas 1.9 and 1.13 of that source. Theorem 16.2.1 and Corollary 16.3.6 of (Sabbah and Schnell 2026) give a pure object \(B\in\mathrm{pHM}_Z(T,\mathbb R,w)\) extending this same variation with its real polarization. Use its prime holomorphic filtered component.

Both underlying complex regular holonomic modules are the intermediate extension \(\operatorname{IC}_Z(\mathbb H_\mathbb C)\). For the Saito filtered module use strict support and regularity in Remark 1.11 of (Saito 1990); for the Sabbah–Schnell component use (Sabbah and Schnell 2026, sec. 14.2.13 and Theorem 14.7.1). Regular Riemann–Hilbert and uniqueness of intermediate extension (Sabbah and Schnell 2026, Theorem 13.2.11 and Section 13.2.12) give the unique unfiltered identity extending the chosen identity on \(Z^\circ\). It remains to prove that this identity preserves \(F\) at the boundary.

On a smooth support \(U\) of dimension \(e\), write \(\mathscr H\) for the flat holomorphic bundle of the variation, with decreasing Hodge filtration. Its right filtration is \[ F_p(\omega_U\otimes\mathscr H)=\omega_U\otimes F^{-p-e}\mathscr H. \tag{113}\] Indeed the left filtration is \(F_p^L=F^{-p}\), and the side change is \(F_p^R=\omega_U\otimes F_{p+e}^L\). The same formula is recorded in (Sabbah and Schnell 2026, sec. 16.3.12). The right closed transfer (103) introduces no further shift. Thus the two right filtrations agree away from the boundary of \(Z^\circ\).

Work locally on \(T\). Choose a holomorphic \(g\) whose zero set contains that boundary but no local component of \(Z\), and split into the finitely many local strict-support factors if necessary. Such a \(g\) exists by prime avoidance: the boundary has smaller dimension than each local support branch. If the boundary is empty there is nothing to prove. Use the graph embedding in \(T\times\mathbb C_t\), and let \(Q_g\) denote the transferred module. Put \(b=\dim T\). Saito’s left convention uses increasing \(F^L\) and decreasing \(V_L^\alpha\) with \(t\partial_t-\alpha\) nilpotent on its \(\alpha\)-grade. On the graph ambient the conversion is \[ F_p^R=\omega_{T\times\mathbb C}\otimes F^L_{p+b+1}, \qquad V_L^\alpha\longleftrightarrow V^R_{-\alpha-1}. \tag{114}\] The second rule follows because transposition sends \(t\partial_t\) to \(-t\partial_t-1\). In particular \(V_L^{>-1}\) becomes \(V^R_{<0}\). The sign change of a normal \(\partial_t\) does not change a generated sum.

We verify the filtered surjectivity needed in Saito’s reconstruction. Put \(\Psi=\psi_{g,1}Q\) and \(\Phi=\phi_{g,1}Q\). Strict support gives a surjective \(\mathrm{can}:\Psi\to\Phi\) and an injective \(\mathrm{Var}\), by (Saito 1990, Proposition 1.5). Forgetting \(F\), this is the nilpotent middle-extension monodromy quiver of holonomic modules. Its monodromy filtrations centered at zero obey \[ \mathrm{can}(M_k\Psi)=M_{k-1}\Phi \tag{115}\] by (Sabbah and Schnell 2026, Lemma 3.3.13). The centers of the weight filtrations in Saito’s pure definition are \(w-1\) for \(\Psi\) and \(w\) for \(\Phi\). Thus (115) says \(\mathrm{can}(W_i\Psi)=W_i\Phi\). Condition (1.8.2) of Definition 1.8 and the direct-factor Lemma 1.15 of (Saito 1990) put both \(\operatorname{gr}_i^W\Psi\) and \(\operatorname{gr}_i^W\Phi\) in Saito’s pure real category of weight \(i\). The induced surjection on each grade is a filtered pure morphism, hence is \(F\)-strict by its Proposition 1.10.

At a stalk, take \(y\in F_p\Phi\) and choose \(i\) with \(y\in W_i\Phi\). Strictness on the \(i\)th grade lifts its class by an element of \(F_p\Psi\cap W_i\Psi\). Subtract its image and repeat on the next lower weight. The finite monodromy filtration terminates this procedure. It proves \[\mathrm{can}(F_p\Psi)=F_p\Phi\quad\text{for all }p.\] The finite lifting argument obtains filtered surjectivity from pure strictness on each weight grade.

Saito’s filtered middle-extension formula (Saito 1990, equation (1.7.2)) now applies: the local pure factor satisfies its specializability conditions, the zero set of \(g\) does not contain its support, and \(\mathrm{can}\) is filtered surjective. After (114), that formula is \[ F_pQ_g^R= \sum_{v\geq0} \left(V^R_{<0}Q_g^R\cap (j_{\mathrm{op}})_*F_{p-v}(Q_g^R|_{t\ne0})\right)\partial_t^v, \tag{116}\] where \(j_{\mathrm{op}}:T\times\mathbb C^*\hookrightarrow T\times\mathbb C\) and the intersection is inside \((j_{\mathrm{op}})_*j_{\mathrm{op}}^{-1}Q_g^R\). The strict-support module and its \(V\)-pieces embed there by restriction.

For the Sabbah–Schnell component, strict real specializability, pure support, and strictness of \(\mathrm{can}\) follow respectively from Definition 14.2.2, Theorem 14.2.19, and Corollary 14.2.23 of (Sabbah and Schnell 2026). Its unfiltered intermediate-extension property makes \(\mathrm{can}\) surjective, so it is a filtered middle extension. Definition 10.6.1, Remark 10.6.2, and Proposition 10.6.5 of the same source give precisely (116) for its right component: for \(\alpha<0\), the filtered \(V_\alpha\) is the intersection with the filtration off the divisor, and the full module is generated from \(V_{<0}\) by normal derivatives.

The unfiltered intermediate-extension identity preserves the canonical \(V\)-filtration and, by (113), preserves \(F\) off the divisor. The two right sides of (116) are therefore identical. This proves equality of \(F\) on the graph. Formula (103) recovers the module before graph transfer, and its filtration, as the sections annihilated by the normal ideal. Hence the identity is filtered on \(T\). The local identities glue by uniqueness of the unfiltered identity. The comparison uses the actual generic variation and its filtration; the two theories’ internal twist symbols need not have the same weight convention.

A pure object of Sabbah–Schnell is an object of \(\mathrm{WHM}(T)\) with its one-step weight filtration, by Section 14.2.14 of (Sabbah and Schnell 2026). Its Theorem 16.3.10, on the smooth projective \(T\), gives negative-ample graded de Rham vanishing for either filtered component. Sections 8.4.1–8.4.9 of that source identify the perverse right de Rham convention with the Spencer complex ending in the module in degree zero, with its filtration grading preserved. The theorem therefore gives (112) for every homogeneous degree \(p\) in our convention. This proves the lemma. ◻

Proof. On compact \(T\) the locally finite strict-support decomposition of each pure quotient of Proposition 34 is finite. Apply Lemma 35 to each factor. The finite filtration of \(M\) is \(F\)-strict, so its short exact sequences remain exact after \(\operatorname{gr}^F\operatorname{DR}_T\). Their hypercohomology long exact sequences give (112) for \(M\) itself. This deduction uses the comparison for the pure quotients and the strict filtration of \(M\).

By (98), the \(p=1\) right graded de Rham complex has exactly two terms: \[ \operatorname{gr}^F_1\operatorname{DR}_T M= \left[F_0M\otimes T_T\xrightarrow{\ \sigma\ }\operatorname{gr}^F_1M\right] \quad\text{in degrees }-1,0. \tag{118}\] Since \(N\) is invertible, its tensor is exact, and negative hypercohomology gives \[0=\mathbb H^{-1}\bigl(T,N^{-1}\otimes\operatorname{gr}^F_1\operatorname{DR}_T M\bigr) =H^0(T,N^{-1}\otimes\ker\sigma) =\operatorname{Hom}_{\mathcal O_T}(N,\ker\sigma).\] The two-term calculation uses no local freeness of \(\ker\sigma\), so it applies to torsion as well. ◻

Proof. Use (88) together with \[ 0\longrightarrow\mathcal T_{j+1,k-1}\longrightarrow\mathcal T_{j,k} \longrightarrow\mathcal A_j\longrightarrow0\quad(k>1). \tag{121}\] All are valid for negative \(j\) because \(I\) is invertible. Write \(\partial_{j,k}:g_*\mathcal T_{j,k}\to R^1g_*\mathcal A_{j+k}\) for the connecting map of the first sequence. The shorter orders make \(g_*\mathcal T_{j,k}\to g_*\mathcal A_j\) surjective. For \(k>1\), its kernel is the image of \(g_*\mathcal T_{j+1,k-1}\), by left exactness in the second sequence. The shorter order \(k-1\) in degree \(j+1\) lifts that image through \(g_*\mathcal T_{j+1,k}\), which maps into \(g_*\mathcal T_{j,k+1}\). The connecting map vanishes on the kernel. For \(k=1\), the leading map is the identity and its kernel is zero. This proves the unique factorization (119). If it is zero, exactness of the first sequence gives the desired epimorphism.

Here is the derivation calculation on germs. A leading section in degree \(j\) lifts to a section of \(\mathcal T_{j,k}\) after a base shrink. Choose local representatives \(x_\alpha\in I^j\) on an ambient open cover near \(E\). Their differences belong to \(I^{j+k}\), and their classes modulo \(I^{j+k+1}\) represent \(\delta_k(x)\). For a second section \(z\) of degree \(l\), take simultaneous representatives. On an overlap, \[x_\beta z_\beta-x_\alpha z_\alpha =x_\alpha(z_\beta-z_\alpha)+z_\alpha(x_\beta-x_\alpha) +(x_\beta-x_\alpha)(z_\beta-z_\alpha).\] The last term lies in \(I^{j+l+2k}\subset I^{j+l+k+1}\). The first two terms reduce to multiplication by the leading coefficients. Their Čech classes therefore give \[\delta_k(xz)=x\delta_k(z)+z\delta_k(x).\] This uses the ordinary multiplication action of layer sections on their first cohomology, and gives the asserted graded derivation.

For the chain rule, choose simultaneous shorter representatives \(G_{i,\alpha}\in\mathcal O_Z\) of the degree-zero sections \(t_i\). Their differences lie in \(I^k\). Shrink the local opens so that \(F(G_{1,\alpha},\ldots,G_{b,\alpha})\) is defined. Its holomorphic Taylor difference, modulo the square of the differences, is \(\sum_iF_{t_i}(G_\alpha)(G_{i,\beta}-G_{i,\alpha})\). The square lies in \(I^{2k}\subset I^{k+1}\), and the coefficient restricts to \(F_{t_i}(t)\) on \(E\). This gives (120) before any restriction to a generic parameter. It is valid when the target contains torsion. The connecting map and all representative calculations are natural for restrictions and the prescribed ambient root isomorphisms. ◻

Proof. All statements are on germs, so choose a common parameter shrink and simultaneous shorter lifts of the finitely many sections \(x,t_i\). Choose local ambient representatives near \(E\), indexed by \(\alpha\), with \[ x_\alpha\in I^{-a-k},\quad x_\beta-x_\alpha\in I^{-a},\qquad G_{i,\alpha}\in\mathcal O_Z,\quad G_{i,\beta}-G_{i,\alpha}\in I^k. \tag{126}\] They exist by the smaller orders in Lemma 37; for \(k=1\) the shorter truncation is already the layer. Pull them to \(\widehat Z\), suppressing pullback symbols, and put \[\eta_\alpha=x_\alpha\tau,\qquad Q_{i,\alpha}=\psi_i(w)-G_{i,\alpha},\qquad P_\alpha=\prod_i Q_{i,\alpha}.\] On \(H\) the \(G_{i,\alpha}\) restrict to the coordinate functions of \(h\). The \(Q_i\)’s and a reduced equation of \(H\) are therefore the regular graph-support equations. The logarithmic insertion and \(D_E=\rho^*E\geq H\) give the precise pole bounds \[\begin{align*} \eta_\alpha&\in\omega_{\widehat Z}(H+kD_E),\\ \eta_\beta-\eta_\alpha,\quad \eta_\alpha(G_{i,\beta}-G_{i,\alpha}) &\in\omega_{\widehat Z}(H),\tag{127}\\ \eta_\alpha(G_{i,\beta}-G_{i,\alpha})(G_{l,\beta}-G_{l,\alpha}), \quad(\eta_\beta-\eta_\alpha)(G_{i,\beta}-G_{i,\alpha}) &\in\omega_{\widehat Z}(H-kD_E)\subset\omega_{\widehat Z}. \end{align*}\] The last inclusion uses \(k\geq1\). Thus the quadratic and cross difference terms have no pole on the first-factor support \(H\).

In algebraic local cohomology consider the local sections \[S_\alpha=\left[\frac{\eta_\alpha\wedge dw}{P_\alpha}\right] \quad\text{of }K^\flat.\] The \(H\)-pole is already in \(\eta_\alpha\). These are finite-pole generalized fractions with a fixed ordering of the graph equations. To expand their differences, first quotient the localization along \(H\) by forms regular there, and then localize for the \(Q_i\)’s. Each polar class is killed by a power of a reduced equation of \(H\), and every \(G_{i,\beta}-G_{i,\alpha}\) is a multiple of that equation. The change from \(Q_{i,\alpha}\) to \(Q_{i,\beta}=Q_{i,\alpha}-(G_{i,\beta}-G_{i,\alpha})\) therefore has a finite geometric expansion on each polar class. By (127) only its linear terms survive. This proves the exact generalized-fraction equality on an overlap \[ S_\beta-S_\alpha= \left[\frac{(\eta_\beta-\eta_\alpha)\wedge dw}{P_\alpha}\right] +\sum_i\left[ \frac{\eta_\alpha(G_{i,\beta}-G_{i,\alpha})\wedge dw} {Q_{i,\alpha}P_\alpha}\right]. \tag{128}\] There is no infinite series or division by a Jacobian in this identity.

Let \(C\) be the first Čech cochain in (128). Let \(E_i\) have the numerator of its \(i\)th summand but only the denominator \(P_\alpha\), and let \(E_i^{(l)}\) denote that cochain with the additional denominator \(Q_{l,\alpha}\). The simple fractions \(C,E_i\) are cocycles: their reductions are respectively the cocycles for \(\delta_k(x)\) and \(x\delta_k(t_i)\); on a triple overlap the possible change of the representative of \(x\) is a cross term with no \(H\)-pole by (127). The description of the residue insertion on representatives and complete-intersection adjunction identify their classes with \(\widetilde c,\widetilde e_i\) in \(F_0M^\flat\). More explicitly, the residue of a log form \(\eta\) on \(H\), pulled with \(dw/dt\), is represented by \([\eta\wedge dw/P_\alpha]\); the determinant of the graph conormal gives exactly that frame. Thus this identification holds also at ramification. The cochains \(E_i^{(l)}\) need not individually be cocycles.

The numerator of \(E_i\) is independent of \(w\). The right canonical action is minus differentiation of the coefficient of the volume form, so at the cochain level \[ E_i\partial_{w_j}=\sum_l J_{lj}E_i^{(l)}. \tag{129}\] Write \(\check d\) for the Čech differential. Equation (128) is \(\check dS=C+\sum_iE_i^{(i)}\). Multiply it by \(\det J\) and use \(JJ^\#=(\det J)\mathrm{id}\) together with (129). The result, still at the cochain level, is \[ \check d(S\det J)=C\det J+ \sum_{i,j}(E_i\partial_{w_j})J^\#_{ji}. \tag{130}\] The placement of \(J^\#_{ji}\) after the right derivative is part of this exact formula.

These cochains lie in the end term of the relative right de Rham complex for the product projection, with first-factor coordinates held fixed. That term has no outgoing relative differential, so a Čech coboundary there is a total coboundary in direct image. The action in (129) induces the base right action there. Passing (130) to \(M^\flat\) gives (123). This uses the natural map from these Čech classes to direct image, and does not require that all direct-image classes be computed by this cover.

The term \(\widetilde c\det J\) is in \(F_0\). Moving a holomorphic function past a right vector field changes an operator by order zero. Taking \(\operatorname{gr}^F_1\) of (123) gives (124). With unchanged coordinates \(J=\mathrm{id}\), the exact formula itself gives (125). If \(b=0\), there are no \(Q_i\); the undivided difference of the \(\eta_\alpha\) in the support module of \(H\) is the cochain for \(c(x)\) and is a total coboundary. This proves the last assertion as well. ◻

Proof of Proposition 30. Assume the shorter orders and use Lemma 37. First suppose \(b>0\), and fix an arbitrary point \(t_*\in T=\mathbb P^b\). Proposition 33 gives the coordinate-power map \(\psi:T'=\mathbb P^b\to T\), the line \(R=\mathcal O_{T'}(1)\) with \(R^r\simeq\psi^*\mathcal O_T(1)\), and a single compact SNC graph support \(H'\subset\mathscr W\) with submersive projection \(p':\mathscr W\to T'\). It is unbranched over \(t_*\). Put \[K'=\mathcal H^{b+1}_{[H']}(\omega_{\mathscr W}),\qquad M'=\mathcal H^1p'_+K'.\] The global graph proposition verifies every hypothesis of Proposition 34: the support has pure dimension \(\dim V-1\), globally smooth finitely many components, and proper smooth closed intersections with one ambient Kähler form. Thus \(F_0M'=R^1h'_*\omega_{H'}\hookrightarrow M'\), in the absolute canonical convention, and Corollary 36 applies.

Take the local leading section \(x=u^{-m}\). On a local graph chart choose a frame \(\bar p\) of \(R\) with \(\bar p^r=\psi^*\ell\) for a downstairs frame \(\ell\). The local formula \[ \bar p^m\longmapsto \sum_{j,i}J^\#_{ji}\widetilde e_i(u^{-m})\otimes\partial_{w_j} \tag{131}\] defines a global morphism \[ R^m\longrightarrow F_0M'\otimes T_{T'}. \tag{132}\] We verify the transition, including on the branch locus.

Let \(t'=t'(t)\) and \(w'=w'(w)\) be changes of coordinates, and put \(A_t=\partial t'/\partial t\), \(B_w=\partial w'/\partial w\). These matrices are invertible. If \(\bar p'=\lambda\bar p\), then \(\lambda^r=\psi^*\mu\) for the downstairs unit \(\mu\) changing the frame of \(\mathcal O_T(1)\). Locally even at a branch point, \(\lambda\) descends as a holomorphic unit: take an \(r\)th root of \(\mu\) on a small downstairs neighborhood, and observe that the remaining ratio has \(r\)th power one and hence is locally constant. Denote the resulting downstairs unit also by \(\lambda\).

Compare the local root pairs by sending their new boundary frame to \(\lambda p_1\). Lemma 29 extends this power-compatible comparison to the ambient germs. Its pullback agrees with the transition of the global root near the graph, by the uniqueness and properness argument in Proposition 33. It therefore identifies both full covers, their tautological forms, and the fixed functorial principalizations. Under this identification \(u'^{-m}=\lambda^m u^{-m}\). In the definition of \(e_i(x)\), the factor \(x\) multiplies outside the derivation. Hence no derivative of \(\lambda\) occurs. Naturality and the chain rule (120) transform the downstairs column of \(e_i\)’s by \(\lambda^m A_t\).

The absolute graph factor changes by \(dw'/dt'=(\det B_w/\det A_t)\,dw/dt\). Consequently, for the pulled columns and Jacobians, \[\begin{align*} \widetilde e'&=\frac{\det B_w}{\det A_t}\lambda^m A_t\widetilde e, &J'&=A_tJB_w^{-1},\tag{133}\\ (J')^\#&=\frac{\det A_t}{\det B_w}\,B_wJ^\#A_t^{-1}, &(J')^\#\widetilde e'&=\lambda^m B_wJ^\#\widetilde e. \tag{134}\end{align*}\] The adjugate identity holds for every \(J\): it follows for invertible \(J\) from the inverse formula and then for all \(J\) as a polynomial identity. Only the coordinate changes \(A_t,B_w\) are inverted. Since the vector basis changes by \(B_w^{-1}\), the last equality is exactly the transition for the image of the frame \(\bar p^m\) in (131). This proves (132) on all of \(T'\). At ramification the classes are the natural Čech pullbacks under the resolved ambient germ comparisons; a flat base-change isomorphism for \(R^1h_*\omega_H\) is not used.

By Lemma 38, the image of (132) lies in \(\ker\sigma\). The line \(R^m=\mathcal O_{\mathbb P^b}(a+k)\) is ample because \(a+k>0\). Corollary 36 makes the global map zero. At a point above \(t_*\), the map \(\psi\) is locally biholomorphic and \(J^\#\) is invertible. The ambient product comparison then identifies the graph family and its Čech classes with the downstairs family. Thus the zero map implies \(e_i(u^{-m})=0\) as a downstairs germ near \(t_*\), including a class supported at that parameter. The point \(t_*\) was arbitrary, and the root comparisons allow any chart at it. It follows that these classes vanish locally everywhere.

The injection \(\tau_*\) and the invertible layer frame \(u^{-m}\) now imply \(\delta_k(t_i)=0\) on every chart: multiplication by that frame is an isomorphism from the layer \(\mathcal A_k\) to \(\mathcal A_{-a}\), including on first direct images. For any local \(x\) of degree \(-m\), all \(e_i(x)\) vanish. The exact unchanged coordinate identity (125) gives \(c(x)=0\) in \(M^\flat\). The injection (98) brings this back to \(R^1h_*\omega_H\), and the residue injection gives \(\delta_k(x)=0\). When \(b=0\), Lemma 38 gives \(c(x)=0\) directly, and the same two injections give this conclusion; there are no parameter coordinates to treat.

In particular \(\delta_k(u^{-m})=0\). The derivation rule for the invertible Laurent frame gives \[0=\delta_k(u^{-m})=-m u^{-m-1}\delta_k(u),\] so \(\delta_k(u)=0\). For any local \(F\in g_*\mathcal O_E\), the section \(Fu^{-m}\) also has degree \(-m\), and \[0=\delta_k(Fu^{-m})=u^{-m}\delta_k(F).\] Thus \(\delta_k(F)=0\). Every local graded section is \(Fu^j\), so \(\delta_k\) is zero in every integer degree. Lemma 37 closes the current order. Induction from \(k=1\) proves Proposition 30. ◻

Completion of Theorem 2. Define global divisorial subsheaves of meromorphic functions on the normal space \(V\) by \[ J_j=\mathcal O_V\left(-\left\lceil jG/r\right\rceil\right)\quad(j\in\mathbb Z). \tag{135}\] They agree in each root chart with \((\pi_*I^j)^{\mu_r}\). Indeed, at a prime over \(S_i\), an invariant meromorphic function of downstairs order \(v_i\) has order \((r/d_i)v_i\). Membership in \(I^j\) is equivalent to \[v_i\geq\left\lceil jd_i/r\right\rceil.\] At all other primes it is regular. The full invariant meromorphic algebra is the downstairs algebra, and normality extends the codimension-one test. Finite coherent pushforward and averaging by \(\mu_r\) are exact. This also proves coherence of the divisorial sheaves in (135) and of their quotients locally, and their valuation descriptions glue globally.

Since \(0<d_i\leq r\), the sheaf \(J_1=\mathcal O_V(-S)\) is the ideal of the reduced divisor \(S\). The same coefficient inequality gives \(J_1J_j\subset J_{j+1}\) for every \(j\). Hence \[\mathcal B_j=J_j/J_{j+1}\] is a coherent \(\mathcal O_S\)-module, and \(\mathcal B_0=\mathcal O_S\). Taking invariants of Proposition 30 preserves its epimorphisms: lift an invariant local section upstairs and average the lift. For \(l<n\), define the sheaf of complex vector spaces \[\mathscr F_{l,n}=f_*(J_l/J_n)\quad\text{on }T, \qquad F_j=f_*\mathcal B_j.\] Set also \(\mathscr F_{n,n}=0\). The quotients \(J_l/J_n\) are considered on the underlying space of \(S\), just as before. The invariant lifting surjections give exact sequences of complex vector space sheaves \[ 0\longrightarrow\mathscr F_{j+1,n}\longrightarrow\mathscr F_{j,n} \longrightarrow F_j\longrightarrow0\quad(j<n). \tag{136}\] In particular \(\mathscr F_{l,n}\) has a finite filtration with the full quotients \(F_j\), \(l\leq j<n\). Each \(F_j\) is a coherent \(\mathcal O_T\)-module by properness of \(f\); coherence as an \(\mathcal O_T\)-module is not asserted for \(\mathscr F_{l,n}\).

Cartier translation by \(G\) in (135), together with the actual line identity \(\mathcal O_V(G)=L\) defined by \(s\), gives \[J_{j-r}=J_j\otimes\mathcal O_V(G),\qquad \mathcal B_{j-r}=\mathcal B_j\otimes L|_S.\] Projection formula and (81) consequently give \[ F_{j-r}=F_j\otimes\mathcal O_T(1),\qquad F_0=f_*\mathcal O_S\ne0. \tag{137}\] This is an identity of the actual holomorphic lines and sheaves.

For \(N>0\), each \(-Nr\leq j<0\) is uniquely \(j=l-tr\) with \(0\leq l<r\) and \(1\leq t\leq N\). By (137), \(F_j=F_l(t)\). Analytic GAGA and Serre’s coherent finiteness and vanishing theorems (Serre 1956, sec. 3, no. 12, Theorems 1 and 3), (Serre 1955, sec. 66, Theorems 1 and 2) imply, for every \(i>0\), \[ \sum_{j=-Nr}^{-1}h^i(T,F_j) \leq C_i:=\sum_{l=0}^{r-1}\sum_{t\geq1}h^i(T,F_l(t))<\infty. \tag{138}\] These constants are independent of \(N\), and they are zero for \(i>\dim T\). The long exact sequences of the finite filtration (136) give the same bounds for \(h^i(T,\mathscr F_{-Nr,0})\), as well as finiteness and vanishing above that fixed dimension. Serre vanishing is being applied only to the coherent quotients \(F_l(t)\).

There is at least one section of \(F_0(t)\) for every \(t\geq1\). Choose a point of the nonempty \(S\) and a section of \(\mathcal O_T(t)\) nonzero at its image; its pullback is nonzero there. Thus \(\sum_{j=-Nr}^{-1}h^0(T,F_j)\geq N\). Additivity of the finite Euler characteristics in (136), and the uniform higher bounds for both the quotients and \(\mathscr F\), give a constant \(C\) independent of \(N\) with \[ h^0(T,\mathscr F_{-Nr,0})\geq N-C. \tag{139}\] For example one can take \(C=2\sum_{i=1}^{\dim T}C_i\).

Finally, \(J_{-Nr}=\mathcal O_V(NG)\) and \(J_0=\mathcal O_V\). Degree-zero direct image on the underlying support gives \[H^0(T,\mathscr F_{-Nr,0})= H^0(V,\mathcal O_V(NG)/\mathcal O_V).\] The exact sequence on \(V\) loses at most the fixed finite dimension \(h^1(V,\mathcal O_V)\) when lifting these quotient sections to \(H^0(V,\mathcal O_V(NG))\); finiteness follows from proper coherent direct image to a point. Equation (139) therefore makes these section spaces unbounded. For some \(N\) the line \(\mathcal O_V(NG)\) has two independent sections. Their quotient is a nonconstant meromorphic function on the irreducible normal \(V\), so its Iitaka dimension is at least one. Since \(\mathcal O_V(G)=L=\mathcal O_V(qA)\) is the actual adjoint identity, this proves \(\kappa(V,A)\geq1\), as claimed. ◻