Proof. The first assertion about currents follows from the definition of minimal multiplicity and Siu decomposition. Homogeneity and subadditivity follow by scaling and adding testing currents. The subtraction identity is the corresponding property in the big cone, followed by a small Kähler perturbation. More explicitly, for \(\alpha_\varepsilon=\alpha+ \varepsilon\{\omega\}\), subtract \(F_\varepsilon=\min\{F,N(\alpha_\varepsilon)\}\) coefficientwise. In the big cone all positive currents contain this divisor, so subtracting it translates each minimal multiplicity by its coefficient. As \(\varepsilon\downarrow0\), \(F_\varepsilon\to F\). The class of \(F-F_\varepsilon\) can be absorbed in a Kähler perturbation tending to zero (there are only finitely many components). This gives the upper bound for the asserted identity; subadditivity applied after adding \(F\) gives the reverse bound. Weak compactness of positive currents gives pseudo-effectivity of the limit. This is also the subtraction property of the divisorial decomposition in (Boucksom 2004, sec. 3).

For the strict-transform formula, pull almost-positive testing currents to \(Y\). Near the generic point of \(D'\), the map is an isomorphism, so the generic order is unchanged. This proves one inequality. Conversely, push a testing current on \(Y\) to \(X\). To control its negative error, write the error as a small multiple of the fixed positive current \(\mu_*\omega_Y\), and add a fixed smooth Kähler representative of \(C\omega_X-\{\mu_*\omega_Y\}\) for \(C\) sufficiently large. The result is a positive test in a Kähler perturbation tending to zero. The pushforward of \(\omega_Y\) has no divisorial order at the generic point of \(D\), where \(\mu\) is an isomorphism. Thus this test has the same generic order as the original one at \(D'\). Regularization, if needed, returns to tests with analytic singularities. This proves the other inequality and the formula. It is the analytic form of the strict-transform comparison in (Boucksom 2004, Lemma 5.8).

Finally, choose analytic-singularity tests for a modified-nef class whose generic order at the chosen prime tends to zero. Subtract that generic divisorial order before restricting to a resolution of the prime. The remaining singular set does not contain its generic point, so restriction gives an almost-positive current there. The subtracted multiple and the Kähler error tend to zero. Taking the limit proves the last assertion; see also (Boucksom 2004, Propositions 2.4 and 3.8). ◻

Proof. It suffices by Lemma 4 to prove \(E\leq N(\alpha)\). Write \[E-N(\alpha)=U-V, \qquad U,V\geq0,\] with no common component, and suppose \(U\neq0\). Its components are \(p\)-exceptional. Put \(n=\dim Y\) and \(\ell=\max\{\dim p(U_i):U_i\text{ a component of }U\}\). Normality gives \(\ell\leq n-2\). Let \(\eta\) be the pullback of a Kähler form on \(X\), let \(\omega\) be Kähler on \(Y\), and set \[q(\gamma,\delta)= \int_Y\gamma\,\delta\,\eta^{\ell}\omega^{n-\ell-2}.\] The restriction property in Lemma 4 gives \(q(Z(\alpha),U)\geq0\). The term \(q(p^*M,U)\) is zero: on each component of \(U\), it contains \(\ell+1\) factors pulled back from an image of dimension at most \(\ell\). Distinct effective divisors have nonnegative intersection against the semipositive and Kähler factors in \(q\). Hence \[0\leq q(Z(\alpha),U)=q(U,U)-q(V,U)\leq q(U,U).\] On the other hand, \(q(U,\eta)=0\) by the same dimension count, whereas \(q(\eta,\eta)>0\). The mixed Hodge index theorem, applied first with Kähler factors and then by a semipositive limit, says that \(q\) is negative semidefinite on \(\eta^\perp\). It follows that \(q(U,U)=0\) and that \(U\) is in the radical of \(q\): the radical assertion follows from negative semidefiniteness on \(\eta^\perp\), and then from \(q(U,\eta)=0\). But \[q(U,\omega)=\sum_i u_i\int_{U_i} \eta^{\ell}\omega^{n-\ell-1}>0.\] Indeed, a component with image dimension \(\ell\) has strictly positive integrand on a dense open. This is a contradiction. The mixed Hodge index statement used here is the mixed Hodge–Riemann relation (Dinh and Nguyen 2006, Theorems A and C); the limit preserves the assertion that there is at most one positive direction. ◻

Proof. Subadditivity and nefness give \(N(\mu^*\alpha)\leq\mu^*R\). By the strict-transform formula, the difference \(U=\mu^*R-N(\mu^*\alpha)\) is effective and exceptional. Moreover \(Z(\mu^*\alpha)=\mu^*\beta+\{U\}\) is modified nef. Apply Lemma 5 to this class. It gives \(U\leq N(Z(\mu^*\alpha))=0\), proving the assertion. ◻

Proof. The divisor current \([\operatorname{div}(s)]/m\) is a positive current in \(c_1(L)\), so Lemma 4 gives the order bound. In the stated decomposition, every section therefore divides holomorphically by the canonical section of \(mR\). Conversely, multiplication gives a section of \(mL\). The actual rational line identity makes these operations inverse after clearing denominators. ◻

Proof. Write \(X\) for the manifold. The assertion is immediate if \(\dim X=0\), so assume \(\dim X>0\). Every positive current \(T\) in \(c_1(L)\) contains \([R]\). The residual \(T-[R]\) is positive with zero cohomology class, so its mass against a Kähler form to the power \(\dim X-1\) is zero. Thus it vanishes. For the second assertion, add \([F]\) to any positive current in \(c_1(L-F)\). Uniqueness gives \(F\leq R\), and subtraction in Lemma 4 gives the asserted negative part. A torsion line has zero Chern class and becomes trivial after taking a positive multiple, so the last statement is the same argument. ◻

Proof. With compatible canonical choices there is an actual rational divisor identity \[K_Y+B_Y=p^*J+E_p,\qquad E_p=\sum_{E\text{ exceptional}}a(E;X,\Delta)E\geq0.\] Suppose \(q:V\to Y\) realizes the decomposition for the left side. Lemma 5, applied over \(X\), gives \[N\bigl(q^*(K_Y+B_Y)\bigr) =N(q^*p^*J)+q^*E_p.\] Cancelling \(q^*E_p\) in the actual rational line identities proves the first assertion. This argument also shows that further resolutions using the reduced-exceptional convention do not change the assertion \(\mathcal G_n\).

If \(J\) is nef, its pullback is nef, so its negative part is zero. The semiample line in the decomposition is therefore \(q^*p^*J\) itself, up to rational line isomorphism. Choose an actual Cartier multiple generated on \(V\). Normality gives \((p q)_*\mathcal O_V=\mathcal O_X\), and projection formula identifies its sections with the sections downstairs. A base point downstairs would make every pullback section vanish on its nonempty fiber. There is no such point, so that multiple of \(J\) is globally generated. ◻

Proof. At \(x\in X\), choose a holomorphic frame of \(L\) and put \[R=\mathcal O_{X,x},\qquad K=\operatorname{Frac}(R),\qquad A=(\nu_*\mathcal O_Z)_x.\] The generic algebra \(A\otimes_RK\) is a product of finite field extensions of \(K\), of total dimension \(d\). The local meromorphic coefficient \(a\) of \(s\) belongs to this algebra. Since \(Z\) is irreducible and \(s\) is not identically zero, its restriction is nonzero on every generic local component. Thus \(a\) is nonzero in every field factor, and the determinant of multiplication by \(a\) is nonzero. Under a change of frame by a unit \(g\), this determinant changes by \(g^{-d}\). The determinants therefore glue to the asserted meromorphic section. The valuation formula for a norm over a discrete valuation ring gives the displayed divisor identity at every prime of the normal space \(X\).

If \(s\) is holomorphic, then \(a\in A\) is integral over \(R\). Its norm is integral over \(R\) and belongs to \(K\), hence belongs to \(R\) by normality. This does not require \(A\) to be locally free. If \(s\) is nonzero at every point over \(x\), then \(a\) is a unit in the finite semilocal algebra \(A\); applying the same argument to \(a^{-1}\) shows that its norm is a unit in \(R\).

For a meromorphic function \(f\), use instead the determinant of \(T\) minus multiplication by its local coefficient. These monic polynomials agree on overlaps, and Cayley–Hamilton shows that they annihilate \(f\). All these constructions use fractions of local analytic germs, so they remain available when the global meromorphic functions on \(X\) are constant. ◻

Proof. Replace \(Y\) by the smooth model in the hypothesis and pull \(F\) back. Write \(e^*L+F\sim_{\mathbb Q}R=N(e^*L+F)\). Pull back a positive current in \(c_1(L)\) and add \([F]\). Lemma 8 gives \(F\leq R\). Put \(D_Y=R-F\). Subtraction in Lemma 4 gives \[e^*L\sim_{\mathbb Q}D_Y=N(e^*L)\geq0.\] In particular \([D_Y]\) is the unique positive current in \(c_1(e^*L)\).

Choose \(m\) so that \(mL\) and \(mD_Y\) are integral and the latter is the divisor of a holomorphic section \(s\) of \(e^*(mL)\). Factor \(e\) through its normal Stein space: \[Y\xrightarrow{h}Z\xrightarrow{\nu}X, \qquad \nu\text{ finite of degree }d=\deg e.\] The connected birational map \(h\) descends \(s\) to a holomorphic section \(s_Z\) of \(\nu^*(mL)\), by normality and projection formula. Lemma 10 gives a holomorphic section of \(mdL\) whose divisor is \[\operatorname{div}(\operatorname{Nm}_\nu(s_Z)) =\nu_*\operatorname{div}(s_Z)=m e_*D_Y.\] Thus \(D=d^{-1}e_*D_Y\) is effective and satisfies \(L\sim_{\mathbb Q}D\).

Both \(e^*D\) and \(D_Y\) are positive divisor currents in \(c_1(e^*L)\), so uniqueness gives \(e^*D=D_Y\). We already know \(N(L)\leq D\). If this inequality were strict at a prime \(A\subset X\), choose analytic-singularity tests for \(L\) whose generic orders at \(A\) tend to \(\nu_A(L)\), and pull them to \(Y\). For any prime \(A'\) dominating \(A\), the generic orders are multiplied by \(\operatorname{mult}_{A'}e^*A\). The pulled-back Kähler errors can be enlarged to Kähler errors upstairs tending to zero. Consequently \[\nu_{A'}(e^*L) \leq \operatorname{mult}_{A'}(e^*A)\,\nu_A(L) <\operatorname{mult}_{A'}(e^*A)\,\operatorname{coeff}_A(D),\] contrary to \(N(e^*L)=D_Y=e^*D\). This proves \(D=N(L)\). ◻

Proof. The good-model assertion is the rational-boundary consequence of the full argument in (OpenAI 2026c, Proposition 2.5, Theorem 9.6, and Section 10). That argument establishes the additional inductive and nonvanishing premises of its Proposition 2.5 before applying it, and proves a stronger real-boundary induction. Its only use of Assumption 1 is in its Lemma 6.1, on a resolved projective Albanese fibration with both boundaries zero. Thus its premise is precisely available here.

For clarity, its good-model conclusion gives on a common smooth projective resolution \[u^*(K_X+B)=v^*(K_{X_m}+B_m)+F, \qquad F\geq0\text{ and }v\text{-exceptional},\] as an actual rational divisor identity; the line on \(X_m\) is semiample. This is the comparison in (OpenAI 2026c, Lemma 2.2). Algebraic nefness of a rational line on a projective variety implies analytic nefness, by adding arbitrarily small ample rational classes. Lemma 5 therefore identifies \(F\) with the analytic negative part of the left side. On a smooth projective variety, analytic pseudo-effectivity of a divisor class agrees with algebraic pseudo-effectivity (Boucksom et al. 2013, Proposition 1.2); hence the good-model assertion applies to every projective instance of \(\mathcal G_n\). This proves the stated form of \(\mathcal G_n\). If the original adjoint is nef, the same comparison and normal descent give its semiampleness. ◻

Proof. We use Propositions 13 and 14, whose proofs occupy the rest of the paper. The assertion is immediate in dimension zero. Assume it in smaller dimensions and let \((X,B)\) be as in Definition 3 in dimension \(n>0\).

If \(a(X)=n\), the Kähler Moishezon theorem makes \(X\) projective, and Proposition 12 applies. If \(0<a(X)<n\), the algebraic reduction of \(X\) is a dominant meromorphic map to a projective model of dimension \(a(X)\) (Campana and Peternell 1999, Theorem and Definition 3.1). Proposition 13 applies.

It remains to consider \(a(X)=0\). If \(X\) is simple, use Proposition 14. Otherwise Campana’s maximal covering-family theorem supplies a generically finite evaluation map from an incidence space which itself has a nontrivial fibration; see (Campana 2026, Lemma 17 and Corollary 18). After resolving the incidence, the graph, and the base, we obtain a smooth compact Kähler source \(X'\) and maps \[e:X'\longrightarrow X,\qquad f:X'\longrightarrow Y', \qquad 0<\dim Y'<n,\] where \(e\) is proper generically finite and \(Y'\) is in class \(\mathcal C\) (Fujiki 1982, (1.1.1), (1.5)(3)–(4), (2.2)(1), (3), and (2.3)). The algebraic dimension of \(X'\) is zero. Indeed, a meromorphic function on \(X'\) descends through the birational part of the normal Stein factorization of \(e\). Lemma 10 gives its characteristic polynomial over the finite part, with meromorphic coefficients on \(X\). These coefficients are constant because \(a(X)=0\), so the function is constant on the irreducible space \(X'\). Bimeromorphic modifications do not change this conclusion.

Choose a reduced SNC divisor \(B'\) containing the inverse image of \(\mathop{\mathrm{Supp}}B\), after a further resolution. Pullback of logarithmic differentials gives the actual rational identity \[K_{X'}+B'=e^*(K_X+B)+F,\qquad F\geq0.\] For example, this follows locally by pulling back logarithmic top forms for the reduced support of \(B\); decreasing its coefficients to those of \(B\) only increases the error. Thus the adjoint upstairs is pseudo-effective. Proposition 13 applies to \((X',B')\). Its semiample positive part is torsion, since a nonconstant semiample map would give a nonconstant meromorphic function on \(X'\). Proposition 11 now gives the required decomposition for \(K_X+B\). This exhausts the cases and proves the induction. ◻

Proof of Theorem 2. Apply Theorem 15 on a log resolution of the given lc pair, with the reduced-exceptional boundary. The adjoint is pseudo-effective because the original one is nef. Proposition 9 then gives generation of an actual Cartier multiple at every point of the original normal space. ◻

Proof. These are the semiampleness, proper relative model and finite-model assertions of Theorem 70, proved below by dimension induction independently of \(\mathcal G\). Finite marked weak-model geography is recorded in Lemma 69. Projective analytic model constructions used in that induction are supplied by Proposition 40. None of these statements promotes an undetected generalized-ray contraction to a projective map. ◻

Proof. For dlt input, decrease the floor by a sufficiently small positive rational multiple. The resulting pair is klt and remains \(f\)-antiample. For this klt pair the base point free hypothesis holds for \(L\), because \(L\) has zero fiber degrees and the negative adjoint is relatively ample. Over a smaller base neighborhood every sufficiently high power of \(L\) is therefore generated. The associated morphism is constant on each connected fiber: its tautological line has degree zero on every fiber curve, and every positive-dimensional projective image contains a curve. Its graph then factors through \(Z\); the graph projection is finite bimeromorphic and \(Z\) is normal. Descend two consecutive generated powers \(L^k,L^{k+1}\) to lines \(M_k,M_{k+1}\). The line \(M_{k+1}\otimes M_k^{-1}\) pulls back to \(L\). Projection formula identifies this quotient with \(f_*L\), so these local descents and their evaluation maps agree on overlaps. Pull the descended line to the positive side of a flip. This is the argument of (OpenAI 2026b, Lemma 7.4). ◻

Proof. We give the reduction to the local finite-generation input, including why it yields a single finite program over the compact base. Let \(\mathcal N_{\mathbb R}\) be the finite-dimensional space of degrees of global rational lines on curves over \(V\), and put \(r=\dim\mathcal N_{\mathbb R}\). If \(r=0\), then \(J\) is already curve-nef. Assume \(r>0\). Choose rational lines \(H_1,\ldots,H_r\) whose degree classes form a basis and such that both \(H_i\) and \(J+H_i\) are relatively ample. Such a basis exists because the relatively ample classes whose sum with \(J\) is also relatively ample form a nonempty open set containing sufficiently positive classes. Define the rational simplex of adjoints \[\Pi=\operatorname{conv}\{J,J+H_1,\ldots,J+H_r\}.\] We will choose a general direction \(H=\sum_i\theta_iH_i\) with \(\theta_i>0\) and \(\sum_i\theta_i=1\). Its entire scaling segment lies in this simplex: \[J+tH=(1-t)J+\sum_i t\theta_i(J+H_i)\in\Pi \qquad(0\leq t\leq1),\] and it lies in the relative interior for \(0<t<1\).

We produce the klt representatives for these exact vertices on each sufficiently small Stein neighborhood separately. A relative Cartier line and its inverse have local effective representatives there: their proper direct images are coherent of generic rank one, so Cartan generation supplies nonzero local sections. This uses that \(\pi\) is bimeromorphic. Represent each \(H_i\) by a divided general free effective divisor \(G_i\sim_{\mathbb Q}H_i\), using a sufficiently divisible relatively generated multiple. These finitely many general divisors preserve klt with \(B\). Choose a relatively ample integral line \(A\), and local effective representatives \(A^+\sim A\), \(A^-\sim -A\) by the preceding argument. For one sufficiently small positive rational \(\epsilon\), all boundaries \[\Delta_0=B+\epsilon(A^++A^-),\qquad \Delta_i=B+G_i+\epsilon(A^++A^-)\] are effective and klt. Their adjoints represent \(J\) and \(J+H_i\), respectively, and \(\epsilon A^+\) is a common effective relatively ample summand. In particular, every rational adjoint in \(\Pi\) is relatively big.

On this Stein member, choose one simultaneous projective log resolution of the finite family. Give each exceptional prime a coefficient in \((\max\{0,c_0,\ldots,c_r\},1)\), where the \(c_j\) are its crepant coefficients for the vertex pairs. If \(E\geq0\) is resolution-exceptional with \(-E\) resolution-ample, then subtracting a sufficiently small multiple of \(E\) from the pullback of \(\epsilon A^+\) leaves a relatively ample rational divisor. A sufficiently small positive rational multiple of this divisor can be removed from every boundary while leaving all coefficients effective and below one; the strict exceptional margins ensure this along the exceptional primes. A divided general free representative of this common summand preserves the simultaneous SNC condition. The smooth multigraded theorem now applies to these vertices. Exceptional corrections leave the local adjoint rings unchanged by projection to a normal space; clearing the finitely many line identifications simultaneously makes them multiplicative. This is the reduction in (OpenAI 2026b, sec. 7.2), using (Das et al. 2024, Theorem 3.1).

Here are the two consequences of finite generation that we need. The normalized main relative Proj of any rational adjoint \(\Theta\in\Pi\) extracts no prime divisor, and its trace is a relatively ample rational line. Indeed, resolve the degree-one base ideal after a common Veronese and write \[p^*(m\Theta)\sim q^*H_\Theta+G,\qquad G\geq0,\] with \(H_\Theta\) tautological and relatively ample. Generation says that all sections in degree \(k\) vanish at least along \(kG\). If a component of \(G\) were not \(q\)-exceptional, relative generation of \(q_*\mathcal O(G)\otimes H_\Theta^k\) would supply a section with smaller vanishing there. If a \(p\)-exceptional prime \(Q\) were not \(q\)-exceptional, the same argument with \(q_*\mathcal O(Q)\otimes H_\Theta^k\) would contradict \(p_*\mathcal O(Q)=\mathcal O_T\). This proves nonextraction. There are also only finitely many marked normalized main Proj models, where the marking records their common bimeromorphic open over \(V\). To see this, choose finitely many homogeneous generators locally. Proj charts of each diagonal ring use homogeneous monomials as denominators; the supports of those monomials in the finite set of generators determine the degree-zero localizations and their gluing. Only finitely many support patterns occur. Normalization and taking the main component preserve this finiteness. A finite cover of the compact base gives finitely many global marked models, since marked local identifications agree on the common dense open and glue uniquely. These are (OpenAI 2026b, Lemmas 7.5 and 7.6).

For completeness, we construct the scaling program controlled by this finite list. On a working model, let \(A_0\) be a relatively ample line. The closed curve cone in the dual of its global degree space has compact slice \(A_0\cdot z=1\): for every line \(D\), both \(kA_0+D\) and \(kA_0-D\) are relatively ample for \(k\) large, which bounds every coordinate. Local cone truncations on the finite Stein cover express the negative part of this slice, after any positive ample truncation, using finitely many actual curve classes. A negative extremal ray can therefore be separated by a rational nef support annihilating just that ray. A positive multiple of the support minus the klt adjoint is relatively ample. Relative base point freeness contracts exactly the ray. If the contraction is divisorial, its single exceptional prime and Lemma 18 prove the global strong property downstairs. If it is small, the relative Proj of the adjoint is its small positive model. For any global line \(D\) on the negative model, killing its ray degree and applying the same lemma gives an actual rational identity \[D\sim_{\mathbb Q}cJ+f^*D_Z,\qquad c\in\mathbb Q.\] It transforms to the positive side, proving the global strong property there. This is the continuation construction of (OpenAI 2026b, Proposition 7.7); it also preserves projectivity over \(V\) and compact Kählerness.

The transforms of \(H_1,\ldots,H_r\) continue to span the degree spaces: use pullback in a divisorial step and the preceding decomposition of each line in a small step. A line of zero global relative degrees stays so by Lemma 18; curves in the contraction bases can be lifted through projective morphisms. There are only countably many possible finite sequences of curve rays, and each determines its contractions and flips uniquely. We may therefore choose \(H\) in the relative interior of \(\operatorname{conv}\{H_1,\ldots,H_r\}\) so that it avoids all ties between independent ray degrees in advance. At a positive nef wall, write \(w=J+tH\) for the traces on the current model, and choose a preceding parameter \(s>t\) for which \(J+sH\) is relatively ample. On the zero face of the normalized compact curve slice, \[J=-\frac{t}{s-t}(J+sH)\] in degrees. Thus \(J\) is uniformly negative on that face. Choose the positive ample truncation so that its compact remainder \(K\) misses the face. The finite rational polyhedral test and the general choice of \(H\) make the face a single ray \(R\). The wall class has a positive minimum on \(K\) and is positive on every other truncated curve generator. Hence a sufficiently small open neighborhood of \(w\) inside the annihilator \(R^\perp\) consists of nef classes with zero face exactly \(R\).

Suppose first that \(R^\perp\) has positive dimension. It is a rational hyperplane because \(R\) is represented by an integral curve. Choose a rational simplex of support classes in that neighborhood containing the degree class of \(w\) in its relative interior, and represent each vertex by a rational line \(N\). For each such \(N\), compactness gives an integer \(a>0\) for which \(aN-J\) is relatively ample. Relative base point freeness contracts exactly \(R\) and descends \(N\) to a relatively ample rational line on the contraction base. These bases agree because the contracted curves agree. The difference between \(w\) and the corresponding convex combination of these rational lines has zero relative degree. In the finite rational span of the lines involved, the degree map has rational kernel, since rational lines have rational degrees on integral curves. The difference is therefore a real combination of rational lines of zero relative degree. Lemma 18 descends these lines, and lifting curves from the base shows that their descents still have zero degree over \(V\). They do not affect relative ampleness. Convexity now shows that \(w\) descends to a relatively ample real line class over \(V\). If \(R^\perp=0\), the same zero-degree descent applies to \(w\); the contraction base has no curves over \(V\), hence is finite over \(V\), and relative ampleness is automatic.

After a flip, let \(w^+\) be the pullback of this wall class to the positive side. For sufficiently small \(\delta>0\), \[J^++(t-\delta)H^+ =\left(1-\frac{\delta}{t}\right)w^+ +\frac{\delta}{t}J^+\] is relatively ample over \(V\): the wall is pulled back from a relatively ample class on the contraction base, and \(J^+\) is relatively ample over that base. After a divisorial contraction, openness of the ample cone on the base gives the same nonempty interval of relative ampleness.

Each working model is consequently one of the marked ample models already counted. Indeed, for a fixed finite prefix and an interior parameter \(0<t<1\), replace the direction and parameter by nearby rational ones. They remain in \(\Pi\), and all strict signs in that prefix and the final relative ampleness persist; the associated relative section ring has this working model as its Proj. A repeated marked working model would have the same trace of \(B\), because all steps are nonextracting, and hence the same discrepancies. This contradicts the strict increase in Equation (2) at an intervening nontrivial step. The finite list therefore forces termination. The last adjoint is curve-nef, since the continuation construction applies whenever it is not. Composing Equation (2) proves the final comparison. ◻

Proof. Choose a small rational \(\epsilon>0\) for which \((S,\Delta-\epsilon D)\) is effective and klt. On a projective log resolution \(r:W\to S\), use the strict lowered boundary and give every exceptional prime a coefficient strictly between its crepant coefficient and \(1\), and at least zero. The resulting klt adjoint has the actual form \[K_W+\Gamma\sim_{\mathbb Q}r^*(K_S+\Delta-\epsilon D)+F, \qquad F\geq0,\] where every \(r\)-exceptional prime has positive coefficient in \(F\). Apply Proposition 19 over \(S\). On its endpoint the trace \(F_Y\) is effective, exceptional over \(S\), and relatively nef. Negativity makes \(F_Y=0\). Since all initially exceptional primes had positive coefficient, \(Y\to S\) is small and the lowered adjoint is crepant. Restoring \(\epsilon D\) gives the full crepant identity.

A projective small morphism to a smooth germ is an isomorphism: transport a relatively ample line to the smooth germ, where it is Cartier; smallness makes the original line its pullback, contradicting relative ampleness on a nontrivial fiber. Thus \(h\) is an isomorphism over an SNC open meeting all lc centers of the dlt pair. The full crepant comparison then proves dlt on \(Y\). ◻

Proof. Write \(H=J\) in the first case and \(H=J+bP\) in the second. Fix \(\omega\) with \(\{H\}+\omega\) Kähler. For each \(0<\eta<1\), choose a positive rational \(\epsilon\) so small that, with \(F=\lfloor B\rfloor\), \[(T,B-\epsilon F)\ \text{is klt},\qquad \eta\omega+\epsilon\{F\}\ \text{is Kähler}.\] For \(\lambda\in[\eta,1]\) use the presentations \[\{H\}+\lambda\omega =\{K_T+B-\epsilon F\} +\bigl(b\{P\}+\lambda\omega+\epsilon\{F\}\bigr),\] omitting \(b\{P\}\) in the first case. Their b-data are globally nef on the fixed initial carrier, and their boundary-plus-nef traces are globally modified big. These are effective gklt data.

We first construct a step whenever the unscaled working trace is not nef. Choose a positive shift strictly below the current wall and use the gklt presentation of its current trace \(A\). The ray at the wall is \(A\)-negative. On a compact slice of the analytic curve cone, the cone theorem makes the \(A\)-negative part locally polyhedral after a sufficiently small positive Kähler truncation. Separating the chosen extremal ray from the other finitely many rays in that truncation gives a Kähler class \(\omega_R\) such that \(A+\omega_R\) is nef and its zero face is precisely this ray; see the supporting construction in Proposition 63. Add the pullback of this new Kähler class to the fixed nef carrier. The resulting adjoint still has globally nef data and modified-big boundary-plus-nef trace, so the nef assertion of Proposition 17 supplies its supporting contraction. This uses a new supporting direction on the current space; it does not require the trace of the original scaling direction to be Kähler. We verify a rational ordinary detector before asserting projectivity or constructing a flip. In the second case nefness of \(P\) makes \(J\) negative on the ray; the first case has this property directly. Lower the floor by a small rational amount, retaining negativity of \(J_\epsilon\), so that \(-J\cdot R\leq2(-J_\epsilon\cdot R)\). The ordinary klt cone length bound gives a rational curve generator \(C\) with \[0<-J\cdot C\leq4d.\] In the second case, if \(P\cdot C>0\), integrality of \(mP\) gives \(P\cdot C\geq1/m\), contradicting \[H\cdot C\geq-4d+b/m>0.\] Thus in that case the ray is \(P\)-trivial. The lowered ordinary adjoint is a global rational detector. The detected projectivity and flip constructions in Lemma 31 and Proposition 33 supply the projective ordinary step. The difference of \(J\) and a positive rational multiple of the lowered adjoint has zero contracted degree; exact descent in Lemma 18 shows that restoring the floor preserves the positive-side sign. This is the ordinary replacement of Proposition 34, and the full pair remains dlt by Equation (2).

In the second case, Lemma 18 descends the actual Cartier line \(mP\) without changing \(m\). Analytic nefness descends and pulls back under these projective contractions (Hacon and Xie 2026, Lemma 2.44). A generated multiple descends as a generated line by projection formula and pulls back as one. Thus the same length estimate and the same \(b\) apply at the next step. The corresponding assertion on a final Mori ray follows from exactly the same estimate. For a semiample \(P\), the map of its fixed generated multiple is constant on each connected projective contracted fiber: otherwise its image contains a curve of positive degree. The map therefore factors through the normal contraction base, including in the fiber-type case.

The traces used in this construction are linear by the degree-zero descent description preceding Lemma 18. At a wall the scaled class is pulled back from its base. The comparison for an earlier parameter is a nonnegative multiple of the ordinary adjoint comparison. Consequently the gklt presentations on any fixed positive segment remain gklt along its program, and every working model is a weak model for a member of that segment. The finite marked weak-model assertion in Proposition 17 applies. A marked model cannot recur: its boundary trace would be the same, whereas a nontrivial intervening ordinary step strictly increases a discrepancy. There are therefore only finitely many steps with threshold at least \(\eta\).

For decreasing cutoffs, any already-constructed finite prefix remains valid. The floor perturbations cancel in the displayed classes and continue to cancel under their linear transforms; their nonpositive comparisons preserve the new gklt data. Starting at its last nef wall therefore extends the prefix to the smaller cutoff. This gives one compatible scaling program. A positive limiting threshold would lie in a finite positive truncation, so an infinite program has limit zero. Pseudo-effectivity excludes a negative Mori endpoint. If \(H\) is not pseudo-effective, its pseudo-effective threshold in the initial Kähler direction is positive. Every nef working trace, together with its effective exceptional comparison, makes the initial scaled class pseudo-effective, so the scaling thresholds are bounded below by that positive number. The program is therefore finite. Its endpoint cannot be nef, since that comparison would make \(H\) pseudo-effective; it is the stated Mori fiber contraction. ◻

Proof. For \(\lambda\) in the working interval of \(T_i\), every earlier threshold is at least \(\lambda\). Each earlier step is therefore negative or trivial for this parameter. Composing its comparisons gives Equation (3). Since \(\alpha_i(\lambda)\) is nef, Lemma 5 identifies the exceptional divisor with the negative part.

On the fixed resolution \(U\), choose a Kähler class \(\Omega\) and \(c>0\) such that \(c\Omega-r^*A\) is Kähler. Monotonicity of each minimal multiplicity under addition of a nef class gives \[\nu_Q(r^*\{H\}+\lambda c\Omega) \leq \nu_Q(r^*(\{H\}+\lambda A)) \leq \nu_Q(r^*\{H\}).\] The left side tends to the right side by the definition using small Kähler perturbations. This proves the convergence on \(U\), and the same argument applies on a resolution representing any chosen divisorial place. ◻

Proof. Lemma 6 makes the stated equality valid on any higher resolution. Choose \(b\) as in Lemma 21. Adding \(bP\) leaves the same negative part. Indeed, on a smooth resolution write \(\alpha=\pi^*\{J\}\), \(p=\pi^*\{P\}\), and \(e=\pi^*E=N(\alpha)\). Subadditivity and homogeneity of negative multiplicities give \[\begin{align*} N(\alpha+bp)&\leq e,\\ (1+b)e=N((1+b)\alpha) &\leq N(\alpha+bp)+bN(\{e\}) \leq N(\alpha+bp)+be. \end{align*}\] Thus \(N(\alpha+bp)=e\).

Run the scaling for \(H=J+bP\). Let \(Q\) be the strict transform on a fixed resolution of a prime component of \(E\). Its multiplicity in \(N(\pi^*\{H\})\) is positive. The limit in Lemma 22 makes its multiplicity positive for every sufficiently small positive parameter. If that prime still survived on the corresponding working model, Equation (3) would make its multiplicity zero, because the comparison there is exceptional over that model. It must therefore be contracted after finitely many steps. The same conclusion holds if a nef endpoint is reached before taking the limit. There are only finitely many components of \(E\), so after a finite prefix all have disappeared.

Throughout the program \(P\) descends and remains nef. The actual identity \(J_i\sim_{\mathbb Q}P_i+E_i\), with \(E_i\) the codimension-one pushforward of \(E\), follows from the reflexive trace convention. Once \(E_i=0\), we have \(J_i\sim_{\mathbb Q}P_i\); since \(P_i\) is nef, the program stops. Conversely, a divisorial negative step can contract only a prime of \(E_i\): its ray is \(P_i\)-trivial, so the effective divisor \(E_i\) has negative degree on every contracted curve, and those curves are contained in its support. This proves the exact assertion about contracted primes. ◻

Proof. For a modification of a smooth compact Kähler manifold, the degree-two modification theorem gives these decompositions, and the exceptional classes have type \((1,1)\). Suppose they hold at one step. Its contraction base and both working spaces have rational singularities: slight lowering of the rational Cartier floor makes the ordinary pair klt, and the detected contraction has a base with rational singularities. These contractions are bimeromorphic, so a common resolution and Leray give \(R^if_*\mathcal O=0\) for \(i>0\); this verifies the additional hypothesis in Lemma 30.

The ray of this detected contraction is one ray for Bott–Chern degrees. This also suffices for degree-two classes. Indeed, pull a class \(\gamma\in H^2(V,\mathbb R)\) to a smooth resolution \(p:U\to V\) and take its Hodge decomposition. By the decomposition already proved for this working model in Equation (5), its \((1,1)\)-part is \(p^*\beta+\{E\}\) for a Bott–Chern class \(\beta\) and an exceptional real divisor \(E\). The other Hodge components have zero degree on curves. On every \(p\)-contracted curve the degree of \(E\) is therefore zero, so both signs of negativity make \(E=0\). Lifting any curve of \(V\) through the projective resolution now shows that \(\gamma\) and \(\beta\) have the same curve degrees. Thus subtracting a multiple of \(\{J\}\) from either a degree-two or a Bott–Chern class can annihilate every curve of the contracted fiber. By Lemma 30, a degree-two class, or a Bott–Chern class, with these zero degrees is pulled back from the base. Subtract a multiple of \(\{J\}\) to make any given class annihilate that ray, descend the remainder, and use the trace \(\{J^+\}\) to define its transform. On a common resolution the difference is a multiple of the exceptional adjoint comparison in Equation (2). For a flip the two exceptional prime sets on that resolution are the same. For a divisorial contraction the target resolution has just the one additional exceptional prime of the step. The old decomposition therefore spans the new one and proves surjectivity of the transform.

The sums are direct. If the pullback of a class from \(V\) is an exceptional divisor class, that divisor has zero degree on all curves over \(V\). Applying exceptional negativity to both signs makes the divisor zero; injectivity of pullback gives that the class is zero. The pullback injectivity in degree two and in Bott–Chern cohomology is also Lemma 30, applied to the resolution. Passing to higher smooth resolutions proves the assertion for every \(p\).

For the line assertion, let \(\mathcal{H}\) be a holomorphic line on \(U\), and put \(\mathcal{Q}=(p_*\mathcal{H})^{**}\). Global strong \(\mathbb Q\)-factoriality gives an invertible sheaf \(\mathcal{M}=\mathcal{Q}^{[m]}\) for some \(m>0\). The coherent evaluation comparison of (OpenAI 2026b, Lemma 7.2) gives \[\mathcal{H}^{\otimes m}\simeq p^*\mathcal{M}\otimes\mathcal O_U(E), \qquad E\ \text{an integral \(p\)-exceptional divisor}.\] Indeed \((p_*\mathcal{H}^{\otimes m})^{**}=\mathcal{M}\), since the two sheaves agree at the general points of all divisors of \(V\). Local generators of \(p_*\mathcal{H}^{\otimes m}\) in a frame of \(\mathcal{M}\) compare their evaluations with the pulled-back coefficients by meromorphic quotients. Those quotients agree on the isomorphism locus and hence glue; their zeros and poles are exceptional. Taking Chern classes proves the assertion without choosing a meromorphic section of \(\mathcal{H}\). ◻

Proof. A pluricanonical section makes \(K_X\) pseudo-effective. By \(\mathcal G_d\), on a smooth modification \(\mu:\widetilde X\to X\) there is an actual identity \[\mu^*K_X\sim_{\mathbb Q}P+E,\qquad P\ \text{semiample},\quad E=N(\mu^*\{K_X\}).\] Lemma 7 identifies the divisible section spaces with those of \(P\). Hence \(\kappa(P)=0\); a semiample line of Iitaka dimension zero has a trivial positive multiple. Proposition 11, applied with \(e=\mu\) and \(F=0\), gives an effective rational divisor \(E_X=N(K_X)\) with \(K_X\sim_{\mathbb Q}E_X\). Lemma 6 then gives \(N(\mu^*\{K_X\})=\mu^*E_X\). Apply Proposition 23 with \(P=0\). It reaches a model with torsion canonical line. It is terminal: exceptional places over the initial smooth space have log discrepancy greater than \(1\), and a divisor contracted from that space starts with log discrepancy \(1\) and acquires a strict increase in Equation (2). Finally apply Lemma 24. ◻

Proof. Resolve the dlt pair, giving new exceptional primes coefficient one. The resolved adjoint is the pullback of \(J\) plus an effective exceptional divisor. Apply \(\mathcal G_d\) to this smooth pair. Lemma 5 subtracts the exceptional resolution error from its negative part. On a smooth higher model \(r:\widetilde S\to S\) we obtain the actual identity \[ r^*J\sim_{\mathbb Q}P+E,\qquad P\ \text{semiample},\qquad E=N(r^*\{J\}). \tag{7}\] Every component of \(E\) is exceptional over \(S\) and has center disjoint from the floor, by Equation (6). Choose \(m\) divisible enough that \(mJ,mP,mE\) are integral and \(mP\) is generated. Lemma 7 shows that multiplication by the section of \(mE\) identifies the complete section spaces. Over a neighborhood of the floor, \(E\) is absent upstairs. The line \(mr^*J\) is generated there, and these sections descend to \(S\) by normality. Thus \(mJ\) is generated on a neighborhood \(U\) of the entire floor.

Let \(\rho_\Gamma:\Gamma\to S\) be the normalized graph of the map defined by the complete system \(|mJ|\), and write \(\varphi:\Gamma\to\mathbb P(H^0(S,mJ)^*)\) for its morphism to projective space. The map \(\rho_\Gamma\) is projective and is an isomorphism over \(U\). The rational line \[P_\Gamma=\frac1m\varphi^*\mathcal O(1)\] is semiample. The generated moving system \(|mP|\) induces a morphism \(\psi:\widetilde S\to\Gamma\) whose pullback of \(P_\Gamma\) is \(P\). Choose a small positive rational \(\epsilon\) for which \((S,\Delta-\epsilon D)\) is effective and klt. On a projective log resolution \(W_0\to\Gamma\to S\), give every exceptional prime a nonnegative coefficient strictly above its crepant coefficient for the lowered pair and strictly below \(1\). Its klt adjoint equals the pullback of \(J-\epsilon D\) plus an effective error positive on every prime exceptional over \(S\).

Run Proposition 19 over \(\Gamma\), and write \(g:Y\to\Gamma\) for the endpoint morphism and \(h=\rho_\Gamma g:Y\to S\). The surviving error \(F_Y\) is effective and has positive coefficient on every prime of \(Y\) exceptional over \(S\). Over \(U\), where \(\Gamma=S\), it is also exceptional and relatively nef, since the lowered adjoint is relatively nef and its other term is pulled back from \(S\). Local exceptional negativity makes it zero there. Hence \(h\) is small over \(U\), and it is an isomorphism over the smooth part of \(U\) by the argument in Corollary 20. Restore the floor. The full pair on \(Y\) is dlt near it, by the crepant small comparison and the SNC open, and is klt elsewhere. No exceptional prime has center meeting the floor, so \(h^*D\) is its strict transform and \[J_Y\sim_{\mathbb Q}h^*J+F_Y.\]

Put \(P_Y=g^*P_\Gamma\). On a common smooth model \(a:\widehat S\to\widetilde S\), \(b:\widehat S\to Y\) over \(S\), the maps \(\psi a\) and \(g b\) agree on the common dense open and hence everywhere by separatedness of \(\Gamma\). Thus \(a^*P\sim_{\mathbb Q}b^*P_Y\). Define \(E_Y^0=b_*a^*E\). This is an effective rational divisor, exceptional over \(S\) and supported away from the floor. Pushing Equation (7) to \(Y\) gives \[h^*J\sim_{\mathbb Q}P_Y+E_Y^0.\] The global strong property on \(Y\) makes \(E_Y^0\) rational Cartier. The divisor \(a^*E-b^*E_Y^0\) is \(b\)-exceptional and rationally linearly trivial. Exceptional negativity applied to both signs therefore gives \(a^*E=b^*E_Y^0\).

Define \(E_Y=E_Y^0+F_Y\). Adding the restored-boundary error gives \[ J_Y\sim_{\mathbb Q}P_Y+E_Y,\qquad E_Y\geq0, \tag{8}\] where \(E_Y\) is exceptional over \(S\), is supported away from the floor, and contains every prime exceptional for \(h\). It is rational Cartier by the global strong property. On \(\widehat S\), Lemma 6 gives \[N(a^*r^*\{J\})=a^*E=b^*E_Y^0.\] Exceptional translation for \(b^*F_Y\) over \(S\) then gives \(N(b^*\{J_Y\})=b^*E_Y\). The hypotheses of Proposition 23 are now satisfied. Each contracted curve lies in \(\mathop{\mathrm{Supp}}E_Y\), since its \(P_Y\)-degree is zero and its \(E_Y\)-degree is negative. Nontrivial connected projective fibers are covered by curves, so this contraction and its positive replacement are unchanged over a neighborhood of the floor. The same assertion persists with the pushed-forward effective divisor at each step. All primes exceptional over \(S\) are contracted and no prime coming from \(S\) is contracted. The endpoint \(M\) is therefore small with \(S\), has the asserted morphism over \(U\), and has \(J_M\sim_{\mathbb Q}P_M\) semiample.

Pull \(\gamma\) from \(S\) to \(Y\). Although \(Y\) may be singular, its subsequent program consists of the detected ordinary analytic-ray steps in Proposition 23. At step \(i\), choose \(c_i\in\mathbb R\) so that \(\gamma_i-c_i\{J_i\}\) has degree zero on its contracted ray. All curves in a fiber have class on that same analytic ray. The working spaces have rational singularities after slightly lowering the Cartier floor, and relative vanishing gives rational singularities on the contraction base. Lemma 30 therefore descends this class uniquely to the base. Pull it to the positive side and add \(c_i\{J_{i+1}\}\), defining \(\gamma_{i+1}\). The pullback difference is \(c_i\) times the ordinary adjoint comparison, hence is an actual real exceptional divisor class. No smooth-start cohomology splitting is needed for this construction.

Telescope these comparisons on a common resolution. The original map \(Y\to S\) was a morphism, and \(\gamma\) was pulled back through it. Since \(S\) and \(M\) match in codimension one, the resulting divisor is exceptional over both. Each step is an isomorphism over the chosen neighborhood \(U\) of the floor. There the class remains the pullback from \(S\); the corresponding comparison divisor has zero class and is exceptional. Negativity applied to both signs makes it zero there. Thus its support is disjoint from the inverse image of \(U\), as claimed. ◻

Proof. Let \(P_M=J_M\) denote the semiample rational line from Lemma 26, and use its transform \(A_M\). On a common resolution \(p:W\to S,\ q:W\to M\), let \(F_A\) be the real divisor in the common exceptional span defined by \[q^*A_M-p^*A=\{F_A\}.\] It is effective: \(-F_A\) is \(q\)-nef, because \(p^*A\) is nef and the other term is pulled back from \(M\); apply exceptional negativity. Its support is disjoint from \(p^{-1}(U)\). In particular \(A_M\) is pseudo-effective, and its negative multiplicity at every prime of \(M\) is zero. Indeed, \(q^*A_M=p^*A+\{F_A\}\), and Lemma 5 applied over \(S\) gives \(N(q^*A_M)=F_A\).

Let \(D_M\) be the rational Cartier trace of \(D\). Choose \(\tau>0\) so small that \(A+\tau\{D\}\) is Kähler. For a fixed sufficiently small \(t>0\), consider on \(M\) the boundary \(\Delta_M-t\tau D_M\) and the nef b-class represented on \(W\) by \(t p^*(A+\tau\{D\})\). Its trace on \(M\) is \(t(A_M+\tau\{D_M\})\), so its generalized adjoint is \[ \{P_M\}+tA_M. \tag{10}\] These are effective gklt data. Over \(U\) the b-class descends to \(M\), so its generalized discrepancy contribution is zero there, and subtracting \(t\tau D_M\) raises every zero log discrepancy of the dlt pair. Away from the floor the ordinary pair is klt; on one fixed log resolution its finitely many subunit coefficients stay subunit for small \(t\). This proves gklt. The generalized boundary is big. On a common resolution the pullback of the trace of \(A+\tau\{D\}\) minus \(p^*(A+\tau\{D\})\) is effective exceptional by the same negativity argument as for \(F_A\). The latter pullback is nef and big, so the trace is big; the remaining boundary is effective.

Choose an integer \(m>0\) such that \(mP_M\) is generated, and let \(g:M\to Z\) be the Stein factor of its morphism. Then \(Z\) is normal projective and \(mP_M=g^*H\) for an ample generated Cartier line \(H\) on \(Z\). Set \(L=\{P_M\}+tA_M\). The preceding preparation makes \(L\) an effective gklt adjoint with globally nef b-data and globally modified-big boundary-plus-nef trace. It is pseudo-effective because \(P_M\) is nef and \(A_M\) is modified nef. Apply the proper relative assertion of Proposition 17 over \(Z\). This produces a chosen nonextracting good log terminal model \(\phi:M\dashrightarrow V\) and a morphism \(g_V:V\to Z\), with \(L_V\) nef over \(Z\). The hypothesis here is properness: the semiample morphism \(M\to Z\) need not be projective. Put \[mP_V=g_V^*H,\qquad A_V=(L_V-\{P_V\})/t.\] The relative construction preserves the actual line pulled back from \(Z\), and its forward Bott–Chern transport preserves the displayed linear relation; these are the transforms of \(P_M,A_M\).

The map \(\phi\) is small. On a common projective resolution \(a:W'\to M\), \(c:W'\to V\), its comparison is \[a^*L=c^*L_V+\{E\},\qquad E\geq0,\] where \(E\) is exceptional over \(V\) and has positive coefficient at the strict transform of each prime contracted from \(M\). The forward trace \(L_V\) is pseudo-effective, so exceptional translation gives \[N(a^*L)=N(c^*L_V)+E\geq E.\] But \(L\) is modified nef on \(M\); the coefficient of \(N(a^*L)\) at a strict prime of \(M\) is zero. Thus no prime of \(M\) is contracted. Nonextraction then proves smallness.

Fix \(b>2dm\). We claim that \(L_V+b\{P_V\}\) is absolutely nef. Otherwise the cone theorem for the effective gklt adjoint \(L_V\) gives an \(L_V\)-negative rational extremal generator \(C\) such that \[(L_V+b\{P_V\})\cdot C<0,\qquad 0<-L_V\cdot C\leq2d.\] Indeed the cone summand on which \(L_V\) is nonnegative also has nonnegative \(P_V\)-degree, so a negative ray summand must account for any failure of nefness. If \(P_V\cdot C=0\), ampleness of \(H\) makes \(C\) vertical over \(Z\), contradicting relative nefness of \(L_V\). Otherwise Cartier integrality gives \(P_V\cdot C\geq1/m\), contradicting \[(L_V+b\{P_V\})\cdot C\geq-2d+b/m>0.\] This proves the claim. Both \(\{P_V\}\) and \((1+b)\{P_V\}+tA_V\) are nef. By convexity, \[\{P_V\}+sA_V\ \text{is nef for }0\leq s\leq\frac{t}{1+b}.\] Thus \(\delta=t/(1+b)\) works. The actual line \(P_V\) has the same generated multiple as \(P_M\), since both are pulled back from \(H/m\). Its class is the transform of \(J\). The relative comparison and linear transport give exceptional-divisor comparisons for \(P_M,A_M\). Since \(S,M,V\) agree in codimension one, their final comparison divisors are exceptional over both \(S\) and \(V\), as required. ◻

Proof. We first choose a scaling with strict positive walls and Kähler interior classes. We then reduce the transformations on each log canonical stratum to small diagrams away from its smaller strata. Finally, the single nef interval supplied above rules out infinitely many such diagrams.

Choice of scaling. For every possible finite prefix of the detected ordinary truncation scalings from \(X\), Lemma 24 makes the transform of \(H^{1,1}(X,\mathbb R)\) onto the working Bott–Chern space surjective. There are countably many such sequences: curve rays belong to countably many integral homology classes, and a ray determines its contraction and its flip uniquely. On each working space, two distinct curve rays \(C,C'\) with nonzero \(L\)-degrees give a proper hyperplane of initial directions defined by \[(L\cdot C)(\omega\cdot C')-(L\cdot C')(\omega\cdot C)=0,\] where the degrees use the traces on that working space. Whenever both rays define finite walls, this equation says that their wall parameters agree. Avoid all these hyperplanes, and also avoid \(\omega\cdot C=0\) whenever \(L\cdot C=0\). A general Kähler direction, enlarged to make the initial scaled class Kähler, has these properties.

At a wall the scaled class is pulled back from the contraction base. Immediately after a flip, curves in an opposite-side fiber have positive \(L\)-degree. Another negative ray at the same wall would satisfy the excluded wall equation with that opposite-side ray. Immediately after a divisorial contraction, lift any putative new wall curve through the projective morphism. Its lifted ray is distinct from the contracted ray and satisfies the same wall equation; if its \(L\)-degree were zero, its \(\omega\)-degree would also be zero. Both possibilities were excluded. Hence consecutive positive thresholds strictly decrease, and each working interval has nonempty interior. Its interior nef class is big: the class on \(X\) is a positive Kähler perturbation of the pseudo-effective class \(L\), and bigness is preserved under the birational comparison. No rational curve can have zero interior degree. An \(L\)-negative or \(L\)-positive such curve would violate nefness at one of the two nearby parameters; the zero-\(L\) case was excluded. The gklt presentation of a positive truncation and (Hacon et al. 2026, Theorem 4.3) therefore make the interior class Kähler. By Lemma 21, infinitely many thresholds would tend to zero.

Reduction to small transformations on a stratum. Suppose the sequence is infinite. The homological divisor count (OpenAI 2026b, Lemma 7.8) discards all divisorial ambient steps after a finite prefix. We next reduce the remaining flips to small transformations on a stratum. For a working ambient pair \((X_i,B_i)\), ordinary dlt adjunction on the normalization \(S\) of an lc center gives an effective dlt pair \((S,\Delta_S)\) and the actual rational-line residue identity \[K_S+\Delta_S\sim_{\mathbb Q}(K_{X_i}+B_i)|_S.\] Its lc centers are the proper nested lc centers. The coefficients of these differents form a DCC set depending only on the original coefficients and the chain length. At one restriction they have the form \[1-\frac1r+\frac{\sum_j k_jb_j}{r}, \qquad 0\leq\sum_j k_jb_j\leq1,\] which preserves DCC under iteration. These assertions, including the actual residue comparisons, are (OpenAI 2026b, Lemma 7.16). Restricting Equation (2) along a strict adjunction chain gives an effective comparison on strata, strictly positive for a place whose center maps into the ambient exceptional locus (OpenAI 2026b, Lemma 7.17). Both lemmas are dimension-free.

Identify a surviving lc center with its strict transform on the common isomorphism open. There are finitely many lc centers at each stage, and their surviving list stabilizes. A new zero-discrepancy place was already a zero-discrepancy place, and strictness keeps the general point of its center outside the surgery. Starting with the smallest centers, induct on their dimensions. Fix a surviving center and suppose the loci already avoid all its proper subcenters. We will prove that the loci eventually avoid this center as well. The induced diagrams on its normalized strata are then isomorphisms near their non-klt loci. On each stratum, let \(D_S\) be the sum of the restrictions of the ambient floor components not used in its generic adjunction chain. Strong \(\mathbb Q\)-factoriality of the ambient space makes \(D_S\) rational Cartier, and the nested-center description gives \(\mathop{\mathrm{Supp}}D_S=\mathop{\mathrm{Supp}}\lfloor\Delta_S\rfloor\). Slightly lowering this effective divisor makes the stratum pair klt.

After a further tail the induced transformations on this stratum are small and their boundary transforms agree. Indeed, a prime extracted in a restricted transformation has discrepancy strictly less than \(1\) before extraction: its new boundary is effective and strict comparison increases its discrepancy. Its center is away from the unchanged non-klt locus. At the start of this tail there are only finitely many such places. On a fixed compact log resolution, the positive SNC weights \(1-b_j\) have a positive minimum, and the Jacobian inequality bounds every place below the strict cutoff \(1\); include also the finitely many nonexceptional positive-boundary primes. This is (OpenAI 2026b, Lemma 7.9), with its lc assertion away from the non-klt locus. Discrepancies do not decrease, so all later extractions belong to this same finite list. Repeated extraction of one place would give a strictly decreasing sequence of its boundary coefficients, contrary to the adjunction DCC property.

Once extractions stop, (OpenAI 2026b, Lemma 7.8) makes divisorial contractions finite. Its dimension-free count is the dimension of the span of prime divisor cycles in \(H_{2d-2}\) of the compact \(d\)-dimensional stratum. Restriction to the common isomorphism open uses Borel–Moore homology: removing codimension at least two from the target has no effect in this degree, while a lost prime has nonzero class by its positive Kähler volume. The count strictly drops. The coefficients on the finitely many matched boundary primes then stabilize by monotonicity and DCC. Neither morphism of a restricted diagram can still contract a divisor to the common intermediate space, since strict comparison would change the discrepancy of that matched divisor. The diagrams are therefore small, with identical boundary transforms.

We justify the passage from an isomorphic restricted diagram to absence of ambient surgery near the stratum, also in higher codimension. Its identified boundaries give the same adjunction data, not only isomorphic underlying spaces. Choose a common projective log resolution preserving the general points of the surviving lc centers, and restrict its adjoint comparisons along the strict adjunction chain described above. The generic point of each surviving center lies outside the surgery: a zero-discrepancy place remains such a place, whereas a center contained in the surgery would acquire strictly larger discrepancy. Thus the strict transform \(S_W\) of the normalized stratum is not contained in the comparison support. For a zero-dimensional stratum this already excludes any intersection with the surgery, so suppose its dimension is positive. Its restriction is consequently a well-defined effective divisor. Strict adjunction identifies its class with the difference of the two pulled-back stratum adjoints. When the restricted diagram is an isomorphism with the same boundary, that difference is zero. An effective nonzero divisor on the compact Kähler resolution of \(S_W\) has positive mass against a Kähler power, so the restricted divisor is zero.

If an ambient step nevertheless met the stratum, take the first such step. The full-support assertion following Equation (2), and surjectivity of \(S_W\) onto the stratum, force its comparison support to meet \(S_W\). Noncontainment makes this a nonzero effective restriction, and the cumulative comparison dominates it. This contradicts the vanishing just proved. Earlier steps are isomorphisms on a neighborhood of the stratum by the choice of this first step. The ambient diagram is therefore an isomorphism near the whole stratum. This verifies the support conclusion used in the induction on proper subcenters; for a higher-codimension stratum it is noncontainment in the support, not merely being different from a divisor component, that is needed.

Excluding the remaining walls. It remains to show that only finitely many of these small diagrams are nontrivial. Write \(S\) for the first stratum in this last tail and \(J=K_S+\Delta\) for its adjoint. Let its restricted original scaling be \(\{J\}+\lambda H\). Choose an interior parameter \(\lambda_0>0\) on this working space, above the remaining walls, and set \[A=\{J\}+\lambda_0H.\] It is Kähler, as the restriction of the interior ambient Kähler class. The exact change of parameter is \[ \{J\}+tA=(1+t)(\{J\}+\lambda H),\qquad \lambda=\frac{\lambda_0t}{1+t},\qquad t=\frac{\lambda}{\lambda_0-\lambda}. \tag{11}\] It applies for \(0\leq\lambda<\lambda_0\). The new walls \(t_i\) strictly decrease toward zero, and the traces \(\{J_i\}+tA_i\) are nef on their successive restricted working intervals.

On a common resolution of a nontrivial small stratum step, let \(D_i\) be the real exceptional divisor defined by \[p_i^*\{J_i\}-q_i^*\{J_{i+1}\}=\{D_i\}.\] The strict adjoint comparison makes \(D_i\) an effective nonzero divisor exceptional over both strata. Equality of the scaled pullbacks at its wall gives the exact Bott–Chern equality \[ p_i^*(\{J_i\}+tA_i)-q_i^*(\{J_{i+1}\}+tA_{i+1}) =\left(1-\frac{t}{t_i}\right)\{D_i\}. \tag{12}\] For \(t\) in a later working interval all preceding factors on the right are nonnegative. Telescoping on a common resolution expresses the pullback of \(\{J\}+tA\) as the pullback of its nef working trace plus an effective divisor exceptional over that working stratum. Lemma 5 identifies the latter divisor with the negative part.

Push the resulting positive currents to one fixed smooth resolution of \(S\) and let \(t\downarrow0\). Closedness of the pseudo-effective cone proves that \(J\) is pseudo-effective. The negative multiplicities converge valuation by valuation by the limit statement in Lemma 22. The comparison divisors have no strict prime of \(S\), since the maps are small. Their coefficients also vanish at every place whose center meets the floor: each finite composition is unchanged on a neighborhood of the floor by the induction on smaller centers. Passing to the limit proves Equation (6).

The stratum has dimension less than \(n\), and the divisor \(D_S\) constructed above is rational Cartier with support equal to its floor. Lemmas 26 and 27 give one small model \(V\) on which all transformed classes of \(\{J\}+tA\) are nef for \(0\leq t\leq\delta\). On a common smooth resolution \(p:W\to S,\ q:W\to V\), define the real divisor \(F(t)\) in the common exceptional span by \[ p^*(\{J\}+tA)-q^*(\{J_V\}+tA_V)=\{F(t)\}. \tag{13}\] Uniqueness of the representing exceptional divisor makes \(F(t)\) coefficientwise affine in \(t\). It is effective on \([0,\delta]\): \(-F(t)\) is \(p\)-nef because the class pulled back from \(V\) is nef, so exceptional negativity applies. It is exceptional over \(V\) as well. Lemma 5 gives \[N\bigl(p^*(\{J\}+tA)\bigr)=F(t) \qquad(0\leq t\leq\delta).\] Every divisorial negative multiplicity is thus affine on this whole interval. This remains true on higher resolutions by Lemma 6.

Choose a nontrivial stratum wall \(t_i\in(0,\delta)\). On a common resolution of \(S,V\) and the adjacent strata, their nef working models compute the same negative part on their respective open intervals. Their two affine expressions agree at \(t_i\). But Equation (12) says that, at a prime with coefficient \(d_i>0\) in \(D_i\), their difference is \[\left(1-\frac{t}{t_i}\right)d_i.\] Their slopes differ by \(-d_i/t_i\). A single affine function on \([0,\delta]\) cannot agree with both on two open intervals. This contradiction excludes every such wall. Infinitely many walls would tend to zero, so the nontrivial diagrams on this stratum are finite.

There are finitely many surviving strata. Induction on their dimensions makes all sufficiently late ambient flipping and flipped loci disjoint from the floor. This proves the theorem. ◻

Proof. The signed representation persists under codimension-one pushforward. A curve disjoint from the transformed floor has degree zero for each component in this representation, and hence for the transformed adjoint. Every negative operation must therefore meet the floor. After the finite prefix supplied by Theorem 28 no further negative operation is possible. The continuation in Lemma 21 makes the last adjoint nef. ◻

Proof. These are, respectively, the two cases of (Das and Hacon 2026, Lemma 2.6(1),(2)). The first case is birational descent between spaces with rational singularities and does not require a projectivity criterion. The second case uses relative vanishing for the ordinary rational klt pair and the projective relative curve space. In particular, it can be applied to a projective log-Fano contraction before constructing generalized data on its base. ◻

Proof. Fix a Kähler class \(\omega\) on \(T\) and normalize \(\overline{\mathrm{NA}}(T)\) by \(\omega\cdot z=1\). The resulting slice \(S\) is compact. Its unique point on \(R\) has negative \(D\)-degree, so \(d=c_1(D)\) is negative on a neighborhood of that point in \(S\). On the complementary compact set, \(\alpha\) has a strictly positive minimum, while \(d\) is bounded. For all sufficiently small \(s>0\), the class \(\alpha-sd\) is therefore strictly positive on \(S\) and is Kähler by cone duality.

Choose an integer \(m>0\) making \(mD\) integral. Then \[c_1(-mD)+f^*((m/s)\gamma)=(m/s)(\alpha-sd)\] is Kähler. On a neighborhood in the base with a potential for \(\gamma\), absorb that potential in a smooth metric on the same global line bundle \(-mD\). Its curvature is positive on the fibres. The analytic positive-line-bundle criterion and the fibrewise relative-ampleness criterion make \(-mD\) relatively ample. The line bundle was global throughout, so no gluing of unrelated local polarizations is required. ◻

Proof. Suppose first that \(\pi\) is bimeromorphic. Grauert’s proper direct-image theorem makes \(\mathcal F=\pi_*\mathcal L\) coherent. On the nonempty open subset \(U^\circ\) where \(\pi\) is an isomorphism, it is a line bundle. Choose \(u_*\in U^\circ\). Cartan’s theorem A says that global sections generate \(\mathcal F_{u_*}\), so one global section has nonzero image in \(\mathcal F_{u_*}/\mathfrak m_{u_*}\mathcal F_{u_*}\). Under \[H^0(U,\pi_*\mathcal L)=H^0(Y,\mathcal L)\] it gives a section \(s\) nonzero at the unique point above \(u_*\). Its zero divisor is effective Cartier and represents \(\mathcal L\). The section may vanish elsewhere; only its meromorphic inverse is being used to obtain a meromorphic trivialization.

For the general case, fix \(u\in U\) and \(y\in\pi^{-1}(u)\), and choose a \(\pi\)-ample line bundle \(\mathcal H\). Relative Serre generation, after shrinking to a Stein neighbourhood of \(u\), gives an integer \(n\) for which both \(\mathcal L\otimes\mathcal H^n\) and \(\mathcal H^n\) are relatively generated. Their proper direct images are coherent. Cartan’s theorem A and the evaluation maps supply sections \(s\) and \(t\) of these two bundles which are both nonzero at \(y\). The quotient \(s/t\) is the required nonzero meromorphic section of \(\mathcal L\). At no point is \(Y\) assumed Stein. ◻

Proof. Since \(f\) has a global relatively ample bundle and all its vertical curves lie on \(R\), numerical invariance of ampleness on the projective fibres shows that \(-L\) is \(f\)-ample.

A local rational ordinary adjoint. Fix \(z\in Z\) and a relatively compact Stein neighbourhood \(U\). Take a projective log resolution \(p:V\to T_U\) carrying \(\mathbf M\), and put \(\rho=f p\). Write its actual structure boundary as \(B_V\), so \[[K_V+B_V]+[\mathbf M_V]=p^*A.\] Every coefficient of \(B_V\) is less than one. Apply Lemma 32 to obtain a Cartier representative \(D_L\) of \(L\) on \(T_U\). The same birational direct-image argument applies to the coherent rank-one reflexive canonical sheaf: it supplies a canonical form nonzero at a chosen point of the common smooth isomorphism locus. Use this meromorphic form on \(T_U\) and its pullback to \(V\) to choose compatible canonical representatives, and set \[N=t p^*D_L-K_V-B_V.\] For every \(\rho\)-vertical curve, its degree equals that of \(\mathbf M_V\). Thus \(N\) is \(\rho\)-nef.

The relative bigness of \(N\) uses the additional fact that \(\rho=f\circ p\) is bimeromorphic: both the small contraction \(f\) and the resolution \(p\) are bimeromorphic. It does not follow from relative nefness alone. In fact every real Cartier divisor is relatively big for this projective bimeromorphic map over the Stein neighbourhood. Here is the required ample-plus-effective decomposition explicitly. Write \(N=\sum_j r_jN_j\), where \(r_j>0\), \(\sum_jr_j=1\), and the \(N_j\) are rational Cartier divisors in its finite Cartier span. Choose a \(\rho\)-ample line bundle and a Cartier representative \(P\) for it using Lemma 32. If \(q_jN_j\) is integral, apply the birational case of that lemma to \(\mathcal O_V(q_jN_j-P)\). Its nonzero holomorphic section has an effective Cartier zero divisor \(E_j\), giving the actual relation \(q_jN_j\sim P+E_j\). This use depends on \(\rho\) being bimeromorphic; the section is allowed to vanish along exceptional fibres. Therefore \[N\sim_{\mathbb R} \underbrace{\left(\sum_j\frac{r_j}{q_j}\right)P}_{H} +\underbrace{\sum_j\frac{r_j}{q_j}E_j}_{E}.\] Here \(H\) is \(\rho\)-ample and \(E\geq0\), proving \(\rho\)-bigness. For a map with positive-dimensional general fibres the direct image used above can be zero; no such bigness claim is made for that setting. Make the finitely many section choices on a slightly larger Stein neighbourhood, and now shrink \(U\) so its closure is compact in that neighbourhood. Properness makes the inverse image of this closure compact. A log resolution of the finitely many resulting divisors therefore gives a uniform positive bound for the following choice. Choose \(\epsilon>0\) small enough that \((V,B_V+\epsilon E)\) is sub-klt. The divisor \((1-\epsilon)N+\epsilon H\) is relatively ample. Express it as a positive combination of rational ample divisors and take general members of sufficiently high multiples. Relative Bertini (Das et al. 2024, Theorem 2.20), applied on a log resolution of \(B_V+E\), gives an effective \(\Theta\sim_{\mathbb R}N\) such that \((V,B_V+\Theta)\) is sub-klt. The high multiples make the added coefficients arbitrarily small. All choices are made near the compact fibre; shrink \(U\) once to retain them.

Put \(\Delta=p_*(B_V+\Theta)\geq0\). Write the actual linear equivalence as \(\Theta=N+\sum_j a_j\operatorname{div}_V(g_j)\). The bimeromorphic map \(p\) identifies meromorphic function fields, so \(g_j=p^*h_j\). Pushing down gives \(K_{T_U}+\Delta=tD_L+\sum_j a_j\operatorname{div}_{T_U}(h_j)\); in particular this divisor is real Cartier. Pulling back this same identity gives \[K_{T_U}+\Delta\sim_{\mathbb R}tD_L, \qquad p^*(K_{T_U}+\Delta)=K_V+B_V+\Theta.\] Thus \((T_U,\Delta)\) is klt. Any finitely many base line bundles in a relative linear equivalence can first be trivialized by shrinking \(U\). Record the resulting equality as \[ K_{T_U}+\sum_i d_iD_i=tD_L+\sum_j b_j\operatorname{div}(g_j), \qquad d_i>0, \tag{14}\] using the positive support of \(\Delta\). After a further relatively compact shrinking, the displayed divisors have finitely many prime components: their locally finite supports meet a compact inverse image. Equality of coefficients in (14) is a finite rational affine system in \((d_i,t,b_j)\). The klt condition is open within this system, as seen on one log resolution; the pullbacks of its adjoints vary linearly by the displayed identity. A nearby rational solution therefore gives a rational effective klt boundary \(\Delta_q\) and \(t_q\in\mathbb Q_{>0}\) with \[ K_{T_U}+\Delta_q\sim_{\mathbb Q}t_qD_L. \tag{15}\] The individual local primes \(D_i\) need not be \(\mathbb Q\)-Cartier: the identity ensures that the total adjoint is \(\mathbb Q\)-Cartier.

One global flip algebra. The adjoint in (15) is \(f_U\)-antiample. Its ordinary flip and local finite generation follow from (Fujino 2022, Theorems 1.14 and 1.18). Clearing the displayed linear equivalence identifies a Veronese of its canonical algebra with a Veronese of \(\mathcal R:=\bigoplus_{m\geq0}f_*L^m\) over \(U\). Hence \(\mathcal R\) is locally finitely generated by (Fujino 2022, Lemma 2.26). The graded pieces are coherent. Over a connected normal base open the inverse image is irreducible; products of nonzero sections of line bundles are nonzero there. Thus its graded section algebra is an integral domain, without choosing any global meromorphic frame of \(L\). Compactness supplies one sufficiently divisible Veronese \(\mathcal R^{(r)}\) generated in degree one. Its relative Proj \(T^+\) restricts to the ordinary flip over every such \(U\). It is therefore normal, locally klt, and small over \(Z\), and has the global relatively ample bundle \(\mathcal O_{T^+}(1)\). On the common big open this bundle is \(L^r\). Reflexivity gives \[(L^+)^{[r]}\simeq\mathcal O_{T^+}(1).\] This proves that \(L^+\) is a rational line bundle before any assertion of factoriality. Relative positivity over the compact Kähler base makes \(T^+\) compact Kähler.

For any global rank-one reflexive sheaf \(\mathcal F^+\) on \(T^+\), take its coherent reflexive transform \(\mathcal F\) on \(T\) using a common resolution. Some \(M=\mathcal F^{[m]}\) is a line bundle. Choose \(c\in\mathbb Q\) so that \((M+cL)\cdot C=0\), and clear its denominator to obtain an integral bundle \(J=n(M+cL)\) numerically trivial over \(f\). The local rational klt pairs (15) have antiample adjoint, so Lemma 18 yields one global line bundle \(J_Z\) with \(J=f^*J_Z\). On \(T^+\), pull back \(J_Z\), undo the rational twist \(ncL^+\), and compare on the common big open. After clearing the already established index of \(L^+\), reflexivity gives an invertible positive reflexive power of \(\mathcal F^+\). This proves global strong factoriality.

The original generalized pair. The local log-Fano pairs above give \(R^if_*\mathcal O_T=0\) for \(i>0\) by (Fujino 2022, Theorem 5.2). They make \(T\) locally klt, hence rational. Leray for a projective resolution then shows that \(Z\) is rational. Lemma 30(i) therefore gives \(f^*H^{1,1}_{\mathrm{BC}}(Z,\mathbb R)=R^\perp\), with injective pullback. Thus \(A=t c_1(L)+f^*\eta\) for a unique \(\eta\). Define \(A^+=t c_1(L^+)+(f^+)^*\eta\); it is relatively Kähler.

To construct its actual discrepancy divisor, let \(\mathcal I\) be the ideal defined by \[\operatorname{im}(f^*f_*L^r\longrightarrow L^r) =\mathcal I\otimes L^r.\] Principalize \(\mathcal I\) on a common resolution \(p:W\to T\), \(q:W\to T^+\) carrying \(\mathbf M\). If \(\mathcal I\mathcal O_W=\mathcal O_W(-F_L)\), the tautological quotient gives the actual bundle identity \[q^*\mathcal O_{T^+}(1)=p^*L^r\otimes\mathcal O_W(-F_L).\] Hence \(E_L=F_L/r\) is effective and exceptional over both sides, and \([E_L]=p^*c_1(L)-q^*c_1(L^+)\). If \(B_W\) is the original structure boundary, put \(B_W^+=B_W-tE_L\). Then \[[K_W+B_W^+]+[\mathbf M_W]=q^*A^+.\] Its pushforward is \(B^+\), and its discrepancies do not decrease. The local ordinary comparison in (15) has discrepancy divisor \(t_qE_L\), so its strictness proves the stated strictness for \(tE_L\) as well. Thus the original pair remains gklt.

Finally, write any \(\theta\in H^{1,1}_{\mathrm{BC}}(T,\mathbb R)\) uniquely as \(\theta=f^*\eta_\theta+s c_1(L)\) and set \(\theta^+=(f^+)^*\eta_\theta+s c_1(L^+)\). Its correction is the actual divisor \(sE_L\). For a rational line \(M\), the number \(s\) is rational. Applying Lemma 18 to a Cartier multiple of \(M-sL\) gives \(M=sL+f^*M_Z\) as actual rational lines. Reflexive extension on the common big open gives \(M^+=sL^++(f^+)^*M_Z\). Consequently the cohomological transport agrees with the reflexive transform on Chern classes. ◻

Proof. We reduce to fixed ordinary pairs, and construct one global program. Choose finitely many Stein open sets \(U_\ell\subset S\) and semianalytic Stein compact subsets \(W_\ell\Subset U_\ell\) whose interiors cover \(S\). Such compacts are obtained by intersections with closed polydiscs in local embeddings of \(S\). In particular they satisfy condition (P) of (Fujino 2022): the source is normal, the base is Stein, the compact is Stein, and its intersection with any analytic subset defined near it has finitely many connected components. This remains true on every normal projective model. Shrinking \(U_\ell\) around \(W_\ell\) preserves the finite cover.

Actual ordinary replacements. On each chart represent the finitely many line bundles in the data by Cartier divisors. For a projective map to a Stein space this can be done by twisting a line bundle and its prospective meromorphic frame by sufficiently positive relative lines and using nonzero sections. The generic-rank-one direct-image argument is an alternative only when the map is bimeromorphic. Relative bigness gives \(N\sim_{\mathbb R}P+E\), with \(P\) relatively ample and \(E\geq0\). For sufficiently small \(\epsilon>0\), \((W,B_W+\epsilon E)\) is sub-klt near the compact inverse image. The line \((1-\epsilon)N+\epsilon P\) is relatively ample. General divided relative sections give an effective representative \(T\) such that \[\Theta:=\epsilon E+T\sim_{\mathbb R}N, \qquad (W,B_W+\Theta)\text{ is sub-klt}.\] All supports and coefficient margins are fixed near that compact. With compatible canonical representatives, push the finite principal-divisor expression for \(\Theta-N\) down and pull it back again. For \(\Delta_\ell=p_*(B_W+\Theta)\) this gives \[\Delta_\ell\geq0, \qquad K_X+\Delta_\ell\sim_{\mathbb R}J|_{X_{U_\ell}}, \qquad p^*(K_X+\Delta_\ell)=K_W+B_W+\Theta.\] In the last equality the chosen principal correction is included in the representative of \(K_X+\Delta_\ell\). Thus \((X_{U_\ell},\Delta_\ell)\) is ordinary klt and its boundary is relatively big. Lemma 11.15 of (Fujino 2022), with its big part equal to \(\Delta_\ell\) and its remaining part zero, replaces it, in its actual real linear equivalence class, by a klt boundary with an effective general relatively ample rational summand. Its condition on non-klt centres is vacuous. Choose the reserved ample summand to be \(\mathbb Q\)-linearly equivalent to a small rational multiple of a fixed global \(f\)-ample line. Its later strict transform is consequently \(\mathbb Q\)-Cartier by the transport of that global line, without a local factoriality claim.

We make these choices for a finite rational polytope of actual global line data. Start with a finite rational span containing \(J\), and add relatively ample rational lines spanning the global relative degree space. The above construction is valid in a neighbourhood of \(J\): retain a small part of the ample summand and use openness of ampleness and the strict klt inequalities. It is also valid along a segment from \(J\) to a sufficiently positive adjoint, by adding divided general ample divisors. Use finitely many such representatives and one log resolution on each chart. More explicitly, represent a finite simplex inside the open ample neighbourhood used above; varying its positive coefficients represents all nearby global line parameters on the same finite support. Form the joint finite affine system consisting of the divisor identities on all charts, with common global line coordinates and separate boundary and principal-divisor coordinates. This system is rational. Allow all boundary coefficients to vary, including those originally contributed by the real boundary \(B\); fix only the reserved rational ample part and the zero-support constraints. Its projection to the global line coordinates contains the just constructed neighbourhood, so it has a rational affine section. Choose that section sufficiently close to the constructed real affine lift, by rational approximation in its affine space of sections, so that the strict klt and effectiveness margins are retained along the compact segment. In the joint solution space the strict klt and effectiveness margins permit a rational polytope containing the scaling segment. This supplies on every chart a rational polytope of ordinary klt boundaries with a fixed effective relatively ample rational summand, representing the restrictions of the same global line parameters. These boundaries, including the scaling representatives, are fixed before the program starts. Their divisors need not agree on different charts.

One global scaling construction. Apply the global degree-space construction in the proof of Proposition 19 to \(J\) and a general ample scaling direction in the chosen span. Here are the modifications that allow real line data and a nonbirational map to \(S\). The global relatively ample line still gives a compact normalized slice of the curve cone. The fixed ordinary chart pairs give finite negative-ray truncations. A general direction avoids equal wall parameters for independent ray-degree functionals, exactly as in that proof. A positive wall therefore selects a single ray in the global degree space. A rational nef support is represented by a global rational line; a large multiple minus the chartwise ordinary adjoint is relatively ample. Ordinary relative base point freeness contracts the ray using the evaluation of this one global line.

If the contraction is small, choose a global rational line having negative ray degree and construct its flip by its one global graded algebra. The local ordinary proof of Proposition 33 applies to this algebra: all curves of the contraction have proportional degrees for global lines, so the detector and the driving line are positively proportional over the contraction; ordinary rational replacement gives local finite generation. This part of that proof needs only the already projective contraction and these line degrees. It does not require a prior assertion about the full Bott–Chern cone. The relative Proj is the next global model. In a divisorial step the exceptional prime and Lemma 18 give the same continuation as in Proposition 19. Killing the degree of any global line by a rational multiple of the detector, then applying that lemma, preserves global strong \(\mathbb Q\)-factoriality in both cases. The global lines continue to span the relative degree spaces. The rational support-simplex argument in that proposition makes the wall a descended relatively ample real line on its contraction base and gives a nonempty interval of ampleness immediately after each step. This argument uses relative line data and projectivity, not birationality of the map to \(S\).

We check explicitly the persistence of the ordinary pairs. Expand the line parameters in finitely many global rational lines. Their reflexive transforms are actual rational lines by the preceding strong property. Pushing forward the fixed principal-divisor identities gives \[K_{X_i}+\Delta_{\ell,i}\sim_{\mathbb R}J_i|_{(X_i)_{U_\ell}}.\] Hence these ordinary adjoints are \(\mathbb R\)-Cartier; no local factoriality of individual prime divisors is being assumed. On a common resolution their comparison minus the appropriate positive multiple of the detector comparison is exceptional and relatively numerically trivial. Negativity in both signs makes this difference zero. Every chosen step is consequently negative for the fixed ordinary pairs. They remain effective and klt, and their boundaries remain relatively big. The same equality of comparison divisors, after adding the fixed base pullback, preserves the generalized pair and its discrepancy inequalities.

For termination, every already constructed working model is a weak log canonical model on each chart of a boundary in the fixed polytope: choose a parameter in its nonempty ample scaling interval. Indeed, actual descent at a preceding wall \(\lambda_j\) makes the comparison for \(J+tH\) equal to \((1-t/\lambda_j)\) times its \(J\)-comparison. For a parameter in the current interval all preceding \(\lambda_j\) are at least \(t\), so these comparisons are effective. Theorem E of (Fujino 2022) gives finitely many marked models on each chart. That theorem does not require locally \(\mathbb Q\)-factorial targets. There are therefore finitely many possible global marked working models. Indeed, isomorphisms between the restrictions of two existing marked models agree on their common dense open, and hence on overlaps. A repeated marked working model would have the same boundary transform and discrepancies, contrary to strict discrepancy increase at an intervening nontrivial step. The global program is finite. Relative pseudoeffectivity excludes a Mori fibre endpoint, so \(J_m\) is nef over \(S\). This is a finite-cover argument for a single constructed program; it does not glue independent minimal models.

Semiampleness and the global polarization. On the endpoint the fixed ordinary pair is klt, its effective boundary is relatively big, and its adjoint is nef over \(W_\ell\). Lemma 11.16 of (Fujino 2022) applies to this already existing weak model. More explicitly, Lemma 11.15 replaces the endpoint boundary by \(A_\ell+B_\ell\), where \(A_\ell\geq0\) is relatively ample, \(B_\ell\geq0\), and the pair is klt. Theorem 8.3 then gives semiampleness near \(W_\ell\): its log canonical, klt-existence, real Cartier, ample-summand and nef-over-compact hypotheses have all been checked.

The actual equivalences identify these local semiample adjoints with restrictions of \(J_m\). Their ample models agree on overlaps by (Fujino 2022, Lemma 11.3). To retain a global polarization, shrink the initial parameter polytope around \(J\) after the finite program: all its finitely many strict step signs persist, so the transformed ordinary boundaries remain klt and big. On each chart the pushforward of its initial common ample rational summand is effective and relatively big. Apply (Fujino 2022, Lemma 11.13) to this common big rational part and the small endpoint boundary polytope. All pairs are klt, so the condition on non-klt centres is vacuous. The lemma gives a common general relatively ample rational summand by one \(\mathbb Q\)-linearly trivial translation. In particular the rational parametrization is preserved; separate applications of Lemma 11.15 to individual parameters would not suffice here. In this finite rational span Theorem 11.17 of (Fujino 2022) makes the endpoint nef region a rational polytope. Intersect the finitely many inverse images of these regions and express \(J_m\) as a positive combination of rational points in its minimal face. The corresponding global rational lines are semiample on every chart by the same ordinary base point free argument. After clearing denominators and taking common sufficiently divisible powers over the finite cover, each is relatively generated globally: generation on the charts is generation of its global coherent relative evaluation map. The product of these finitely many relative morphisms, followed by Stein factorization, gives a projective \(g:X_m\to Z\), a projective \(\rho:Z\to S\), and descended global lines whose positive combination \(H\) is relatively ample. It realizes the glued ample model and gives \(J_m\sim_{\mathbb R}g^*H\). Adding back \(f_m^*\beta\) proves the asserted Bott–Chern identity. Composing the effective step comparisons proves the final discrepancy statement. ◻

Proof. Lemma 11.15 and Theorem 8.3 of (Fujino 2022) give local semiampleness. For an existing weak model this is also Lemma 11.16. To globalize the polarization, take a finite rational span of the global lines defining \(J\). On each chart the ample part of the normalized boundary allows small perturbations in this span. The joint rational affine construction in the preceding proof gives a rational family with a common rational ample part; all boundary coefficients are allowed to vary. Theorem 11.17 then gives a rational nef region on each chart. The finite-intersection, rational-vertex and global-evaluation argument in that proof gives one projective contraction and its actual descended ample real line. Only its local ample models are identified by uniqueness. ◻

Proof. The line \(N\) is fixed; it need not be a general scaling direction. In particular several extremal rays may have the same wall parameter. Choose \(\epsilon>0\) such that \(D+2\epsilon N\) is still not pseudoeffective over \(S\). On the finite Stein-compact cover choose divided general effective representatives of \(N\) so that all the resulting ordinary pairs representing \(D+tN\), \(0\leq t\leq T\), are klt. For \(\epsilon\leq t\leq T\) reserve a common effective relatively ample rational summand. The joint rational-family construction in Proposition 40 puts these fixed pairs in the polytopes required by Theorem E of (Fujino 2022).

Suppose a finite prefix has reached \(X_i\), and put \(\mu_i=\lambda_{i-1}\), with \(\mu_0=T\). Its transformed \(D_i+\mu_iN_i\) is nef. Define \[\lambda_i=\min\{t\in[0,\mu_i]:D_i+tN_i \text{ is nef over }S\}.\] The feasible set is a nonempty closed interval. Inductively \(\lambda_i>2\epsilon\): all previous steps have nonpositive comparison for \(D+2\epsilon N\), so its transform is still not pseudoeffective. A first threshold at most \(2\epsilon\) would make this transform nef by convexity between the current and preceding thresholds, a contradiction.

There is a \(D_i\)-negative extremal ray \(R_i\) with \((D_i+\lambda_iN_i)\cdot R_i=0\). To see attainment without a generality assumption, let \(t<\lambda_i\) tend to \(\lambda_i\). A ray negative for \(D_i+tN_i\) has positive \(N_i\)-degree, because \(D_i+\mu_iN_i\) is nef. When \(t>\lambda_i/2\), it is also negative for \(D_i+(\lambda_i/2)N_i\). The latter has fixed ordinary klt big-boundary representatives on the charts, since \(\lambda_i/2>\epsilon\). Normalize each of these boundaries by (Fujino 2022, Lemma 11.15) as \(A_\ell+B'_\ell\), with \(A_\ell\geq0\) relatively ample and the new pair klt. The \(A_\ell\)-truncation of the cone theorem for \(K+B'_\ell\) gives finitely many rays negative for the entire adjoint in question. The finite cover therefore gives finitely many global negative rays. One of these rays attains the wall, as required.

Choose one such ray, even if the wall vanishes on other rays. The rational-support argument for an individual negative ray in Proposition 19 gives a global rational nef support annihilating exactly \(R_i\), and ordinary relative base point freeness contracts it. For the real ordinary boundary one may first choose a nearby rational klt boundary that is still negative on \(R_i\). The contraction therefore has a rational ordinary antiample adjoint. Its flip, when small, is constructed from one global detector algebra as above.

If the contraction is of fibre type, stop. Otherwise the wall \(w_i=D_i+\lambda_iN_i\) has degree zero on \(R_i\). In the finite rational span of its actual line factors, this zero-degree subspace is rational: its defining functional has rational values on rational lines. Thus \(w_i\) is a real combination of rational lines of zero ray degree. Lemma 18 descends those lines to the contraction base. Their combination is nef over \(S\), by lifting curves through the projective contraction. Pulling it to the positive side shows that the transformed wall stays nef. The next threshold therefore satisfies \(\lambda_{i+1}\leq\lambda_i\); equality is permitted. No interval of ampleness between successive thresholds is required.

If \(E_i\geq0\) is the ordinary \(D_i\)-comparison on a common resolution, this actual wall descent gives, for every real \(t\), \[p_i^*(D_i+tN_i) \sim_{\mathbb R}q_i^*(D_{i+1}+tN_{i+1}) +(1-t/\lambda_i)E_i.\] For \(t\leq\lambda_i\) the error is effective. This proves all the nonpositivity and fixed-pair persistence assertions used in the induction, including the lower bound on thresholds.

At every working model use the wall parameter \(t=\lambda_i\), rather than an interior ample parameter. Since \(\lambda_i\leq\lambda_j\) for every earlier step, the displayed comparisons show that \(X_i\) is a weak log canonical model of the initial fixed chart pairs representing \(D+\lambda_iN\). Its trace is nef by definition. Theorem E counts all these marked weak log canonical models, including the ones at repeated wall parameters. The finite chart cover gives a finite global marked list. Repetition would contradict the strict ordinary \(D\)-discrepancy increase at an intervening step. Thus this program with the specified \(N\) is finite. Non-pseudoeffectivity excludes a nef endpoint for \(D\), so the last contraction is a Mori fibre contraction. The comparison at \(t=1=\lambda_i\) is crepant: such a step supplies no strict discrepancy improvement for \(D+N\), while it is still strictly negative for \(D\). ◻

Proof. Choose one resolution \(p:\widehat V\to V\) witnessing Definition 16, and write \(K_{\widehat V}+\widehat B=p^*(K_V+B)\) using the actual meromorphic identification. Its unit components are the strict transforms \(\widehat S_i\) of the finitely many components \(S_i\) of \(S\). Every component \(\widehat Z\) of an intersection of distinct \(\widehat S_i\)’s maps birationally onto a stratum \(Z\), because the map is an isomorphism at its general point. Conversely every stratum is obtained this way. The general SNC locus makes both the component and its index set unique. Compactness makes the collection finite. The empty intersection is \(\widehat V\) itself. Write \(p_Z=p|_{\widehat Z}:\widehat Z\to Z\), so \(p_V=p\).

Here are the adjunction and normality details, including the facts used at deeper intersections. For a chosen set of indices, keep their coefficients one and allow each unused coefficient to be either one or \(1/2\). Write \(B^{\mathbf e}=B-\sum_i(1-e_i)S_i\). Global \(\mathbb Q\)-factoriality makes this an actual rational-line operation, and \[\widehat B^{\mathbf e}=\widehat B- p^*\sum_i(1-e_i)S_i.\] The subtracted divisor is effective. All exceptional coefficients remain below one, and the remaining unit components are exactly the retained strict transforms. Successive SNC adjunction on \(\widehat Z\) gives \[(K_{\widehat V}+\widehat B^{\mathbf e})|_{\widehat Z} =K_{\widehat Z}+\widehat B_{\widehat Z}^{\mathbf e}.\] Here and below such a restriction equality means equality of the meromorphic residue maps in degree \(q\), and hence equality of the actual lines. If two orders of residue differ by an ordering sign, its \(q\)-th power is one.

We induct on the number of chosen indices. At each stratum already constructed, we retain normality, effectivity of every weighted different, and the crepant SNC residue comparison on \(\widehat Z\). For this chosen comparison the exceptional coefficients are below one, the unit components are the retained strict intersections, and the full-weight pair is dlt. These assertions hold for \(Z=V\). Lowering unused unit components preserves dlt; lowering all of them makes the effective pair at the current stratum klt. It has rational singularities and is Cohen–Macaulay by the dimension-free floor-connectedness lemma (OpenAI 2026a, Lemma 2.1). If just one unused coefficient is retained, the unit divisor of the induced SNC boundary is a disjoint union \(R\) of smooth divisors on \(\widehat Z\). The same lemma gives \[(p_Z)_*\mathcal O_R=\mathcal O_{p_Z(R),\mathrm{red}}.\] The fibers are connected, so the images of different connected components of \(R\) cannot meet. Each component maps birationally to its image; the displayed equality, factored through finite normalization, makes that image normal. It is compact Kähler by restriction of the local Kähler potentials from \(V\).

For every allowed weight choice, push forward the SNC different to define the rational different on this new stratum. In codimension one, the residue of a local frame of the ambient invertible line identifies its divisorial adjoint with the restricted line. Reflexive extension on the normal stratum gives Equation (35) everywhere. Pulling it back agrees with the original SNC residue on a dense open and therefore everywhere, proving crepancy as an actual meromorphic identification.

The different is effective. At a general point of any of its primes, the normal surface-slice calculation of (OpenAI 2026a, Lemma 6.2), which is stated in every dimension, preserves exactly the residue order and reduces it to adjunction on a normal surface germ with effective boundary and a smooth marked curve. For completeness, on a minimal resolution of that surface, write the crepant boundary as the effective strict transform plus an exceptional divisor \(E\). For each exceptional curve \(D\), \[E\cdot D=-(K_{\widetilde T}+B^{\rm str})\cdot D\leq0.\] Indeed \(B^{\rm str}\cdot D\geq0\), and adjunction gives \(K_{\widetilde T}\cdot D=2p_a(D)-2-D^2\geq0\); minimality excludes a smooth rational exceptional \((-1)\)-curve. The negative-definite exceptional intersection matrix gives \(E\geq0\): if \(E=P-N\) with disjoint nonnegative parts and \(N\ne0\), then \(E\cdot N=P\cdot N-N^2>0\), contrary to the preceding inequalities. The strict transform is finite birational over the smooth marked curve, and hence isomorphic to it. Adjunction to that smooth strict transform thus has nonnegative coefficient. This is the original coefficient by the surface-slice residue calculation.

The full-weight pair is dlt and has the asserted floor. On an SNC model, at the general point of the center of a divisorial valuation \(v\), take normal parameters \(x_1,\ldots,x_a\), giving an unmarked parameter boundary weight zero. The SNC discrepancy inequality is \[a(v;\widehat Z,\widehat B_{\widehat Z}) \geq \sum_{i=1}^a(1-b_i)v(x_i).\] Every \(v(x_i)>0\). Thus a zero-discrepancy center is an intersection of unit components. Those intersections meet the isomorphism locus, while every exceptional component has coefficient below one. To produce a dlt resolution for the restricted map, principalize its nonisomorphism locus and resolve together with the ordered SNC boundary. That locus contains no entire log canonical center, so the displayed discrepancy inequality makes every new exceptional coefficient strictly below one. This is the dlt resolution criterion. It also identifies the unit primes downstairs with the next incident strata.

Finally let \(R_{\rm all}\) be the whole unit union on \(\widehat Z\). Sections on its smooth components which agree on all their SNC intersections glue by the elementary equalizer for the coordinate ideals of an SNC union. Even iterated residues make the equalities independent of the chosen chain. Apply the floor-connectedness lemma to the full-weight pair and this full unit union. It gives \((p_Z)_*\mathcal O_{R_{\rm all}}=\mathcal O_{C_Z}\). Projection formula therefore descends the glued section uniquely to the restricted invertible line on the reduced \(C_Z\). The proof with \(Z=V\) gives the assertion for \(S\). ◻

Proof. We use the dimension-free torsion-free restriction lemma (OpenAI 2026a, Lemma 7.1). In its notation, for an effective compact Kähler lc pair \((T,G)\), klt away from its reduced floor \(C\), and a connected map \(f:T\to P\) to a normal compact base from which an actual positive adjoint multiple is pulled back, it asserts that \(R^1f_*\mathcal O_T(-C)\) is torsion-free. The ideal sequence gives \[ \mathcal{Q}:=\mathop{\mathrm{coker}}(\mathcal O_P\longrightarrow f_*\mathcal O_C) \lhook\joinrel\longrightarrow R^1f_*\mathcal O_T(-C). \tag{38}\] For \(T=Z\) and \(P=Y_Z\), if \(C_Z\) is vertical, \(\mathcal{Q}\) has proper support and is zero. Serre vanishing for the kernel of \(\mathcal O_P\to f_*\mathcal O_C\), tensored with \(N_Z^k\), extends every restriction in a uniform large-degree tail. If \(C_Z\) dominates, the same display says that a section extends as soon as its values are constant on the reduced floor fiber over a dense open of \(Y_Z\): its class in the torsion-free \(\mathcal{Q}\otimes N_Z^k\) then vanishes, and the local lifts glue. They are unique because \(C_Z\) dominates \(Y_Z\), so \(\mathcal O_{Y_Z}\to(f_Z)_*\mathcal O_{C_Z}\) is injective. This uses neither reduced special scheme fibers nor base change at their points.

Suppose now that \(C_Z\) dominates \(Y_Z\). There is an effective rational Cartier divisor \(D\) with \(\mathop{\mathrm{Supp}}D=\mathop{\mathrm{Supp}}C_Z\): restrict to \(Z\) the globally \(\mathbb Q\)-Cartier floor primes of \(V\) not containing \(Z\). For small rational \(\epsilon>0\), \((Z,B_Z-\epsilon D)\) is effective klt. Apply Corollary 20 to obtain a projective small strong \(\mathbb Q\)-factorialization \(\rho:Z_0\to Z\), crepant also for the full pair. Write \(\Delta_0\) for its full boundary and \(D_0=\rho^*D\). The full adjoint \[P=K_{Z_0}+\Delta_0\] is the actual semiample rational line pulled back from \(Y_Z\), and the lowered klt adjoint is \(J_{\rm low}=P-\epsilon D_0\). Put \(d=\dim Z\), choose \(m>0\) with \(mP\) Cartier and generated, and take \(b>4md\). Consider \[H=J_{\rm low}+bP=(1+b)P-\epsilon D_0.\] On a general fiber of the map to \(Y_Z\), its class is \(-\epsilon\{D_0\}\). This is the negative of a nonzero effective divisor because the floor dominates \(Y_Z\), so its pairing with a Kähler power is negative. If \(H\) were pseudo-effective, a positive current representing it would restrict to a positive current on almost every resolved smooth fiber, contradicting this pairing. Thus \(H\) is not pseudo-effective.

Apply Lemma 21 to the lowered klt pair and the nef rational line \(P\). Its \(H\)-program with Kähler scaling has ordinary \(J_{\rm low}\)-negative steps, is finite, and ends in a projective Mori contraction. Every step, including the final Mori ray, is \(P\)-trivial. The same actual Cartier line \(mP\) descends at each birational step. Consequently its semiample map to the fixed base \(Y_Z\) persists on every working model. That map is constant on each connected final Mori fiber and hence factors through the Mori contraction. We obtain \[T\xrightarrow{u}W\xrightarrow{g}Y_Z.\] Both maps have connected fibers. The full adjoint on each model is the trace of \(P\). On a common resolution of a birational step, its natural meromorphic canonical comparison is an exceptional divisor with zero class, since the two actual lines descend from the same line on the contraction base. Exceptional negativity applied to both signs makes that divisor zero. Thus the full pair remains crepant, including its meromorphic adjoint identifications. The full transformed pair \((T,G)\) is effective lc and is klt away from its full floor \(C^+\): there it agrees with the transformed lowered klt pair. The program is nonextracting, so the transformed divisor \(D^+\) has \(\mathop{\mathrm{Supp}}D^+=\mathop{\mathrm{Supp}}C^+\). Since the full adjoint is pulled back from \(Y_Z\), the lowered adjoint on a \(u\)-fiber is \(-\epsilon D^+\). Hence \(D^+\) is relatively ample over \(W\).

Compare the connected components of a general floor fiber on the original small model and on \(T\). On a common projective log resolution the full crepant subboundary is the same. Its unit union maps to both floors with connected fibers by (OpenAI 2026a, Lemma 2.1). Proper closed surjections with connected fibers preserve connected components, also after restricting over a point of \(Y_Z\). Thus the connected components of the two underlying floor fibers correspond.

Every connected component of the floor fiber on \(T\) meets one common general Mori fiber. Indeed \(C^+\to W\) is surjective because \(C^+\) is the support of the relatively ample \(D^+\). Apply Equation (38) to \(u\). Surjectivity of the floor makes \(\mathcal O_W\to u_*\mathcal O_{C^+}\) injective; the displayed torsion-free cokernel then makes \(u_*\mathcal O_{C^+}\) torsion-free. Let \[C^+\longrightarrow\overline C\longrightarrow W\] be the Stein factorization. The finite space \(\overline C\) is reduced, and every irreducible component of \(\overline C\) dominates \(W\). To see the last implication algebraically, a vertical minimal prime of a finite reduced algebra over a domain would give, by prime avoidance, a nonzero element killed by a nonzero element of the domain. After shrinking \(Y_Z\) to a dense open, the fiber-dimension theorem gives every component of \(\overline C_y\) dimension \(\dim W-\dim Y_Z\). The fiber \(W_y\) is irreducible of that dimension: a resolution of \(W\) still has connected fibers over \(Y_Z\), and a general one is smooth and connected and surjects onto \(W_y\). Finiteness now makes each component of \(\overline C_y\) dominate \(W_y\). Consequently each connected component of \(C^+_y\) meets \(u^{-1}(w)\) for the same general \(w\in W_y\).

The reduced general fiber \(F\) of \(u\) is irreducible by the same resolution argument. If \(\dim F\geq2\), the support of the effective ample Cartier multiple of \(D^+|_F\), whose support is \(C^+\cap F\), is connected. One elementary proof, valid even if \(F_{\rm red}\) is not normal, cuts by general hyperplanes to an integral projective surface. If the ample divisor split into disjoint nonzero Cartier parts \(D_1,D_2\), their pullbacks to a resolution would be orthogonal, while the projection formula gives \(v^*(D_1+D_2)\cdot v^*D_i>0\) for both \(i\). Thus both \((v^*D_i)^2>0\), contradicting the surface Hodge index theorem. If \(\dim F=1\), choose the fiber also off the singular, boundary intersection, and ramification loci. It is a smooth connected curve and \[0=\deg(K_F+G|_F)=2g(F)-2+\deg(G|_F).\] There is a unit point, so \(F\simeq\mathbb P^1\). Its floor has one or two points; if there are two they exhaust the horizontal boundary. In every connected case the preceding common-fiber observation makes the general floor fiber connected, so no comparison is needed. In the remaining case the two unit points represent all its connected components.

Normalize the horizontal floor primes and take their Stein factorizations over \(W\). If there are two primes, each finite Stein map has degree one and is an isomorphism over normal \(W\); their main fiber product gives a proper bimeromorphic comparison. If there is one prime, its normal Stein space is a double cover of \(W\). The exchange on its general fiber extends over the branch locus: the reduced horizontal non-diagonal component of the double fiber product has finite birational projections to the normal cover and hence both projections are isomorphisms. Its graph lifts to the normalized prime. Composing with the program graphs transports the comparison to two original normal floor primes, possibly the same one. All graph projections are projective, and the primes survive birationally because the program extracts no divisors.

This comparison is crepant as a meromorphic residue identity. On a smooth ruled open, order the markings after an étale local cover, choose a base volume \(\xi\), and put the markings at \(z=0,\infty\). A pulled-back \(q\)-pluriadjoint frame has the form \[a(w)(dz/z)^{\otimes q}\otimes\xi^{\otimes q}.\] Its two residues are \(a(w)\xi^{\otimes q}\) and \((-1)^q a(w)\xi^{\otimes q}\), which agree. This is the two-marking Poincaré-residue comparison of (Kollár 2012, sec. 3, Definition 13 and Proposition 14). On a resolved graph both restricted actual lines are pulled back from \(Y_Z\); their meromorphic embeddings agree on this dense open and hence everywhere, including along exceptional primes. Equal residues therefore give a constant value on the entire general floor fiber. The horizontal consequence of Equation (38) extends the section. Normality descends it from the small model. Its restriction is the prescribed section on the original reduced floor, since every original floor prime is covered birationally and the two restrictions agree after that pullback.

It remains to check Equation (37). Let \(\widehat T\) be a common smooth resolution of the parent and Mori models and let \(a,b:H\to\widehat T\) be the two maps from a resolved branch graph. The resolution of the Mori model is an isomorphism near a general entire \(\mathbb P^1\) fiber: its centers have codimension at least two, hence cannot dominate the ruling base. Every holomorphic two-form on \(\widehat T\) comes from the base on that ruled open. In fact its vertical one-form terms vanish on \(\mathbb P^1\), and its remaining coefficients are constant on that compact fiber. Thus \(a^*-b^*\) kills \(H^{2,0}\), and also \(H^{0,2}\) by conjugation. This rational Hodge map on degree two has image of type \((1,1)\). The Lefschetz \((1,1)\) theorem puts its real image in \(\mathop{\mathrm{NS}}(H)_{\mathbb R}\). Apply this to the pulled-back parent class \(c_Z\). The link is therefore an arrow as defined above. ◻

Proof. The center is a dlt stratum by the SNC discrepancy calculation in the proof of Lemma 73. Work over the generic SNC part of \(D\), of codimension \(a\), with normal coordinates \(x_1,\ldots,x_a\) for the unit divisors. The zero-discrepancy divisorial valuations centered there are exactly the primitive rational monomial rays in \(\mathbb R_{>0}^a\). Here is a local verification that also fixes their normalization. The discrepancy inequality in the proof of Lemma 73 says on every log-smooth model that the center of a zero-discrepancy valuation is exactly a unit stratum: an additional unmarked normal parameter, or a parameter with coefficient below one, would give positive discrepancy. Blow up that stratum. If the positive normal orders of the valuation are \(w_1,\ldots,w_b\), the chart indexed by a smallest order \(w_k\) replaces these by \(w_k\) and the positive members of \(w_i-w_k\). The exceptional divisor is again unit, and the center is exactly the new unit stratum. While there is more than one positive order their sum decreases. Eventually the center is the generic point of one unit divisor, where a normalized divisorial valuation has order one. Reversing these ordinary toric blowups proves monomiality and primitive normalization. Conversely this subtraction algorithm for any primitive positive integer vector terminates at order one, and reversing it extracts that monomial valuation. Comparison on a common graph gives uniqueness of the valuation with those weights and of its center on any model.

For a unit component \(E\) meeting the inverse image of the generic SNC part of \(D\), put \[v_E=(\mathop{\mathrm{ord}}_E x_1,\ldots,\mathop{\mathrm{ord}}_E x_a)\in\mathbb Z_{\geq0}^a.\] If the unit components through a unit stratum \(Q\) meeting that inverse image are \(E_1,\ldots,E_b\), define \[\phi_Q:\mathbb R_{\geq0}^b\longrightarrow\mathbb R_{\geq0}^a,\qquad (t_1,\ldots,t_b)\longmapsto\sum_{j=1}^b t_jv_{E_j}, \qquad \sigma_Q=\phi_Q(\mathbb R_{\geq0}^b).\] The source carries its standard lattice \(\mathbb Z^b\). For the fixed resolution \(T\), these cones subdivide the orthant over its interior. On rational rays, \(\phi_Q\) sends a monomial valuation on \(T\) to the same normalized valuation on \(Z\), so it is injective on each cone. Interiors of different cones are disjoint, because the center and the weights of a monomial valuation on \(T\) are unique. Every rational ray in the orthant occurs: view its valuation on \(T\) and use the same SNC description there. There are only finitely many cones over the generic part of \(D\); density of rational rays therefore gives coverage. Taking subsets of unit components gives the faces, and uniqueness of valuations also on faces identifies intersections as common faces. Consequently \(\sigma_R\) is a face of an \(a\)-dimensional cone \(\sigma_{R'}\) with the same downstairs center, for a unit stratum \(R'\subseteq R\).

Let \(E_1,\ldots,E_a\) be the unit components through \(R'\). The integer matrix \(A=(\mathop{\mathrm{ord}}_{E_j}x_i)\) of this full cone is unimodular. Indeed every positive primitive integral vector in the source cone is the normalized vector of a divisorial valuation, so its image is primitive. If a prime divided \(\det A\), a nonzero kernel vector modulo that prime, lifted to a positive primitive integral vector, would have nonprimitive image. Hence \(\det A=\pm1\). The strata \(R'\) and \(D\) have the same dimension. We next prove that their generically finite map has degree one. In local coordinates at a general point of \(R'\), \[x_i=u_i(y,z)\prod_{j=1}^a y_j^{A_{ij}},\] where the \(u_i\) are units, the \(y_j\) are normal coordinates, and \(z\) are coordinates along the stratum. The tangential map is generically étale in characteristic zero. Since \(A^{-1}\) is integral, replace \(y_j\) by \(\widetilde y_j=y_j\prod_i u_i^{(A^{-1})_{ji}}\). Then \(x_i=\prod_j\widetilde y_j^{A_{ij}}\), and the map consisting of \(\widetilde y\) and the downstairs tangential coordinates is locally biholomorphic at a general étale point of \(R'\to D\). Two distinct points over a general point of \(D\) would therefore give two lifts of nearby target points whose normals approach with any fixed positive rate vector in the interior of this cone. Varying the nonzero leading coefficients fills an open set of nearby torus points, so such a target can be chosen in the locus where \(p\) is an isomorphism. This contradiction proves that \(R'\to D\) is birational. Finally, after taking residue along \(R'\), the normal coefficient of \(\bigwedge_i dx_i/x_i\) is \(\det A\) times that of \(\bigwedge_jdy_j/y_j\); terms differentiating the units vanish in that residue. Since \(\det A=\pm1\), the even pluriresidue removes the sign. ◻

Proof. The assertions are immediate for \(d=0\), when every floor is empty. For \(d>0\), choose a common log resolution \(p:T\to Z\), \(p':T\to Z'\) of the arrow. For a target floor prime \(D'\), start with its strict transform \(R\). It is a unit stratum and maps birationally to \(D'\). Its source center \(p(R)\) is a stratum by Lemma 75. If \(R\to p(R)\) is not birational, that lemma replaces \(R\) by a unit stratum inside it which maps birationally onto the same source center. The target center may then shrink. If the map to that center is not birational, apply the lemma on the target side, and continue alternately. Each necessary replacement strictly decreases dimension. The process therefore ends with one unit stratum mapping birationally to lower strata \(D_0,D_0'\) on the two sides.

The image in the common system base remains unchanged throughout. Indeed the two maps from the resolved graph to the Stein targets are identified by the arrow, and a replacement keeps its entire image on the side then being treated. Initially the image was \(f_{Z'}(D')\). The terminal comparison of \(D_0\) and \(D_0'\) is an arrow: restricting the actual meromorphic identification to iterated residues gives crepancy, with the logarithmic determinant sign removed by Lemma 75; restricting the line Chern classes in Equation (37) gives the same condition modulo \(\mathop{\mathrm{NS}}_{\mathbb R}\). Lower invariance thus identifies the transported and assigned residues on \(D_0'\).

This equality determines the entire section on \(D'\). The Stein map of \(L_{D'}=L_{Z'}|_{D'}\) is the Stein factor of \(f_{Z'}|_{D'}\), because the line induced on that Stein space is ample; its target is finite over \(f_{Z'}(D')\). The image of \(D_0'\) in this target is a closed irreducible subset of full dimension, since it still maps onto \(f_{Z'}(D')\), and therefore is the whole target. Both sections on \(D'\) come from it. Equality after restriction to \(D_0'\) proves equality on \(D'\).

The same construction shows that the image of every target floor prime in the system base is the image of a lower source stratum, and the reverse statement follows from the inverse arrow. A floor dominates the system base on one side exactly when it does on the other. This proves the last assertion. ◻

Proof. At the point strata use the same scalar in the canonical zero-form generator \(1\). The even residue convention identifies these generators along every path and under every point arrow. This gives the required system in dimension zero; if there are no points the empty system is understood.

Suppose the system has been constructed through dimension \(d-1\). Replace it by powers and their span in a common degree \(k\) chosen so that Lemma 74 applies on every \(d\)-stratum, Proposition 77 applies on every vertical \(d\)-stratum, and \(\mathcal I_{f_Z(C_Z)}\otimes N_Z^k\) is generated for every \(d\)-stratum \(Z\). These conditions hold in a common divisible tail: there are finitely many strata, and the last condition is Serre’s theorem. For a \(d\)-stratum \(Z\), Lemma 73 glues the lower tuple to a section on its whole \(C_Z\). Its residues satisfy the one link in Lemma 74, if that link is present, because the lower tuple is invariant. That lemma extends the section to \(Z\).

Let \(\mathcal P_d(k)\) be the vector space of all tuples through dimension \(d\) whose lower part is in \(\mathcal V_{d-1}(k)\) and whose new components restrict to that lower part. Call its elements pre-tuples. Its projection to \(\mathcal V_{d-1}(k)\) is surjective, since the finitely many extensions can be chosen independently. This system already generates. To check a point \(z\in Z\), put \(y=f_Z(z)\). If \(y\in f_Z(C_Z)\), choose a floor point over \(y\) and a lower tuple nonzero there. Every extension is the pullback of a section of \(N_Z^k\), so it is nonzero at every point over \(y\), including \(z\). If \(y\notin f_Z(C_Z)\), the chosen generation of \(\mathcal I_{f_Z(C_Z)}\otimes N_Z^k\) supplies a base section nonzero at \(y\). It pulls back to a section vanishing on \(C_Z\); combine it with the zero lower tuple and zero components on the other new strata. This also treats an empty floor. Surjectivity to the lower system preserves generation at lower points.

We impose invariance in dimension \(d\). For a dominating floor, injectivity of restriction and Lemma 76 already make every pre-tuple invariant on that stratum. Verticality is invariant under arrows, so the remaining orbits consist entirely of vertical strata.

For each such orbit choose a representative \(Z\), and let \(G_Z\) be the finite isotropy image on \(H^0(Z,L_Z^k)\) given by Proposition 77. For a pre-tuple with component \(s_Z\), form the norm \[P_Z(s_Z)=\prod_{g\in G_Z}g(s_Z) \in H^0(Z,L_Z^{k|G_Z|}).\] Choose a common positive integer \(a\) divisible by all \(|G_Z|\), use \(P_Z(s_Z)^{a/|G_Z|}\), and transport it to each member of that orbit. This is independent of the chosen arrow: changing that arrow by an isotropy arrow merely permutes the factors in degree \(k\). No finiteness assertion in degree \(ak\) is needed. On strata with dominating floor use \(s_Z^a\), and on all lower strata use the componentwise \(a\)-th power of the assigned tuple.

The resulting tuple is compatible. By Lemma 76, every transported factor in a norm has exactly the prescribed lower restriction. The norm and its indicated power therefore restrict to the \(a\)-th power of that restriction, the same value used on all lower strata and in all other orbits. It is invariant in dimension \(d\) by construction, and remains invariant below it.

These tuples still generate. Fix a point \(z\) in an orbit member. For each of the finitely many transported factors, choose a point over \(z\) on a resolution of its comparison graph. Equality of the pulled-back invertible adjoint lines identifies the fiber values. The factor is nonzero at \(z\) exactly when the representative component \(s_Z\) is nonzero at the corresponding point of \(Z\). Evaluation at each such point is a nonzero linear functional on the generating space \(\mathcal P_d(k)\). A finite union of their proper kernels cannot cover a complex vector space. One pre-tuple makes all factors nonzero and gives a norm nonzero at \(z\). At a lower point a nonzero assigned value stays nonzero after its \(a\)-th power. Taking the linear span of all the constructed tuples therefore gives a compatible invariant generating system in degree \(ak\). This completes the induction on \(d\). ◻

Proof of Theorem 72. Take the system in Proposition 82. Its components on the prime components of \(S\) agree on every common lower stratum, so Lemma 73 descends each tuple uniquely to a section of the actual line \(\mathcal O_V(qkJ)|_S\). At any \(x\in S\), choose a prime component through \(x\) and a tuple nonzero there. It is the pullback of the descended value in the same invertible line fiber at \(x\), so that value is nonzero. The finite-dimensional span of these descended sections generates at every point of the reduced \(S\). This is the claimed global generation of an actual Cartier multiple. ◻

Proof of Proposition 13, assuming Proposition 83. We use the reduced-exceptional boundary after every source modification. The effective discrepancy identity in Proposition 9 preserves pseudo-effectivity, and that proposition transfers the resulting decomposition back to the original pair. A dominant meromorphic map to a space in class \(\mathcal C\) can be resolved on smooth compact Kähler models; taking Stein factorization gives connected fibers. These operations preserve the positive dimensions of the base and the fiber.

We first suppose \(a(X)>0\). If \(a(X)=n\), then \(X\) is Moishezon and Kähler, hence projective, and Proposition 12 applies. Otherwise resolve the algebraic reduction as a fibration \(g:X'\to W\) to a smooth projective variety of dimension \(a(X)\). Put \(J'=K_{X'}+B'\) for the modified log adjoint. Its restriction to a very general smooth fiber is pseudo-effective. Indeed, one may choose a sequence of positive currents in \(c_1(J')+\varepsilon_j[\omega]\), with analytic singularities and \(\varepsilon_j\downarrow0\), and restrict all of them outside a countable union of proper analytic subsets of the base. By \(\mathcal G_{\dim F}\), the restricted adjoint has nonnegative Kodaira dimension.

That dimension must be zero. If it were positive, generic coherent base change would give a divisible \(q\) for which \(\mathcal E=g_*\mathcal O_{X'}(qJ')\) has a fiber system with a positive-dimensional image on very general fibers. A sufficiently large ample twist of the coherent sheaf \(\mathcal E\) is generated at the generic point of the projective \(W\). Evaluation would therefore give two sections of \(\mathcal O_{X'}(qJ')\otimes g^*\mathcal O_W(A)\), for one ample \(A\), whose ratio is nonconstant on a general \(g\)-fiber. This ratio is a meromorphic function on \(X'\). Every meromorphic function on \(X'\) factors through its algebraic reduction, a contradiction. Proposition 83 now applies.

Suppose next that \(a(X)=0\), and resolve the fibration in the hypothesis as \(f:(X',B')\to Y\). The base may be replaced by a smooth compact Kähler model, and \(a(Y)=0\), since its meromorphic functions pull back to \(X'\). Write \(d=\dim X'-\dim Y>0\). The same restriction argument and \(\mathcal G_d\) show that \[\kappa(F,K_F+B'_F)\geq0\] on very general smooth fibers. If this integer is \(j<d\), the relative Iitaka construction of (OpenAI 2026e, Lemma 2.6) gives, after the same source modifications, a factorization \[X''\xrightarrow{g}W\xrightarrow{h}Y\] with connected fibers, \(\dim W=\dim Y+j\), and \(\kappa(G,K_G+B''_G)=0\) on a very general \(g\)-fiber. The cited lemma applies to rational SNC boundaries on compact manifolds in class \(\mathcal C\); its conclusion concerns the restricted log adjoint, and does not require finite generation. A Kähler model of \(W\) has algebraic dimension zero because it is dominated by \(X''\). Thus \(0<\dim W<n\), and Proposition 83 applies again.

It remains to exclude the possibility that \(K_F+B'_F\) is big. The stable-family comparison below requires a horizontal boundary with coefficients strictly less than one, so we first lower the boundary while keeping the restricted adjoint big. Let \(B'_{\mathrm{hor}}\) be the sum of the components of \(B'\) dominating \(Y\). There is a rational \(0<\lambda<1\) such that \(K_F+\lambda B'_{\mathrm{hor},F}\) is big for very general \(F\). To justify the uniform choice, exclude at once the generic base-change and image-dimension exceptional sets for all rational \(\lambda\) and all divisible degrees. On a fiber outside this countable union, openness of the big cone gives one such \(\lambda\), and a single degree has full-dimensional image. Generic base change makes the same choice work very generally. The boundary \(\lambda B'_{\mathrm{hor}}\) is horizontal, rational SNC, and has all coefficients strictly less than one. The stable-family comparison (OpenAI 2026e, Lemma 5.2) therefore supplies a proper generically finite cover \(Y'\to Y\), a surjection \(Y'\to R\) to a smooth projective variety, and a projective stable family \(\mathcal V\to R\) with positive-dimensional general fiber, such that the main transform of \(X'\) is bimeromorphic to \(\mathcal V\times_RY'\).

The main transform still has algebraic dimension zero. Normalize it and factor its proper generically finite map to \(X'\) through the normal Stein space. A meromorphic function descends through the birational part. Lemma 10 gives its characteristic polynomial over the finite part, with meromorphic coefficients on \(X'\). Those coefficients are constant because \(a(X')=0\), so irreducibility makes the function constant. Normalization and modifications preserve meromorphic functions. It follows also that \(a(Y')=0\), so the surjection from \(Y'\) to the projective \(R\) forces \(R\) to be a point. But then the main transform is bimeromorphic to the product of \(Y'\) with a positive-dimensional projective variety. Rational functions on that factor contradict algebraic dimension zero. This excludes the big case and completes the reduction. ◻

Proof. The flattening construction of (OpenAI 2026e, Lemma 2.4) gives the stated reference property, and preserves it after any finite sequence of further base modifications. Its source boundary is the one in (OpenAI 2026e, Lemma 2.1). We may resolve the discriminant and the horizontal divisor of the relative generator so that all relevant strata are smooth over the dense smooth log locus. Proposition 2.7 of (OpenAI 2026e) then gives \(T\) and a rational parabolic Hodge line \(M\). A positive multiple of that line is the highest rank-one Hodge step of a complex summand of a pure real-polarizable variation with an integral lattice. Theorem 3.1 of (OpenAI 2026e) gives the factorization \(M=p^*P_S\) and the stated positivity, after a further modification and a positive rational rescaling. These are identities in \(\mathop{\mathrm{Pic}}\otimes\mathbb Q\), not merely identities of classes.

Here is the order computation that gives Equations (43)–(45). Choose a degree \(m\) for which \(g_*\mathcal O_X(m(K_{X/W}+B))\) has generic rank one and all boundary denominators are cleared. Its reflexive hull is a line on the smooth \(W\). Let \(s\) be a meromorphic relative generating form in a local frame of this line. At the generic point of a prime \(D\), choose a nonvanishing ordinary base volume \(\omega_W\), and set \[l_E=\frac1m\mathop{\mathrm{ord}}_E(s\wedge g^*\omega_W^m),\qquad \delta_E=B_E,\qquad \alpha_D=\min_{E\to D}\frac{l_E+\delta_E}{a_E}.\] The relative-times-base identification in this formula is taken in the \(m\)-th canonical tensor power. The source is resolved over the generic point of \(D\), so all the primes needed for the divisorial test occur in this minimum. In the notation of the proof of (OpenAI 2026e, Proposition 2.7), the parabolic order is \[\beta_D=\min_{E\to D}\frac{l_E+1-a_E}{a_E}, \qquad T_D=\alpha_D-\beta_D.\] Thus the corresponding local frame of \(T+M\) has order \(\alpha_D\). Comparing its pullback with the log adjoint form defines \(A_*\), with \[ (A_*)_E=l_E+\delta_E-a_E\alpha_D . \tag{47}\] Changing the generating form multiplies it by a meromorphic base function. This changes both \(\alpha_D\) and \(\beta_D\) by its normalized base order and leaves the comparison invariant. The local comparisons therefore give the rational divisor and the line identity globally. Equation (47) proves Equation (44). Horizontally the comparison is precisely the zero divisor of the fiberwise log section, so it is effective and has the asserted restriction.

The same order computation proves the relative injection. In a divisible degree \(q\), a meromorphic base coefficient \(\varphi\) gives a regular base section at \(D\) exactly when \[\mathop{\mathrm{ord}}_D(\varphi)+q\alpha_D\geq0.\] Its corresponding upstairs orders at the primes \(E\to D\) are \[a_E\mathop{\mathrm{ord}}_D(\varphi)+q(l_E+\delta_E).\] Their simultaneous nonnegativity is equivalent to the preceding inequality. Conversely, an upstairs section restricts to a multiple of the generator on general connected fibers, so its ratio with the generator descends meromorphically to \(W\). The order inequalities make it regular outside codimension two on \(W\), and normality extends it. Consequently \[g_*\mathcal O_X(qJ)\ \lhook\joinrel\longrightarrow\ \mathcal O_W(qH).\] Applying the left exact functor \(h_*\) proves Equation (45). The reverse global comparison follows from the reference property: its only initially untested poles are on \(g\)-contracted primes, all exceptional over \(X_0\); pushing to \(X_0\), extending in codimension two, and pulling back with the reduced-exceptional boundary removes them. This is the global comparison in (OpenAI 2026e, Proposition 2.7).

Finally consider \(\tau:W'\to W\). The parabolic rational line pulls back under SNC modifications, as in the construction preceding (OpenAI 2026e, Theorem 3.1). The coefficient \(T_D\) at an old strict transform is unchanged. Indeed the sheaf of regular relative adjoint forms inside the common meromorphic generator line is unchanged over the generic point of \(D\) by the log modification formula. Equivalently, the old inequalities above already test all vertical primes there; in SNC coordinates logarithmic pullback preserves those inequalities, so new source exceptional primes impose no stronger one. This keeps \(\alpha_D\) unchanged in compatible frames, while parabolic pullback keeps \(\beta_D\) unchanged. Over a new \(\tau\)-exceptional prime every source prime is exceptional over \(X_0\): a pre-existing one had image of codimension at least two on \(W\), and a new one is exceptional by construction. All therefore have boundary coefficient one. The formula for \(T_D=\alpha_D-\beta_D\) then gives coefficient one on that new base prime. Thus \[T'=\tau_*^{-1}T+\operatorname{Exc}(\tau)_{\mathrm{red}}.\] The usual log discrepancy formula for the SNC pair \((W,T)\) now gives \(K_{W'}+T'=\tau^*(K_W+T)+E_H\), with \(E_H\) effective and exceptional. Adding the pulled-back parabolic line proves Equation (46). ◻

Proof. On a very general smooth fiber \(F\), use the model supplied by \(\mathcal G_{\dim F}\) to write \(\mu_F^*J_F\sim_{\mathbb Q}P_F+N(\mu_F^*J_F)\). Lemma 7 identifies the divisible section spaces with those of \(P_F\), so \(\kappa(P_F)=\kappa(F,J_F)=0\); a semiample line of Kodaira dimension zero is torsion. Proposition 11, applied to the modification \(\mu_F\) with zero added divisor, gives \(J_F\sim_{\mathbb Q}N(J_F)\) on \(F\), with \(N(J_F)\) rational. Lemma 7 then identifies the normalized zero divisor of the prepared relative generator with \(N(J_F)\). Thus \[A_{*,\mathrm{hor}}|_F=N(J_F).\]

We compare these fiber multiplicities with those on \(X\). For each of the finitely many components \(E\) of \(A_{*,\mathrm{hor}}\), choose a countable sequence of small-Kähler-perturbation currents with analytic singularities whose generic orders at \(E\) approach \(\nu_E(J)\). Choose a common very general \(F\) for these sequences and these components. The currents restrict positively after the same perturbations, and analytic singularities ensure that their orders along the components of \(E|_F\) are their generic orders along \(E\). Each restricted order is at least the corresponding perturbed minimal multiplicity on \(F\). Passing to the limit gives \((A_*)_E\leq\nu_E(J)\). This proves \[ A_{*,\mathrm{hor}}\leq N(J). \tag{48}\]

Choose a positive curvature current for the rational line \(J\). Every such current contains its divisorial negative part, hence subtracting \([A_{*,\mathrm{hor}}]\) leaves a positive current. On the inverse image of a dense smooth open \(W^\circ\), with all vertical data removed, Equation (43) identifies the resulting singular metric with one on \(g^*H\). In a frame pulled back from \(W^\circ\) its weight is plurisubharmonic. It is constant on each compact connected smooth fiber, by the maximum principle (with the value \(-\infty\) allowed). Local holomorphic sections of the submersion show that these values form a plurisubharmonic weight on \(H|_{W^\circ}\).

This weight extends across the missing divisors. At the generic point of any such divisor \(D\), choose \(E\to D\) for which \((A_*)_E=0\), using Equation (44). At a general point of \(E\) away from the other comparison divisors, the comparison after subtracting the horizontal part has neither a zero nor a pole. We can choose local coordinates, with the remaining vertical coordinates denoted by \(w\), in which \[g(u,z_2,\ldots,z_k,w)=(u^{a_E},z_2,\ldots,z_k).\] Indeed \(g|_E\) has maximal tangential rank generically and a local equation of \(g^*D\) is a unit times \(u^{a_E}\); the unit has a local \(a_E\)-th root. A smaller such chart covers a neighborhood of the base point. The upstairs plurisubharmonic weight is locally bounded above there, and the comparison unit is bounded. Hence the descended weight is locally bounded above near the generic point of \(D\). The removable singularity theorem for plurisubharmonic functions extends it across \(D\) away from codimension two, and the Hartogs extension for plurisubharmonic functions extends it across the remaining analytic subset. The extensions respect the line transitions because they agree on the dense open. They give a positive singular metric on \(H\), which proves pseudo-effectivity. ◻

Proof. The projective quotient \(S\) in Lemma 84 is a point: otherwise its rational functions pull back nontrivially to \(W\). Thus \(M\sim_{\mathbb Q}0\) and \(H\sim_{\mathbb Q}K_W+T\). By Lemma 85 and \(\mathcal G_{\dim W}\), its pullback to a smooth higher model is a semiample rational line plus its negative divisor. Every semiample line on a space of algebraic dimension zero is torsion, since its generated system has zero-dimensional image. Proposition 11, applied to this modification with zero added divisor, therefore gives Equation (49), including rationality of the divisor. ◻

Proof. We use the construction in the proof of (OpenAI 2026d, Proposition 3.3). Its absolute input is a nef-plus- effective rational decomposition for the canonical divisor of a smooth projective variety with pseudo-effective canonical class. This is supplied here by Proposition 12 and its common-resolution comparison.

Replace the image of \(f\) by its normal Stein factor. This does not change relative numerical classes or contracted curves, and now \(f\) is surjective with connected fibers. Its very general fiber is smooth projective with pseudo-effective canonical divisor, hence is non-uniruled by (Boucksom et al. 2013, Theorem 0.2 and Corollary 0.3). Compactify the projective morphism and resolve away from \(V\). We obtain \(\bar f:\bar V\to\bar Z\), with \(\bar V\) smooth projective and \(\bar Z\) normal projective, unchanged over \(Z\). If \(\dim Z=0\), \(V\) is already projective and the absolute input proves the lemma.

Suppose \(\dim Z>0\), and let \(r:\widetilde Z\to\bar Z\) be a smooth projective resolution. Choose a sufficiently positive very ample divisor \(A_Z\) on \(\bar Z\) and a general \(B\in|2A_Z|\). The pulled-back systems on both \(\widetilde Z\) and \(\bar V\) are base point free. Bertini makes \(r^*B\) and \(\bar f^*B\) smooth nonempty reduced divisors. Their double covers \[\pi_Z:Z^\sharp\to\widetilde Z, \qquad \pi:V^\sharp\to\bar V\] are smooth integral projective varieties. The canonical formulas are \[K_{Z^\sharp}\sim_{\mathbb Q}\pi_Z^*(K_{\widetilde Z}+r^*A_Z), \qquad K_{V^\sharp}\sim_{\mathbb Q}\pi^*(K_{\bar V}+\bar f^*A_Z).\] The first line is big when \(A_Z\) is sufficiently positive. The rational map \(V^\sharp\dashrightarrow Z^\sharp\) has the same very general fibers as \(f\). A covering family of rational curves on \(V^\sharp\) would either dominate a covering family of rational curves on \(Z^\sharp\), or cover its very general fibers by vertical rational curves. Both are impossible. Thus \(V^\sharp\) is non-uniruled and \(K_{V^\sharp}\) is pseudo-effective by BDPP.

Apply the absolute decomposition to \(V^\sharp\). Moving the rational principal difference in the canonical formula into the nef part gives an actual rational divisor decomposition for a pullback of \(\pi^*(K_{\bar V}+\bar f^*A_Z)\). Take a common equivariant resolution of that model and its conjugate under the covering involution. On this smooth projective resolution \(g:T\to V^\sharp\), average the two decompositions. We obtain \[g^*\pi^*(K_{\bar V}+\bar f^*A_Z)=\bar P+\bar N, \qquad \bar P\text{ nef},\quad \bar N\geq0,\] where \(\bar P\) and \(\bar N\) are invariant rational Cartier divisors.

Let \(q:T\to\bar Y\) be the finite quotient by the involution. The invariant composite \(\pi g\) induces a projective birational morphism \(\bar h:\bar Y\to\bar V\), with \(\bar Y\) normal and projective. An invariant rational divisor descends by dividing its coefficient at an upstairs prime by the ramification index. The descended divisor is rational Cartier when the original divisor is: after clearing denominators, choose one Cartier equation on a semilocal neighborhood of the finite orbit above a point (a Cartier divisor is principal on a semilocal ring), and multiply all its translates. The product is invariant and has divisor equal to the group order times the original divisor. It descends to a local equation for a multiple of the downstairs divisor. This argument includes stabilizers and ramification primes.

Accordingly \(\bar P=q^*P\) and \(\bar N=q^*N\) for rational Cartier divisors on \(\bar Y\). Lifting curves through the finite map shows that \(P\) is nef, and coefficientwise descent shows that \(N\) is effective. Finite pullback is injective on rational divisors, so the displayed identity descends. Restrict it to \(Y=\bar h^{-1}(V)\). The term pulled back from \(A_Z\) is numerically trivial over \(Z\), giving \(h^*K_V\equiv_ZP+N\), as required. ◻

Proof of Proposition 87. Lemma 88 and (Tsakanikas and Xie 2024, Theorem 5.2), applied to the generalized pair \((U/Z,0+0)\) with \(U\) smooth, prove relative minimal-model existence for smooth varieties by dimension induction. The hypothesis of that theorem is relative smooth minimal-model existence one dimension lower; the lemma supplies its other hypothesis, an NQC weak Zariski decomposition. The induction begins in dimension zero. Then (Tsakanikas and Xie 2024, Theorem 5.4) supplies minimal models for every pseudo-effective NQC generalized lc pair in the relevant dimension. Their Theorem 2.7 gives existence in the Birkar–Shokurov sense.

The nef data in the statement are NQC, because a nef rational Cartier divisor is a positive multiple of a nef Cartier divisor. In the pseudo-effective case the preceding paragraph therefore gives the Birkar–Shokurov model required by (Tsakanikas and Xie 2024, Theorem 4.2). In the other case, the non-pseudo-effective alternative of that theorem applies directly. To meet its scaling hypothesis, take an effective sufficiently positive ample rational divisor \(A\) whose general components have sufficiently small coefficients. On a fixed log resolution carrying \(\boldsymbol M\), these components are transverse to the boundary and preserve generalized log canonicity, while their ample class can be chosen large enough that \(D+A\) is nef. The theorem supplies a terminating MMP starting on \(V\).

We verify the step types inductively. Suppose \(V_i\) is \(\mathbb Q\)-factorial generalized dlt. By (Tsakanikas and Xie 2024, Remark 2.15), each birational step has the form \[V_i\xrightarrow{g_i}Z_i\xleftarrow{h_i}V_{i+1},\] where \(g_i\) contracts a negative extremal ray and \(h_i\) is small and projective. Suppose \(g_i\) is divisorial, and write \(R\) for its ray. An exceptional prime \(E\) has \(E\cdot R<0\): otherwise it would be effective, exceptional, and \(g_i\)-nef, contrary to negativity. There is only one exceptional prime. Indeed, a combination of two distinct exceptional primes can be chosen to have degree zero on \(R\); negativity applied to both signs of that combination would make it zero.

We show directly that \(Z_i\) is \(\mathbb Q\)-factorial. For a prime divisor \(T\) on \(Z_i\), let \(\widetilde T\) be its strict transform, choose a curve \(C\) spanning \(R\), and put \[a=-\frac{\widetilde T\cdot C}{E\cdot C}\in\mathbb Q.\] Choose a positive integer \(m\) for which \(m(\widetilde T+aE)\) is Cartier. Its line has degree zero on \(R\), so (Xie 2022, Theorem 1.5) gives an actual Cartier line \(\mathcal L_T\) on \(Z_i\) and an isomorphism \[\mathcal O_{V_i}\bigl(m(\widetilde T+aE)\bigr)\simeq g_i^*\mathcal L_T.\] On an open set trivializing \(\mathcal L_T\), the Cartier divisor \(m(\widetilde T+aE)\) is principal upstairs. Pushing this principal divisor identity down through the birational \(g_i\) gives a principal divisor \(mT\) on that open set. Thus \(T\) is rational Cartier. It follows that \(h_i\) is an isomorphism: the pushforward of an \(h_i\)-ample Cartier divisor is rational Cartier on \(Z_i\), and its pullback is the original divisor because \(h_i\) is small. It has degree zero on every contracted curve, so relative ampleness rules out positive-dimensional fibers; a finite birational map to the normal \(Z_i\) is an isomorphism. If \(g_i\) is small, the relatively ample canonical model \(h_i\) is the flip of (Han and Li 2022, Definition 3.2). Thus the chosen birational steps are divisorial contractions and flips. The underlying variety of a generalized dlt pair is klt, and these steps preserve the \(\mathbb Q\)-factorial generalized dlt category (Han and Li 2022, Remark 2.3 and Lemma 3.7). They introduce no prime divisors on the new models. For completeness, on a common resolution of one step \(V_i\dashrightarrow V_{i+1}\) carrying \(\boldsymbol M\), taking the difference of the two generalized discrepancy formulas cancels their common nef divisor. The resulting difference of the adjoint pullbacks is exceptional over the new model and anti-nef over it. The negativity lemma makes this difference effective. Composing these actual rational divisor comparisons gives \(u^*D=v^*D_m+F\) with \(F\) effective and exceptional over the endpoint. The endpoints in the two cases are the nef and Mori fiber endpoints supplied by the terminating program theorem. ◻

Proof. The smooth pair with boundary \(T\) and nef data \(\boldsymbol M\) determined by \(M\) on \(W\) is a rational \(\mathbb Q\)-factorial generalized dlt pair. Its adjoint \(H\) is pseudo-effective by Lemma 85. Proposition 87 gives a chosen terminating program with scaling to a \(\mathbb Q\)-factorial generalized dlt model \(W_{\mathrm{nef}}\) with nef transformed adjoint \[H_{\mathrm{nef}} =K_{W_{\mathrm{nef}}}+T_{\mathrm{nef}}+M_{\mathrm{nef}}.\] Here \(M_{\mathrm{nef}}\) is the trace of the nef b-divisor whose nef representative is \(M\) on \(W\). On a common smooth resolution \(\mu:W'\to W\), \(\sigma:W'\to W_{\mathrm{nef}}\), the actual comparison is \[ \mu^*H\sim_{\mathbb Q}\sigma^*H_{\mathrm{nef}}+E_{\mathrm{nef}}, \qquad E_{\mathrm{nef}}\geq0\ \text{\(\sigma\)-exceptional}. \tag{51}\] The program extracts no divisors. We use this particular model also when the nef data are rationally trivial.

If \(\kappa(H)>0\), Equation (51) preserves the divisible section spaces. For \(\kappa(H)=k\) the nef \(H_{\mathrm{nef}}\) is big, and the second alternative holds with \(W_m=W_{\mathrm{nef}}\) and \(H_m=H_{\mathrm{nef}}\). If \(0<\kappa(H)<k\), the exact global comparison in Lemma 84 gives \(\kappa(J)=\kappa(H)\). Take the Iitaka fibration of \(J\) on the source. Its projective image, followed by Stein factorization and a projective resolution, has dimension \(\kappa(H)\). The log adjoint on very general fibers has Kodaira dimension zero by the relative Iitaka construction (OpenAI 2026e, Lemma 2.6), including its persistence under the reduced-exceptional source convention. This is the first alternative.

Suppose \(\kappa(H)\leq0\) and \(M\sim_{\mathbb Q}0\). We may choose zero nef data in this rational equivalence class. The same generalized dlt model \(W_{\mathrm{nef}}\) is then an ordinary dlt model and \(H_{\mathrm{nef}}\sim_{\mathbb Q}K_{W_{\mathrm{nef}}}+T_{\mathrm{nef}}\). Its adjoint is nef, so Proposition 12 makes \(H_{\mathrm{nef}}\) semiample as an actual rational line. Since \(\kappa(H_{\mathrm{nef}})=\kappa(H)\leq0\), a generated multiple has constant image and is trivial. Thus the second alternative holds with \(W_m=W_{\mathrm{nef}}\) and \(H_m=H_{\mathrm{nef}}\sim_{\mathbb Q}0\).

It remains to suppose \[ \kappa(H)\leq0,\qquad M\not\sim_{\mathbb Q}0. \tag{52}\] Here \(\dim S>0\). For a rational \(c\in(0,1)\), put \(H_c=K_W+T+cM\). We claim \[ H_c\ \text{is not pseudo-effective} \quad(0<c<1,\ c\in\mathbb Q). \tag{53}\] Suppose otherwise for one \(c\), and choose an ample rational line \(U\) on \(S\). We will find \(\delta>0\) for which \(H-\delta p^*U\) has an effective representative, forcing \(\kappa(H)\geq\dim S>0\). Choose a rational \(a>\max\{1,a_0\}\) and a small rational \(\lambda>0\) such that \(K_S+aP_S-\lambda U\) is big. This is possible because \(P_S\) is nef and \(K_S+a_0P_S\) is big. Choose \[0<\eta<\lambda\frac{1-c}{a-1} \quad\text{rational}.\] The rational line \(cP_S+\eta U\) is ample. A sufficiently high general member divided by its degree pulls back to a rational boundary transverse to \(T\) with subunit coefficient. Applying the nonvanishing part of Proposition 12 to this ordinary projective log pair gives \[E_c\sim_{\mathbb Q}H_c+\eta p^*U,\qquad E_c\geq0.\] A nonzero section defining \(E_c\) restricts nontrivially to very general \(p\)-fibers in a fixed divisible degree. The hypotheses of the weak-effectivity statement (OpenAI 2026e, Lemma 3.3) are now satisfied: the source is smooth in class \(\mathcal C\), \(T\) is rational SNC, the base \(S\) is smooth projective, and a relative log system is nonzero. Applied to the big line \(K_S+aP_S-\lambda U\), it gives \[E_a\sim_{\mathbb Q}K_W+T+aM-\lambda p^*U,\qquad E_a\geq0.\] For \[\theta=\frac{1-c}{a-c},\qquad \delta=\theta\lambda-(1-\theta)\eta>0,\] the rational interpolation is \[(1-\theta)E_c+\theta E_a \sim_{\mathbb Q}H-\delta p^*U.\] Multiplying pullbacks of sections of \(U\) by a section of the effective left-hand side yields \(\kappa(H)\geq\kappa(p^*U)=\dim S>0\). This contradicts Equation (52) and proves Equation (53).

We next run a program to fiber type while descending the actual nef line \(H_{\mathrm{nef}}\) through every contraction. Choose the common resolution in Equation (51) to determine the nef data. The negativity lemma gives \[F_M:=\sigma^*M_{\mathrm{nef}}-\mu^*M\geq0 \quad\text{\(\sigma\)-exceptional}.\] Indeed its negative is \(\sigma\)-nef and exceptional. For every rational \(0<e<1\), Equation (51) gives \[ \mu^*H_{1-e}\sim_{\mathbb Q} \sigma^*(H_{\mathrm{nef}}-eM_{\mathrm{nef}}) +E_{\mathrm{nef}}+eF_M . \tag{54}\] If \(H_{\mathrm{nef}}-eM_{\mathrm{nef}}\) were pseudo-effective, this equality and effectivity of the last two terms would make \(\mu^*H_{1-e}\), and hence \(H_{1-e}\), pseudo-effective, contrary to Equation (53).

Fix rational \(0<\varepsilon<1\), choose an integer \(r>0\) with \(rH_{\mathrm{nef}}\) Cartier, and choose an integer \(d_0>2kr\). This coefficient will force every ray of the second program to have zero degree for the descended nef line. Decreasing the nef data to \((1-\varepsilon)\boldsymbol M\) preserves generalized dlt singularities: on the displayed resolution the crepant boundary changes by \(-\varepsilon F_M\), so discrepancies do not decrease. Add \(d_0H_{\mathrm{nef}}\), determined as nef data on \(W_{\mathrm{nef}}\). It changes no discrepancies, and the resulting adjoint is \[(1+d_0)H_{\mathrm{nef}}-\varepsilon M_{\mathrm{nef}} =(1+d_0)\left( H_{\mathrm{nef}}-\frac{\varepsilon}{1+d_0}M_{\mathrm{nef}}\right).\] It is not pseudo-effective by the preceding paragraph. The non-pseudo-effective case of Proposition 87 therefore gives a chosen terminating program to a Mori fiber contraction in the \(\mathbb Q\)-factorial generalized dlt category.

At any stage of this second program, let \(H_i\) be the descended rational line, assuming inductively that it is nef and \(rH_i\) is Cartier, and write \[G_i=K_{W_i}+T_i+(1-\varepsilon)M_i.\] An extremal ray negative for \(G_i+d_0H_i\) is negative for \(G_i\). The pair with adjoint \(G_i\) is still generalized dlt: the added nef b-divisor \(d_0H_i\) descends to the current model, so removing it changes no discrepancy. The generalized length bound (Han and Li 2022, Proposition 3.13) gives a curve \(C\) on that ray with \(0<-G_i\cdot C\leq2k\). If \(H_i\cdot C>0\), Cartier integrality would give \[\frac{d_0}{r}\leq d_0H_i\cdot C <-G_i\cdot C\leq2k,\] a contradiction. Thus \(H_i\cdot C=0\). The exact contraction statement (Xie 2022, Theorem 1.5) descends the Cartier line \(rH_i\) along this contraction as an actual Cartier line. In a flip we pull that same line to the flipped side. The descended line is nef: every curve on the contraction base has a curve above it mapping with positive degree, so its degree is nonnegative; pullback preserves nefness. Consequently \(r\) and nefness persist, and the argument applies inductively to all steps and to the final contraction. Each step is crepant for \(H_i\).

Let \(W_m\) be the last birational model of this second program and let \(H_m\) be the descended line on it. Crepancy for the \(H_i\), together with the absence of extraction, turns Equation (51) into Equation (50) on a common resolution, with an effective divisor exceptional over \(W_m\). The final contraction \(h:W_m\to V\) has connected fibers and, by the same exact descent, \[H_m\sim_{\mathbb Q}h^*H_V\] for a rational line \(H_V\) on the normal projective \(V\). If \(V\) is a point, the second alternative holds with \(H_m\sim_{\mathbb Q}0\).

Assume \(\dim V>0\). To obtain the first alternative, it remains to show that the log adjoint on the new very general fibers has Kodaira dimension zero. Resolve the base program by \(\mu:W''\to W\) and \(\nu:W''\to W_m\), and prepare the source over \(W''\), calling it \(g'':(X'',B'')\to W''\). The diagram we use is \[\begin{tikzcd} X'' \arrow[r,"g''"] & W'' \arrow[r,"\nu"] \arrow[d,"\mu"'] & W_m \arrow[r,"h"] & V\\ & W \end{tikzcd}\] with composite \(f= h\nu g''\). Equation (46) and Equation (50) give \[H''\sim_{\mathbb Q}\nu^*H_m+E'', \qquad E''\geq0\ \text{\(\nu\)-exceptional}.\] To see the exceptionality, the error in Equation (50) is already exceptional over \(W_m\), and every \(\mu\)-exceptional prime is exceptional over \(W_m\) as well: otherwise \(W_m\) would extract a divisor over \(W\). For every degree divisible by one fixed integer, Equation (45), exceptional descent, and projection formula now yield \[\begin{align*} f_*\mathcal O_{X''}(q(K_{X''}+B'')) &\lhook\joinrel\longrightarrow (h\nu)_*\mathcal O_{W''}(qH'')\\ &=h_*\mathcal O_{W_m}(qH_m)=\mathcal O_V(qH_V). \end{align*}\] Here \(\nu_*\mathcal O_{W''}(qE'')=\mathcal O_{W_m}\) by normality and \(h_*\mathcal O_{W_m}=\mathcal O_V\). Generic coherent base change, simultaneously for the countably many divisible degrees, bounds the dimension of every such system on a very general \(f\)-fiber by one. The log adjoint on that fiber is pseudo-effective by restriction of a countable sequence of perturbed currents with analytic singularities. Its dimension is less than \(n\), so the lower-dimensional \(\mathcal G\) supplies a nonzero section in some divisible degree. Taking a common multiple shows that the fiber log Kodaira dimension is zero. All maps in the composite have connected fibers; resolving \(V\) and the source keeps this property and gives a smooth projective base of dimension \(0<\dim V<k\). This is the first alternative. ◻

Proof. Use the chosen base identity in Equation (43). In a rationally trivial case fix a rational trivialization of the positive line and put \(P_0=0\); in the big case put \(P_0=(\nu g)^*H_m\). Define the rational divisor \[A:=A_*+g^*E_W.\] The composed actual line identities give \(J\sim_{\mathbb Q}P_0+A\). The divisor \(A\) may at first have signed coefficients. We first prove \(A\geq0\).

In a rationally trivial case a divisible multiple of the base line has a section whose normalized zero divisor is exactly \(E_W\). Its section under the exact global comparison has normalized zero divisor \(A\). Regularity of this section proves \(A\geq0\), also on the initially untested primes. In the big case choose a Kodaira decomposition \[H_m\sim_{\mathbb Q}A_0+G_0, \qquad A_0\text{ ample},\quad G_0\geq0.\] For rational \(0<t<1\), \[H_m\sim_{\mathbb Q} \underbrace{(1-t)H_m+tA_0}_{\text{ample}}+tG_0.\] Fix a prime \(\Gamma\subset X\). A sufficiently divisible section of the ample summand can be chosen not to vanish at the generic point of \((\nu g)(\Gamma)\); its pullback has zero order at \(\Gamma\). Multiplication by the section of \(tG_0\) and then the global comparison gives a regular upstairs section. Its normalized order is \[A_\Gamma+ t\,\mathop{\mathrm{ord}}_\Gamma((\nu g)^*G_0)\geq0.\] Letting \(t\downarrow0\) proves \(A_\Gamma\geq0\). This applies to every prime in \(A\).

Nefness of \(P_0\) now implies \(N(J)\leq A\). Set \(Q=A-N(J)\geq0\), an effective real divisor. By Equation (48), \(Q\) is vertical. Moreover \[\{P_0+Q\}=\{J\}-\{N(J)\}\] is modified nef. We prove \(Q=0\).

Let \(b:X\to Y\) denote \(g:X\to W\) in the first case and \(\nu g:X\to W_m\) in the second. Thus \(\dim Y=k\), and \(b\) has connected fibers. If \(Q\ne0\), let \[l=\max\{\dim b(\Gamma):\Gamma\text{ a component of }Q\} \leq k-1.\] Choose a Kähler class \(\eta\) on \(Y\) (an ample class when \(Y\) is projective) and a Kähler form \(\omega\) on \(X\). On \(H^{1,1}(X,\mathbb R)\) define \[\langle \alpha,\beta\rangle_l =\int_X \alpha\beta(b^*\eta)^l\omega^{\,n-l-2}.\] In this pairing a divisor or rational line denotes its cohomology class. The exponent is nonnegative since \(l\leq k-1\leq n-2\). The restriction of a modified-nef class to a resolution of every prime is pseudo-effective. Applying this to \(\{P_0+Q\}\) and pairing on each component of \(Q\) gives \(\langle P_0+Q,\Gamma\rangle_l\geq0\). The \(P_0\)-term vanishes by dimension, since it is a pullback from \(Y\) and \(\dim b(\Gamma)\leq l\). Summing with the coefficients of \(Q\) gives \[ \langle Q,Q\rangle_l\geq0. \tag{56}\]

We now separate images of codimension at least two, where the mixed Hodge index theorem gives strict negativity, from divisorial images, where only a multiple of the whole fiber can have square zero.

If \(l<k-1\), the mixed Hodge index theorem contradicts this inequality. In this form \(b^*\eta\) has positive square because \(l+2\leq k\), while \(\langle Q,b^*\eta\rangle_l=0\) by image dimension. The form is a limit of the mixed Hodge index forms obtained by replacing \(b^*\eta\) with \(b^*\eta+\epsilon[\omega]\); it therefore has at most one positive direction. In the orthogonal complement of a positive-square vector it is negative semidefinite, with equality only in the radical of the whole form. But \(\langle Q,\omega\rangle_l>0\), since at least one component of \(Q\) has image dimension \(l\). Thus \(Q\) is not in the radical, and its square is strictly negative, contrary to Equation (56).

It remains to treat \(l=k-1\). Terms from primes whose image has smaller dimension vanish in this form, as do cross terms from different divisorial images. Fix a prime divisor \(D\subset Y\) and take all the primes \(E_1,\ldots,E_s\subset X\) dominating it. Write \(a_i=\mathop{\mathrm{ord}}_{E_i}(b^*D)>0\) and \(C_{ij}=\langle E_i,E_j\rangle_{k-1}\). The divisor \(D\) is rational Cartier, also when \(Y=W_m\). Consequently \[C_{ij}\geq0\ (i\ne j),\qquad \sum_j C_{ij}a_j=0.\] The first assertion is positivity of the proper intersection of distinct effective primes. The second pairs \(E_i\) with the pulled-back divisor \(D\); the additional base factor makes the intersection vanish by dimension. Components of \(b^*D\) mapping into a proper subset of \(D\) make no contribution. The graph with edges \(C_{ij}>0\) is connected. Indeed over a very general point of \(D\) the fiber is connected, all its points lie in the components \(E_i\), and an intersection contributes a positive entry precisely when it dominates \(D\).

For any real vector \(x=(x_i)\), the two displayed properties give the exact identity \[x^{\mathsf T}Cx =-\sum_{i<j}C_{ij}a_i a_j \left(\frac{x_i}{a_i}-\frac{x_j}{a_j}\right)^2.\] Thus the block is negative semidefinite with kernel spanned by \((a_i)\). Equation (56) forces every nonzero block of \(Q\) to equal a positive scalar multiple of \((a_i)\). In particular \(Q\) then contains every prime over that divisorial image \(D\).

In the second case this is impossible. The morphism \(\nu:W\to W_m\) is an isomorphism over the generic point of \(D\), and \(E_W\) is exceptional. Equation (44) therefore supplies a component over \(D\) whose \(A\)-coefficient is zero. Its \(Q\)-coefficient is zero as well, contradicting the preceding conclusion.

In the first case, the same zero minimum shows that each divisorial image of a nonzero block of \(Q\) is a component of \(E_W=N(H)\). Here \(\{Q\}\) itself is modified nef. The class \[b_*(\{Q\}[\omega]^{\,n-k})\] is also modified nef on the smooth \(W\). To see this directly, choose small-perturbation positive currents for \(\{Q\}\) with zero generic divisorial Lelong numbers, wedge them with \(\omega^{n-k}\), and push forward. Such a current has no trace mass on a divisor, so its push has no mass on a base divisor: the inverse image is a union of divisors and subsets of higher codimension, all of zero trace mass. The pushed currents therefore have zero generic divisorial Lelong numbers. Their classes tend to the displayed class; a fixed smooth representative for the vanishing error converts this into the defining small-Kähler-perturbation condition for modified nefness. On the other hand, pushing the actual divisor \(Q\) in this formula kills the components with smaller image and gives a nonzero positive linear combination of the components of \(N(H)\). Boucksom’s exceptionality of the components of a divisorial negative part says that no such combination is modified nef (Boucksom 2004, Definition 3.10 and Theorem 3.12). This final contradiction proves \(Q=0\), and hence Equation (55). ◻

Proof. Put \(D_Z=\lfloor\Delta_Z\rfloor\). We first prove the following assertion for every pair and diagram satisfying the hypotheses of this proposition: \[ \mathbf B(L)\cap D_Z=\varnothing. \tag{58}\] Here \(\mathbf B(L)\) is the intersection of the base loci of all positive Cartier multiples of \(L\). Once this is proved, a log canonical threshold at any remaining stable base locus will create a new floor there and give a contradiction.

Take a higher log resolution if needed and put \[K_U+F=r^*L=:\ell,\qquad C=F^{=1},\qquad E=C-\lfloor F\rfloor,\qquad \Delta_F=F-\lfloor F\rfloor.\] The equality defining \(F\) uses the canonical meromorphic identification. The divisor \(F\) is an SNC subboundary. The integral divisor \(E\) is effective, \(r\)-exceptional, and has no component in common with \(C\): a nonexceptional coefficient of \(F\) is a coefficient of the effective \(\Delta_Z\), while every coefficient of \(F\) is at most one. The image of \(C\) is contained in \(D_Z\). By Theorem 72, a divisible multiple of \(L|_{D_Z}\) is globally generated on the whole reduced floor.

Suppose first that a component \(C_i\) dominates \(Y\). Pulling the floor sections to \(C_i\) shows that \((f|_{C_i})^*H_Y\sim_{\mathbb Q}\ell|_{C_i}\) is semiample. Semiampleness descends under a proper surjection to a normal space in this situation. Factor \(C_i\to Y\) through its normal Stein factor \(Y_i\). Projection formula descends generation through the connected-fiber map. For the finite map \(Y_i\to Y\), at each \(y\in Y\) choose a section of the generated pullback line avoiding every point of the finite fiber; finitely many evaluation kernels cannot cover the section space. Lemma 10 gives a norm section of a fixed power of the line on \(Y\), nonzero at \(y\). These sections generate a fixed multiple of \(H_Y\). Equation (57) and normal birational descent then give semiampleness of \(L\).

We may therefore assume that \(C\) is vertical over \(Y\). Choose a sufficiently divisible integer \(u>1\) for which \(uL|_{D_Z}\) is generated, and consider the integral line \[ \mathcal{N} :=\mathcal O_U(u\ell+E-C) =\mathcal O_U\bigl(K_U+\Delta_F+(u-1)\ell\bigr). \tag{59}\] Although the expression on the right uses rational divisors, its sum is the integral line on the left. The restriction sequence is \[0\longrightarrow\mathcal{N} \longrightarrow\mathcal O_U(u\ell+E) \longrightarrow\mathcal O_C(u\ell+E)\longrightarrow0.\] The direct image of the last sheaf is supported on the proper analytic subset \(f(C)\), so it is torsion. Consequently, the two assertions \[ R^1f_*\mathcal{N}\ \text{torsion-free}, \qquad H^1(Y,f_*\mathcal{N})=0 \tag{60}\] will give surjectivity on global sections: torsion-freeness makes the connecting map to \(R^1f_*\mathcal{N}\) zero, and the \(H^1\)-vanishing then removes the obstruction to lifting a global section.

We prove Equation (60) using analytic injectivity. Equation (57) identifies \(\mathcal{N}\otimes K_U^{-1}\) with \(\Delta_F+(u-1)f^*H_Y\) in \(\mathop{\mathrm{Pic}}(U)\otimes\mathbb Q\); this comparison need not be an isomorphism of the corresponding integral lines. Fix a very ample line \(A_Y\) on \(Y\) and its Fubini–Study metric. A Kodaira decomposition of the big nef \(H_Y\), given arbitrarily small weight as in the proof of Lemma 90, writes \((u-1)H_Y\) as a positive rational multiple of \(A_Y\) plus an effective rational divisor whose fixed singular contribution is arbitrarily small and whose remaining contribution is a general divided ample member. Choose the small weight below the log canonical threshold of its pullback relative to the klt SNC boundary \(\Delta_F\), and choose the ample member generally. Then the resulting boundary on \(U\) is klt. Choose a positive integer \(q_0\) clearing all its denominators and realizing both the chosen Kodaira decomposition and the preceding rational-line comparison as line isomorphisms, in particular \[(\mathcal{N}\otimes K_U^{-1})^{\otimes q_0} \simeq \mathcal O_U(q_0\Delta_F)\otimes f^*\mathcal O_Y\bigl(q_0(u-1)H_Y\bigr).\] The divisor metrics and the Fubini–Study metric define a singular Hermitian metric on \((\mathcal{N}\otimes K_U^{-1})^{\otimes q_0}\). Its \(q_0\)-th root is a metric on the actual line \(\mathcal{N}\otimes K_U^{-1}\), with multiplier ideal \(\mathcal O_U\) and curvature dominating a positive multiple of the pullback Fubini–Study form. The same statements hold after every nonnegative twist by \(f^*A_Y\).

The injectivity theorem (Fujino and Matsumura 2021, Theorem A) applies on the compact Kähler \(U\): multiplication by the pullback of any nonzero section of a positive power of \(A_Y\) is injective on \[H^1(U,\mathcal{N}\otimes f^*A_Y^j) \quad(j\geq0)\] into the correspondingly higher twist. Its assumptions are exactly the semipositive smooth metric on the multiplying line, the preceding positive lower curvature bound, and the trivial multiplier ideal. This injectivity implies Equation (60). For the second assertion, the Leray edge map injects \(H^1(Y,f_*\mathcal{N})\) into \(H^1(U,\mathcal{N})\). Multiplication by a section of a sufficiently high \(A_Y\)-power sends this subspace to zero, since Serre vanishing kills \(H^1(Y,f_*\mathcal{N}\otimes A_Y^j)\) for large \(j\). Injectivity upstairs forces the original subspace to be zero. For the first assertion, suppose \(R^1f_*\mathcal{N}\) has a nonzero torsion subsheaf. A section of a sufficiently high power of \(A_Y\) annihilates a nonzero such subsheaf. After a further sufficiently large twist it has a nonzero global section, and Serre vanishing and Leray identify that section with a nonzero class in \(H^1(U,\mathcal{N}\otimes f^*A_Y^j)\). Multiplication annihilates this class, again contradicting injectivity. This proves both assertions.

The resulting surjectivity extends the floor sections as follows. Pull any section of \(uL|_{D_Z}\) to \(C\), multiply by the canonical section of \(E|_C\), and extend by this surjectivity. The extended section descends to \(Z\), since \(r_*\mathcal O_U(E)=\mathcal O_Z\) by exceptionality and normality. On the strict transforms of the components of \(D_Z\), division by the canonical section of \(E\) recovers the prescribed section. The descended restriction therefore equals it on the whole reduced \(D_Z\), since equality holds generically on every component. Since \(uL|_{D_Z}\) is generated, this proves Equation (58).

We finish by showing that this stable base locus is empty. The base loci of factorial Cartier multiples form a descending sequence of compact analytic subsets and hence stabilize. Sections exist because \(H_Y\) is big and Equation (57) descends their pullbacks. Choose a divisible degree \(v\) whose base locus is \(\mathbf B(L)\), and let \(\mathfrak b\) be its base ideal. If \(\mathbf B(L)\ne\varnothing\), the ideal is a unit near \(D_Z\), while \((Z,\Delta_Z)\) is klt outside \(D_Z\). Its log canonical threshold \[t=\operatorname{lct}_{(Z,\Delta_Z)}(\mathfrak b)\] is therefore a positive rational number, attained by a divisor whose center is contained in \(\mathbf B(L)\). This follows directly on a simultaneous log resolution of the pair and the ideal: the threshold is the minimum of the positive rational discrepancies divided by the positive integral ideal orders.

Choose an integer \(s>t\) and general divisors \(D_1,\ldots,D_s\in|vL|\). On the same resolution their fixed part is the divisor of \(\mathfrak b\); their free transforms are jointly transverse to the SNC data. Hence \[\left(Z,\ \Delta_Z+\frac{t}{s}\sum_{i=1}^s D_i\right)\] is log canonical: the fixed part is at its log canonical threshold and the free components have coefficients \(t/s<1\). There is a new lc place centered in \(\mathbf B(L)\), and the new adjoint satisfies the actual identity \[L_{\mathrm{new}}\sim_{\mathbb Q}(1+tv)L.\] Apply (Hacon et al. 2026, Theorem 3.3) to this compact Kähler lc rational pair. It gives a projective strongly \(\mathbb Q\)-factorial dlt modification \(d:Z'\to Z\). Since the pair is lc, all extracted divisors have log discrepancy zero in our convention, so the modification is crepant. The source is again compact Kähler. Choose its defining dlt log resolution, with exceptional crepant coefficients strictly below one and an isomorphism at the general point of every lc stratum; resolving the remaining data away from those points gives the resolution required by Definition 16. The floor on \(Z'\) has a point over the center of this lc place, hence over \(\mathbf B(L)\). The new nef adjoint is \(d^*L_{\mathrm{new}}\). On a common resolution with \(U\), it satisfies Equation (57) with the big nef line \((1+tv)H_Y\). The previously proved assertion (58) therefore makes its stable base locus disjoint from its floor. On the other hand, normality and projection formula identify all divisible sections under \(d\), and give \[\mathbf B(d^*L_{\mathrm{new}}) =d^{-1}\mathbf B(L).\] This locus contains the stated floor point, a contradiction. Thus \(\mathbf B(L)=\varnothing\); factorial stabilization provides an actual globally generated multiple of \(L\). ◻

Proof of Proposition 83. By Proposition 9 we can begin on the resolved source in the statement. We use a secondary induction on the positive base dimension \(k\). Apply Lemma 84 and then Lemma 85 to obtain the prepared data and the pseudo-effective line \(H\).

If \(a(W)=0\), Lemma 86 and the first case of Lemma 90 give \(J\sim_{\mathbb Q}N(J)\), which is already the required decomposition with trivial positive part. If \(W\) is projective, apply Lemma 89. Its first alternative has a strictly smaller positive base dimension, so the secondary induction and birational transfer finish the proof. In its second alternative, take a common smooth base resolution in Equation (50) and prepare the source over it. Equation (46) adds an effective divisor exceptional over the old base. It is also exceptional over \(W_m\), because the base program extracted no divisors. Thus the second case of Lemma 90 applies.

We have obtained \[J\sim_{\mathbb Q}P_0+A,\qquad A=N(J)\geq0,\] with \(P_0\) nef. A rationally trivial \(P_0\) finishes the proof. In the remaining case \(P_0=b^*H_m\), where \(b:X\to W_m\) is proper and \(H_m\) is big and nef on the projective \(W_m\). Apply Proposition 23 to this nef-plus-negative presentation. It gives a globally strongly \(\mathbb Q\)-factorial dlt compact Kähler model \((Z,\Delta_Z)\) with nef adjoint \(L_Z\). The ordinary dlt endpoint has the resolution property of Definition 16. On a common smooth resolution \(\mu:U\to X\), \(r:U\to Z\) its exact positive-line comparison is \[r^*L_Z\sim_{\mathbb Q}\mu^*P_0 =(b\mu)^*H_m .\] Proposition 91 makes \(L_Z\) semiample. Hence \(\mu^*P_0\) is semiample. Lemma 6 identifies \(\mu^*A=N(\mu^*J)\), so this is precisely \(\mathcal G_n\) on \(U\). Finally Proposition 9 transfers the decomposition back through all the prepared source modifications. ◻

Proof. If the Albanese map of \(X\) is nonconstant, simplicity makes its image \(n\)-dimensional: a positive-dimensional general fiber of smaller dimension would be a forbidden subvariety. At a point where the Albanese map has rank \(n\), some \(n\) invariant one-forms have nonzero wedge after pullback. This gives a nonzero holomorphic section of \(K_X\), contrary to our assumption. Thus \(H^1(X,\mathcal O_X)=0\). The exponential sequence embeds \(\mathop{\mathrm{Pic}}(X)\) into the countable group \(H^2(X,\mathbb Z)\).

The manifold \(X\) is not uniruled, by simplicity. Ou’s non-pseudo-effectivity criterion for the canonical class therefore gives \(L\ge0\) (Ou 2025, Theorem 1.1). We recall how the slope and foliation results in the same paper give the more precise inequality in Equation (61). Let \(\gamma\) be any class in the full dual of the pseudo-effective cone, and use the slope \[\mu_\gamma(\mathcal{E}) =\frac{c_1(\mathcal{E})\cdot\gamma}{\mathop{\mathrm{rk}}\mathcal{E}}\] for torsion-free sheaves. If \(\mu_{\gamma,\min}(\Omega_X)<0\), the first Harder–Narasimhan piece \(\mathcal{T}\subset T_X\) has strictly positive slope. It is saturated and semistable. The tensor slope inequality shows that the bracket \(\bigwedge^2\mathcal{T}\to T_X/\mathcal{T}\) is zero: the minimum slope of its source is at least \(2\mu_\gamma(\mathcal{T})\), whereas the maximum slope of its target is smaller than \(\mu_\gamma(\mathcal{T})\). Consequently \(\mathcal{T}\) is a foliation. Its dual has strictly negative maximum slope and is not pseudo-effective by (Ou 2025, Proposition 4.5). Ou’s foliation theorem (Ou 2025, Theorem 1.4) makes this foliation the tangent foliation of a meromorphic fibration with compact general leaf closures. Its rank is strictly between zero and \(n\): rank \(n\) would give \(\mu_\gamma(T_X)=-L\cdot\gamma/n\le0\). The general leaf closure is therefore a positive-dimensional proper subvariety through a general point, again contradicting simplicity.

We have proved \(\mu_{\gamma,\min}(\Omega_X)\ge0\). In its Harder–Narasimhan filtration all quotient slopes are now nonnegative, so \(\mu_{\gamma,\max}(\Omega_X)\le L\cdot\gamma\). The tensor slope inequality (Ou 2025, Lemma 4.4) gives \[c_1(\mathcal{H})\cdot\gamma \le \mu_{\gamma,\max}(\Omega_X^{\otimes k}) \le kL\cdot\gamma .\] Separation by the dual cone proves Equation (61). For \(k=0\) this is also immediate from the effective zero divisor of a nonzero map \(\mathcal{H}\to\mathcal O_X\).

It remains to choose \(x\) uniformly. For any vector bundle \(\mathcal{V}\), independent global sections are generically pointwise independent. Indeed, choose a maximal pointwise independent subfamily on a dense open set. Expressing any other section in that family by minors gives meromorphic function coefficients. They are constant because \(a(X)=0\), so maximality among a linearly independent family forces all its members to be pointwise independent. Thus evaluation on \(H^0(X,\mathcal{V})\) is injective away from a proper analytic subset (unless that vector space is zero, in which case there is no restriction). Apply this to \(\mathcal{V}=\mathcal{H}^{-1}\otimes\Omega_X^{\otimes k}\) for every \(\mathcal{H}\in\mathop{\mathrm{Pic}}(X)\) and \(k\ge0\). There are countably many such bundles.

For \(x\) outside the resulting exceptional set, write any line bundle on \(Y=\mathop{\mathrm{Bl}}_xX\) uniquely as \(a^*\mathcal{H}\otimes\mathcal O_Y(mF)\), \(m\in\mathbb Z\). A line map from this bundle to \((a^*\Omega_X)^{\otimes k}\), restricted away from \(F\), extends over \(x\) to a section of \(\mathcal{H}^{-1}\otimes\Omega_X^{\otimes k}\) by Hartogs’ theorem. If \(m>0\), this section vanishes at \(x\), since \(a_*\mathcal O_Y(-mF)=\mathcal I_x^m\). The choice of \(x\) rules this out. Thus \(m\le0\), and Equation (61) yields \[c_1(a^*\mathcal{H}\otimes\mathcal O_Y(mF)) \le k a^*L+m\{F\}\le k a^*L.\] If a target has a finite filtration whose graded pieces are direct sums of the indicated cotangent tensors, take the first nonzero associated graded component of the line map and then a nonzero direct-sum component. The preceding argument applies with the tensor order of that component. ◻

Proof. Set \(u_0=\eta/2\). Blowing up \(z'\), the class \(K-u_0\{G'\}\) is Kähler. The Fujita decomposition is unchanged near \(G'\), so it supplies a Kähler current for \(\alpha_{u_0}=b^*\alpha-u_0\{G\}\) that is smooth near \(G\). Its restriction there has class \(u_0c_1(\mathcal O_{\mathbb P^{e-1}}(1))\). The restricted volume is consequently \(u_0^{e-1}\): the current gives this lower bound, and the volume of the restricted class gives the opposite bound.

Let \(\tau\) denote the left side of Equation (65). The classes \(\alpha_u=b^*\alpha-u\{G\}\) are big for \(0\le u<\tau\), since \(\alpha_0\) is big and the pseudo-effective cone is convex. The concave function \(f(u)=\mathop{\mathrm{vol}}(\alpha_u)^{1/e}\) satisfies, by Equation (63), \[f'(u_0)=-\frac{u_0^{e-1}}{f(u_0)^{e-1}}.\] Its tangent line at \(u_0\) must remain positive up to \(\tau\). Therefore \[\tau\le u_0+\frac{f(u_0)^e}{u_0^{e-1}} \le u_0+\frac{\mathop{\mathrm{vol}}(\alpha)}{u_0^{e-1}},\] which is Equation (65). ◻

Proof. Let \(C\subset X^2\) be irreducible through a pair whose coordinates are very general in \(X\). Each projection image is a point or all of \(X\), by properness and simplicity. If exactly one is a point, \(C\) is the corresponding full slice. If both projections are surjective, test their fibers at a general point of \(C\) whose coordinates are still very general. A positive-dimensional fiber must be the full other factor. Thus either \(C=X^2\), or \(\dim C=n\) and both projections are generically finite.

We show that subvarieties of the latter kind cannot sweep \(X^2\). The space of compact \(n\)-cycles on a compact Kähler manifold has countably many irreducible components, each compact (Lieberman 1978, Theorem 1.1). On a component containing integral cycles generically finite over both factors, integrality holds on a dense open set and the two projection degrees remain positive, as can also be tested by intersection with pulled-back Kähler forms. Suppose the incidence over one such component dominates \(X^2\). Resolve the parameter space and the dominating incidence component, obtaining maps \[h:\mathcal Y\longrightarrow\mathcal S,\qquad F_i:\mathcal Y\longrightarrow X\] between smooth spaces. For general \(t\in\mathcal S\), the fiber \(Y_t=h^{-1}(t)\) is smooth, compact, and bimeromorphic to the integral cycle, hence irreducible. Its maps \(f_i=F_i|_{Y_t}:Y_t\to X\) are generically finite. Choose \(t\) also so that \(Y_t\) meets the dense open set where \(d(F_1,h)\) is invertible and \(d(F_1,F_2)\) has rank \(2n\). The first assertion follows from generic finiteness of \(f_1\), and the second from the assumed dominance in \(X^2\).

Our immediate aim is to turn this dominance into a meromorphic frame of \(f_2^*T_X\). Its determinant would meromorphically trivialize \(f_2^*K_X^{-1}\), and hence also \(f_2^*K_X\). On \(Y_t\), the equal-rank bundle map \[d(F_1,h)|_{Y_t}:T_{\mathcal Y}|_{Y_t} \longrightarrow f_1^*T_X\oplus (\mathcal O_{Y_t}\otimes T_t\mathcal S)\] has a meromorphic inverse, given by its adjugate and determinant. For fixed \(v\in T_t\mathcal S\), apply this inverse to \((0,v)\) and then apply \(dF_2\). The result is a meromorphic section of \(f_2^*T_X\). At a general point where \(d(F_1,F_2)\) has rank \(2n\), the resulting map \(T_t\mathcal S\to f_2^*T_X\) is surjective: in local coordinates \((F_1,h)\), it is the derivative of \(F_2\) in the parameter directions with \(F_1\) fixed. Choose \(n\) fixed vectors \(v_i\) whose values there are independent. Their wedge is a nonzero meromorphic section of \(f_2^*K_X^{-1}\) on the irreducible \(Y_t\); its inverse trivializes \(f_2^*K_X\) meromorphically.

Factor the proper generically finite map \(f_2\) as \[Y_t\longrightarrow \overline Y_t \xrightarrow{\nu}X\] by Stein factorization. Here \(\overline Y_t\) is normal and irreducible, the first map is a modification, and \(\nu\) is finite of degree \(N>0\). Meromorphic sections descend across a modification of a normal space, so the section descends to a nonzero meromorphic section of \(\nu^*K_X\). Lemma 10 gives a nonzero meromorphic section of \(K_X^{\otimes N}\). This norm is defined in local finite analytic fraction algebras, so it remains available even though the global meromorphic function field of \(X\) is just \(\mathbb C\). It contradicts our assumption.

Consequently the incidence image of each such cycle component is a proper compact analytic subset of \(X^2\). There are only countably many components, so their union misses a very general pair. This proves the assertion for \(X^2\).

Now let \(W\subset Z\) be irreducible through a very general point. Its image is a point, a full factor slice, or \(X^2\). If its generic relative dimension is one, it is the full inverse image of that image. Otherwise it is a multisection of the restricted \(\mathbb P^1\)-bundle. Over a slice or over \(X^2\), a multisection is a divisor with line bundle \(\mathcal O(k)\otimes\pi^*\mathcal{H}\), \(k>0\). Its homogeneous equation has coefficients in \[H^0\bigl(S,\mathcal{H}\otimes \mathcal{L}_1^{\otimes i} \otimes\mathcal{L}_2^{\otimes(k-i)}\bigr), \qquad 0\le i\le k,\] where \(S\) is the relevant slice or \(X^2\), and a fixed-factor line is understood as constant. A single nonzero monomial cuts out only the axis sections and vertical divisors. A multisection not on an axis therefore has two nonzero coefficients. Their quotient is a meromorphic section of a nonzero power of \(\mathcal{L}_1\otimes\mathcal{L}_2^{-1}\). Over a slice this is, up to inversion and a constant line, a positive power of \(K_X\). Over \(X^2\), restriction to a general factor slice gives the same conclusion. This is impossible. The axes themselves miss a very general point of \(Z\), proving the claim. ◻

Proof. Pull \(Z\) to the Fujita model \(Y_P^2\) from Equation (67), and put \(H=P'_1+P'_2\) on this pulled-back bundle. There is a further smooth projective modification \(\nu:\widehat Z\to\mathbb P_{Y_P^2}(\mu^*\mathcal{L}_1\oplus \mu^*\mathcal{L}_2)\) and a Fujita decomposition of the pullback of \(M\) whose Kähler part is exactly \[ K=\frac12\nu^*H+\Theta , \tag{71}\] where \(\Theta\) is Kähler and has degree \(q\) on a general vertical line. The modification and the divisor part miss that line.

Here is the construction. It suffices to decompose \(H/2+q\xi\) as \(\Theta\) plus an effective divisor, clean on a whole general vertical line. Since \(\mathcal O(1)\) is relatively ample, a small positive class of the form \(\delta(\xi+cH)\) is Kähler for some \(c>0\). Choose \(\delta>0\) so small that \(\delta<q\) and the remaining horizontal part of \(H/2\) is Kähler. Represent the remaining \(\xi\) first as \(\{A_1\}+L_1\) and then as \(\{A_2\}+L_2\). Regularize the pseudo-effective classes \(L_i\) with analytic singularities, paying their arbitrarily small negative errors from that horizontal Kähler part. This gives two Kähler currents in \(H/2+q\xi\), each smooth off its own axis and a proper horizontal analytic set. The maximum of their potentials is locally bounded along an entire general vertical line, since the two axes are disjoint. Analytic regularization preserving a smaller Kähler lower bound (Boucksom 2004, Theorem 2.1(ii)) gives a Kähler current with analytic singularities missing that line. Resolve its singularities and make the small Kähler adjustment described before Lemma 94. The divisor still misses a general vertical line, so the residual class \(\Theta\) has degree exactly \(q\) there. Adding the other half of \(H\) and the pullbacks of the divisor parts \(D_P\) gives Equation (71). Its two summands are nef, with \(\Theta\) Kähler, so every mixed intersection used in the following bounds is nonnegative.

At a very general point of \(\widehat Z\), Lemma 95 lists the possible positive-dimensional subvarieties: the vertical line, the strict transforms of the two full bundles over slices, and \(\widehat Z\). They have multiplicity one there. By Equation (71) and nonnegativity of mixed intersections of nef classes, their top intersections are respectively bounded below by \[\begin{align*} K\cdot(\text{vertical line})&=q, \\ \int_{\text{slice}}K^{n+1} &\ge \frac{n+1}{2^n}\,qv,\tag{72}\\ \int_{\widehat Z}K^d &\ge \frac{d}{2^{2n}}\binom{2n}{n}\,qv^2. \end{align*}\] For example, the middle line is the term containing one factor \(\Theta\) and \(n\) factors from the varying \(P'\); pushforward of \(\Theta\) along a general vertical line is \(q\). The last line is the term with one \(\Theta\) and \(2n\) horizontal factors. These intersections give \(\mathop{\mathrm{vol}}(M)\ge c_nq\). The ordinary Seshadri formula and \(q\ge1\) give \(\epsilon(K,z')\ge c_nq^{1/d}=c_nA_q\) at a very general \(z'\): each possible dimension is at most \(d\), and every numerator in that formula is at least \(c_nq\).

For the upper bound, subtract the axis \(A_1\). For \(0\le u\le q\) put \[M_u=M-u\{A_1\} =P_1+P_2+uL_1+(q-u)\xi .\] The endpoint \(M_q\) is pulled back from \(X^2\) and has zero volume on the \(d\)-fold \(Z\). It is pseudo-effective, and \(M_0\) is big by the preceding construction, so \(M_u\) is big for \(u<q\). The restriction of \(M_u\) to \(A_1\simeq X^2\) is \[(P+uL)_1+(P+(q-u)L)_2.\] The restricted volume along \(A_1\) is at most the volume of this restriction. For big classes \(\alpha_i\) on manifolds of dimensions \(e_i\), \[ \mathop{\mathrm{vol}}(\mathop{\mathrm{pr}}_1^*\alpha_1+\mathop{\mathrm{pr}}_2^*\alpha_2) =\binom{e_1+e_2}{e_1}\mathop{\mathrm{vol}}(\alpha_1)\mathop{\mathrm{vol}}(\alpha_2). \tag{73}\] One can see this directly from the non-pluripolar product formula (Boucksom et al. 2010): the envelope with minimal singularities of a sum on a product is the sum of the two envelopes, by testing the defining inequality on successive slices. Its top product has only the indicated binomial term. Since \(L\le P/r\) and \(r\ge q^2\), monotonicity and Equation (73) bound the restriction volume by \[\binom{2n}{n}(1+u/r)^n(1+(q-u)/r)^n\le C_n.\] Integrating Equation (63) from \(0\) to \(q\) gives \(\mathop{\mathrm{vol}}(M)\le C_nq\). Finally apply Lemma 94 with \(e=d\), \(\eta=c_nA_q\), and \(\mathop{\mathrm{vol}}(M)\le C_nq\). Both terms on the right of Equation (65) are at most \(C_nA_q\), since \(A_q^d=q\). This proves Equation (70). ◻

Proof. The restriction of \(c_1(\Lambda)\) to each fiber is represented by the restriction of the Kähler form \(\beta\). The fiberwise criterion for relative ampleness makes \(\Lambda\) relatively ample after clearing denominators. Relative Serre vanishing then kills the higher direct images for all sufficiently large divisible \(j\). Analytic Grothendieck–Riemann–Roch (Levy 1987), in degrees zero and two, gives \[R_j=\frac{f_*c_1(\Lambda)^e}{e!}j^e+O(j^{e-1}),\qquad c_1(\mathcal F_j) =\frac{f_*c_1(\Lambda)^{e+1}}{(e+1)!}j^{e+1}+O(j^e).\] The degree-zero pushforward is \(w\). Expanding \(\beta=f^*G+c_1(\Lambda)\) shows that \[f_*\beta^{e+1} =f_*c_1(\Lambda)^{e+1}+(e+1)wG,\] because terms with at least two horizontal factors have negative fiber degree after pushforward. This proves the equality in Equation (75); the cohomological GRR equality is an equality in Bott–Chern cohomology by the \(\partial\bar\partial\)-lemma on compact Kähler manifolds. Pushforward of the positive form \(\beta^{e+1}\) is a positive closed \((1,1)\)-current, so \(B_0\) is pseudo-effective.

To prove Equation (76), we need a nef class on the base. The class \(B_0\) is presently only known to be pseudo-effective. We flatten \(f\) to obtain a nef class above the base and compare it with the pullback of \(B_0\). Take a smooth projective flattening modification \(p:Z'_0\to Z_0\), the equidimensional main transform \(\overline U\) of \(U\times_{Z_0}Z'_0\), and a resolution \(U'\to\overline U\). Write \(f':U'\to Z'_0\) and \(q:U'\to U\) for the maps and \(\beta'=q^*\beta\). The class \[B'_0=\frac{f'_*(\beta')^{e+1}}{(e+1)w}\] is nef. It is enough to show that the positive pushforward current representing this class has zero Lelong numbers. This current can be computed by integration on the cycle \(\overline U\) of the smooth form pulled back from \(U\). At a base point, cover the compact fiber by finitely many coordinate neighborhoods, each embedded in a product of base coordinates and ambient coordinates. In the mass over a base ball of radius \(\delta\), after wedging with a base Euclidean form to power \(b_0-1\), the integrand is bounded by a finite sum of projection volume forms using \(b_0-1\) base coordinates and \(e+1\) generic linear combinations of all coordinates of the ambient product, including the remaining base direction. These projections can be chosen finite on the neighborhoods: fixing the \(b_0-1\) base coordinates leaves local dimension at most \(e+1\), by equidimensionality, and generic ambient coordinates finish a finite projection. After shrinking to compact subneighborhoods their degrees are bounded. Change of variables therefore bounds each integral by \(O(\delta^{2(b_0-1)})\) times the measure of the remaining coordinate range. On each compact subneighborhood these ranges, taken over closed base balls, are nested compact sets whose intersection is the image of the central fiber. That image has measure zero in the \(e+1\) coordinates, because the fiber has dimension \(e\). Continuity of finite measure from above shows that the range measures tend to zero. Thus the mass is \[o\bigl(\delta^{2(b_0-1)}\bigr).\] This also covers \(b_0=1\), when it asserts absence of an atom. The pushforward current has zero Lelong numbers at every point, and Demailly regularization (Demailly 1992, Theorem 1.1) proves that its class \(B'_0\) is nef.

Pushforward under \(p\) gives \(p_*B'_0=B_0\). For a modification between smooth compact Kähler manifolds, the kernel of pushforward on real Bott–Chern \((1,1)\)-classes is generated by the classes of its exceptional prime divisors. Since \(p_*p^*B_0=B_0\), it follows that \(B'_0-p^*B_0\) is an exceptional real divisor class. It is \(p\)-nef because \(B'_0\) is nef. Apply relative negativity to this exceptional real divisor. Locally over the base, the usual proof for a projective modification cuts by general hyperplanes to a surface and uses the negative definite intersection matrix of exceptional curves; it forces every coefficient of a relatively nef exceptional divisor to be nonpositive. Thus \[B'_0=p^*B_0-\{D_0\},\qquad D_0\ge0\ \text{\(p\)-exceptional}.\] In particular \[ \int_{Z'_0}(B'_0)^{b_0} =\mathop{\mathrm{vol}}(B'_0)\le\mathop{\mathrm{vol}}(p^*B_0)=\mathop{\mathrm{vol}}(B_0). \tag{77}\] Choose a Kähler class \(\omega'\) on \(Z'_0\) and put \(C=B'_0+\varepsilon\omega'\). For \(0\le k\le b_0\), set \[I_k=\int_{U'}(\beta')^{e+k}(f'^*C)^{b_0-k}.\] These are mixed intersections of nef classes. The first two satisfy \[I_0=w\int_{Z'_0}C^{b_0},\qquad I_1=(e+1)w\int_{Z'_0}B'_0C^{b_0-1} \le(e+1)w\int_{Z'_0}C^{b_0}.\] The mixed nef inequalities make the sequence \(I_k\) log-concave. Every \(I_k\) is positive: on the dense open where \(q\) is a local biholomorphism and \(f'\) is a submersion, \(\beta'\) is positive definite and \(f'^*C\) has rank \(b_0\), so the defining top form is strictly positive. The successive ratios are therefore at most \(I_1/I_0\le e+1\). Thus \[\int_U\beta^{b_0+e}=I_{b_0} \le(e+1)^{b_0}w\int_{Z'_0}C^{b_0}.\] Letting \(\varepsilon\downarrow0\) and using Equation (77) proves Equation (76). ◻

Proof. Choose a real class \(H_0\) on \(Y\) so that \(\zeta_{\mathcal{E}}+\pi_{\mathcal{E}}^*H_0\) is Kähler; relative ampleness of \(\mathcal O(1)\) permits such a choice. Take numbers \(\delta\downarrow0\) with \(b+\delta>0\) rational. Adding \(\delta(\zeta_{\mathcal{E}}+\pi_{\mathcal{E}}^*H_0)\) to the pseudo-effective class in Equation (78) makes it big. A Fujita decomposition on a smooth projective modification \(h:U\to\mathbb P_Y(\mathcal{E})\), with the divisor coefficients rounded upwards as in Equation (64), has the form \[\beta=f^*(A+\delta H_0)+c_1(\Lambda),\qquad \Lambda=(b+\delta)h^*\mathcal O(1)-\mathcal O_U(D') \ \text{in }\mathop{\mathrm{Pic}}(U)\otimes\mathbb Q,\] where \(f=\pi_{\mathcal{E}}h\), \(\beta\) is Kähler, and \(D'\ge0\) is rational. For divisible \(j\), \[f_*\mathcal O_U(j\Lambda)\ \subseteq\ \mathop{\mathrm{Sym}}^{j(b+\delta)}\mathcal{E} .\] Its determinant of rank \(R_j\) consequently maps into \(\mathcal{E}^{\otimes j(b+\delta)R_j}\). The assumed line inequality gives \[\frac{c_1(\mathcal F_j)}{jR_j}\le(b+\delta)D.\] Lemma 97 makes \(A+\delta H_0+\lim c_1(\mathcal F_j)/(jR_j)\) pseudo-effective. Adding the preceding pseudo-effective difference shows that \(A+\delta H_0+(b+\delta)D\) is pseudo-effective. Let \(\delta\downarrow0\). This proves the corollary. In this argument only \(b+\delta\) and the coefficients of \(D'\) are rational; \(A\), \(H_0\), and \(D\) remain real classes. ◻

Proof. First let \(s\) be any rational number in \((0,s_{\max})\) with \(v_E(s)>0\), and use the approximation in Equation (64) for any current used to compute this restricted volume. In Lemma 97, the data for Equation (80) are \[b_0=d,\quad e=d-1,\quad G=2M,\quad \Lambda=s h^*\mathcal O_E(1)-\mathcal O_U(D') \ \text{in }\mathop{\mathrm{Pic}}(U)\otimes\mathbb Q.\] For large divisible \(j\), the associated sheaves satisfy \[ \mathcal F_j=f_*\mathcal O_U(j\Lambda) \subseteq\mathop{\mathrm{Sym}}^{js}\Omega_Z,\qquad R_j=\mathop{\mathrm{rk}}\mathcal F_j,\qquad 0<w\le s^{d-1}. \tag{83}\] The inclusion follows by pushing \(\mathcal O_U(-jD')\subseteq\mathcal O_U\) through \(h\). For the last inequality restrict the decomposition to a general \(\mathbb P^{d-1}\)-fiber: monotonicity of volume bounds the Kähler volume there by \(\mathop{\mathrm{vol}}(s\zeta)=s^{d-1}\).

We claim that the class \(B_0\) of Equation (75) satisfies \[ 0\le B_0\le C_nM . \tag{84}\] Here is the determinant calculation, including its vertical sign. The equality \(H^1(X,\mathcal O_X)=0\) and the Künneth decomposition give \(\mathop{\mathrm{Pic}}(X^2)=\mathop{\mathrm{pr}}_1^*\mathop{\mathrm{Pic}}(X)\oplus\mathop{\mathrm{pr}}_2^*\mathop{\mathrm{Pic}}(X)\). Hence, for some lines \(\mathcal{Q}_{j,i}\) on \(X\) and integer \(\ell_j\), \[\det\mathcal F_j =\pi^*(\mathcal{Q}_{j,1}\boxtimes\mathcal{Q}_{j,2}) \otimes\mathcal O_Z(\ell_j),\qquad c_1(\det\mathcal F_j)=Q_j+\ell_j\xi,\] where \(Q_j=c_1(\mathcal{Q}_{j,1})_1+c_1(\mathcal{Q}_{j,2})_2\). Taking the determinant of the inclusion in Equation (83) gives a nonzero line map into \(\Omega_Z^{\otimes k}\), \(k=jsR_j\). It is first defined where \(\mathcal F_j\) is locally free and extends over the codimension-two complement. Filter this tensor using \[0\longrightarrow\pi^*\Omega_{X^2}\longrightarrow\Omega_Z \longrightarrow \mathcal O_Z(-2)\otimes\pi^*(\mathcal{L}_1\otimes\mathcal{L}_2) \longrightarrow0 .\] A nonzero associated graded component has \(i\) relative factors and \(k_1,k_2\) cotangent factors from the first and second factors of \(X^2\), respectively, where \(0\le i\le k\) and \(k_1+k_2=k-i\). Its relative degree is \(-2i\). Pushing the component to \(X^2\) forces \[m_j:=-2i-\ell_j\ge0,\qquad \ell_j\le-2i\le0.\] A nonzero homogeneous monomial in \(\mathop{\mathrm{Sym}}^{m_j}(\mathcal{L}_1\oplus\mathcal{L}_2)\), with exponent \(0\le m_{j,1}\le m_j\) in the first factor, gives, on the two general slices, \[c_1(\mathcal{Q}_{j,1})\le(i+m_{j,1}+k_1)L,\qquad c_1(\mathcal{Q}_{j,2})\le(i+m_j-m_{j,1}+k_2)L\] by Lemma 93. Each coefficient on the right is at most \(k-\ell_j\). Since \(L\ge0\), we obtain \[ \ell_j\le0,\qquad Q_j\le(jsR_j-\ell_j)(L_1+L_2). \tag{85}\]

Lemma 97 supplies a limit of the entire determinant class divided by \(jR_j\). The projective-bundle decomposition of Bott–Chern cohomology gives separate limits \(Q\) and \(\ell\) of the two displayed components. Since \[B_0=2(P_1+P_2)+(2q+\ell)\xi+Q\ge0,\] testing on a general vertical line gives \(2q+\ell\ge0\). Equation (85) gives \(\ell\le0\) and \(Q\le(s-\ell)(L_1+L_2)\). Use \(\xi\ge0\), \(-\ell\le2q\), \(L\le P/r\), \(r\ge q^2\), and \(s\le C_nA_q\le C_nq\). They give \[B_0\le2(P_1+P_2)+2q\xi+(s+2q)(L_1+L_2) \le C_nM,\] which proves Equation (84).

Volume monotonicity, Lemma 96, and Equation (76) now imply \[ \int_U\beta^{2d-1}\le C_nwq,\qquad v_E(s)\le C_nq\,s^{d-1}. \tag{86}\] For the second inequality, approximate the mass of each current arbitrarily closely by \(\beta\) and use \(w\le s^{d-1}\), then take the supremum over currents. This proves it for rational \(s\); continuity of divisorial restricted volume extends it to every \(s\in(0,s_{\max})\).

At \(s_{\max}\) the class is on the boundary of the pseudo-effective cone, so its volume is zero. The product formula in Equation (73), modification invariance, and the derivative formula in Equation (63) yield \[ 2d\int_0^{s_{\max}}v_E(s)\,\mathrm ds =\mathop{\mathrm{vol}}(C_0) =\binom{2d}{d}\mathop{\mathrm{vol}}(M)^2 \ge c_nq^2 . \tag{87}\] Choose a fixed small \(\theta_n>0\). The contribution of \(0<s<\theta_nA_q\), by Equation (86), is at most \(C_n\theta_n^d q^2\). Fix \(\theta_n\) small enough that this is less than half the last lower bound. The remaining interval has length at most \(C_nA_q\), by Equation (79). Continuity therefore permits a rational \(s\in[\theta_nA_q,s_{\max})\) such that \[v_E(s)\ge c_nq^2/A_q=c_nqA_q^{d-1} \ge c_nq s^{d-1}.\] Choose a current with restricted mass at least three quarters of \(v_E(s)\), and then an approximation as in Equation (81) with \(\int_U\beta^{2d-1}\ge v_E(s)/2\). The first inequality of Equation (86) gives \(w\ge c_ns^{d-1}\), as claimed. ◻

Proof. Remove \(b_V[G_V]\) from \(\rho_V^*T\). Its residual positive current can be restricted to \(G_V\): for analytic singularities removal of the generic divisorial pole leaves a potential that is not identically \(-\infty\) on that divisor. Choose \(z=(x,x,\lambda)\in U_0\) general enough for all restrictions and for Lemmas 93 and 94. On the bundle \[G_V|_Y=\mathbb P_Y(N^*_{V/B}|_Y),\qquad Y=\mathop{\mathrm{Bl}}_xX,\] this restriction is a positive current in the class \[ a^*(2P+qL)-s\{F\}+b_V\zeta_V , \tag{90}\] where \(\zeta_V=c_1(\mathcal O_{\mathbb P(N^*_{V/B}|_Y)}(1))\). Indeed \(M|_{D_\lambda}=2P+qL\), the first projection to \(Z\) is constant on \(Y\), and \(E|_Y=F\). Also \(G_V|_{G_V}=-\zeta_V\), explaining the positive sign of the last term.

The conormal bundle in Equation (90) has exact sequences \[\begin{align*} 0\longrightarrow\mathcal O_Y^{\oplus n} &\longrightarrow N^*_{V/B}|_Y \longrightarrow a^*N^*_{D_\lambda/Z}\otimes\mathcal O_Y(F) \longrightarrow0,\tag{91}\\ 0\longrightarrow\Omega_X &\longrightarrow N^*_{D_\lambda/Z} \longrightarrow\mathcal O_X\longrightarrow0. \tag{92}\end{align*}\] The first constant term is the conormal of \(U_0\) in the first copy of \(Z\), evaluated at \(z\). For the last term in Equation (91), the conormal of the strict transform of \(D_\lambda\) in \(\mathop{\mathrm{Bl}}_zZ\) is \(a^*N^*_{D_\lambda/Z}\otimes\mathcal O_Y(F)\): in a blowup chart, a normal coordinate is divided by an exceptional coordinate, giving precisely the twist by \(F\) for conormals. Equation (92) is the conormal sequence for \(D_\lambda\subset U_0\subset Z\); the diagonal in \(X^2\) has conormal \(\Omega_X\), and the \(\lambda\)-normal direction is constant on \(X\).

Filter a \(k\)-fold tensor of \(N^*_{V/B}|_Y\) by these sequences. A graded term has the form \[\mathcal O_Y(hF)\otimes(a^*\Omega_X)^{\otimes k'} \otimes(\text{a constant vector space}), \qquad 0\le k'\le h\le k.\] For any nonzero line map into that tensor, take a nonzero graded component. Lemma 93 gives \[c_1(\mathcal{H}_Y)\le h\{F\}+k'a^*L \le k(\{F\}+a^*L),\] since both \(\{F\}\) and \(a^*L\) are pseudo-effective. Apply Corollary 98 to Equation (90), with \(D=\{F\}+a^*L\). We obtain \[ a^*(2P+(q+b_V)L)-(s-b_V)\{F\}\ge0. \tag{93}\] If \(b_V\ge s\), the conclusion is immediate. Otherwise \(b_V<s\le C_nA_q\le C_nq\), and \(L\le P/r\) gives \[\left(2+\frac{q+b_V}{r}\right)a^*P-(s-b_V)\{F\}\ge0.\] Use the point bound in Equation (68). Since \(r\ge q^2\), its consequence \[s-b_V\le C_n\left(2+\frac{q+b_V}{r}\right)\] is bounded by a dimensional constant. This proves Equation (89). ◻

Proof. Blow up \(\mathcal J\) and take a common smooth projective resolution with \(U\to E\) from Equation (81). Pull back \(\beta\) and \(D'\). Subtracting a sufficiently small rational multiple of an effective exceptional divisor whose negative is relatively ample makes the pulled-back Kähler class Kähler on this resolution; add that multiple to the effective part. Further upward rational rounding may be arbitrarily small. This gives Equation (94) with all original orders retained. Its general fiber intersection is as close to \(w\) as desired, so arrange \(w^+\ge w/2\). Lemma 97 and the effective divisor give \[ \mathcal F_j^+\subseteq p_+^*\mathop{\mathrm{Sym}}^{js}\Omega_Z,\qquad R_j^+=\frac{w^+}{(d-1)!}j^{d-1}+O(j^{d-2}). \tag{96}\]

We spell out the local order imposed on the polynomials in this inclusion. Near \(z=(x,x,\lambda)\), use coordinates \((\mathbf x,\mathbf a,\lambda)\) on the first \(Z\), where \(\mathbf x\) and \(\mathbf a\) each have \(n\) components and \(U_0=(\mathbf a=0)\). Write the second coordinates as \((\mathbf x+\mathbf h,\mathbf a+\mathbf k,\lambda+\mu)\), with \(\mathbf h,\mathbf k\) each having \(n\) components. Then \[\mathcal V_0=(\mathbf a=\mathbf k=\mu=0),\qquad \Delta_Z=(\mathbf h=\mathbf k=\mu=0).\] In a blowup chart meeting the lines tangent to \(D_\lambda\), take \[h_1=v,\quad h_i=v\alpha_i\ (2\le i\le n),\quad k_i=vy_i\ (1\le i\le n),\quad \mu=vy_{n+1}.\] Here \(E=(v=0)\), while the strict transform is \(V=(a_1=\cdots=a_n=y_1=\cdots=y_{n+1}=0)\). In particular, \[ \mathcal I_V|_E=(a_1,\ldots,a_n,y_1,\ldots,y_{n+1}). \tag{97}\] On a chart of the blowup of \(U_0\), write \(a_1=u\) and \(a_i=u\tau_i\) for \(i\ge2\). Along the general point of \(J_+=(u=0)\), the ideal \(\mathcal J\) is exactly \[ (u,y_1,\ldots,y_{n+1}). \tag{98}\] Here \(u\) measures vanishing along \(J_+\), while the \(y_i\) measure transverse fiber directions. Thus a term of transverse degree \(\ell\) can contribute at most \(\ell\) to the incidence order; the remaining order must come from its coefficient in \(u\). We now make this statement precise.

Suppose locally that the singularities of \(T\) are \(c\log|\mathfrak a|+O(1)\), with the logarithmic normalization for which a divisorial coefficient is \(c\) times the ideal order. If \(k=\mathop{\mathrm{ord}}_V(\mathfrak a)\), then \(b_V=ck\). The normal Taylor coefficients of degree less than \(k\) vanish on a dense open set of \(V\), hence on \(V\) locally. Thus \(\mathfrak a\subseteq\mathcal I_V^k\) also near a general point of \(V\cap E\). Restricting to \(E\) and pulling to \(E^+\) gives the corresponding order in Equation (98). At its general center the displayed generators are regular coordinates. If \(H_{\mathcal J}\) is the exceptional divisor of their blowup, its valuation is the ordinary ideal-adic order: \[\mathop{\mathrm{ord}}_{H_{\mathcal J}}(g) =\max\{k:g\in(u,y_1,\ldots,y_{n+1})^k\}.\] The common resolution dominates this blowup. Pullback preserves the effective difference between \(D'\) and the resolved log divisor, and the later adjustments only increase the effective part. Thus \(D^+\) has coefficient at least \(b_V\) at this valuation. Every polynomial image of a section of \(\mathcal F_j^+\) has integral valuation at least \(\lceil jb_V\rceil\). Locally bounded remainders have zero order at this valuation.

There are \(n\) homogeneous coordinates along \(T_zD_\lambda\) and \(n+1\) transverse homogeneous coordinates in \(T_zZ\). Denote these two groups by \(H_1,\ldots,H_n\) and \(Y_1,\ldots,Y_{n+1}\). For \(m=js\), a local polynomial in \(\mathop{\mathrm{Sym}}^m(p_+^*\Omega_Z)\) has the form \[\sum_{|\alpha|+|\gamma|=m} c_{\alpha,\gamma}(u)\,H^\alpha Y^\gamma.\] Work over the discrete valuation ring \(R\) used to define \(h_j\), choosing the chart parameter \(u\) as uniformizer, and let \(\kappa\) be its residue field. The coefficients \(c_{\alpha,\gamma}\) lie in \(R\). Equation (98) implies, monomial by monomial, \[ \mathop{\mathrm{ord}}_u(c_{\alpha,\gamma}) \ge\max\{0,\lceil jb_V\rceil-|\gamma|\}. \tag{99}\] Indeed, on the chart \(H_1\ne0\), group the dehomogenized polynomial as \[\sum_\gamma P_\gamma(H_2/H_1,\ldots,H_n/H_1)(Y/H_1)^\gamma, \qquad P_\gamma\in R[H_2/H_1,\ldots,H_n/H_1].\] Put \(b_j=\lceil jb_V\rceil\). At the general center of \((u,Y/H_1)\), order at least \(b_j\) forces \(P_\gamma\) to be divisible by \(u^{b_j-|\gamma|}\) whenever \(|\gamma|<b_j\). Otherwise its first nonzero reduction would be a nonzero polynomial over \(\kappa\), which stays nonzero in \(\kappa(H_2/H_1,\ldots,H_n/H_1)\); in the associated graded ring it would give a nonzero term of total \((u,Y/H_1)\)-degree less than \(b_j\). The transverse monomials there are independent. Comparing the internal polynomial coefficients over the base residue field \(\kappa\) now gives Equation (99).

We compare the number of monomials with the rank estimate in Equation (96). The ambient rank and the number of monomials of transverse degree \(\ell\) are \[N_m=\binom{m+2n}{2n},\qquad N_{m,\ell}=\binom{\ell+n}{n} \binom{m-\ell+n-1}{n-1}.\] For fixed \(\delta\in(0,1)\), let \(H_m(\delta)\) count the monomials with \(\ell>(1-\delta)m\). Summing instead over their internal degree gives \[H_m(\delta) \le \binom{m+n}{n}\binom{\lceil\delta m\rceil+n}{n},\qquad \limsup_{m\to\infty}\frac{H_m(\delta)}{N_m} \le\binom{2n}{n}\delta^n.\] On the other hand, \(w^+\ge w/2\) and Equation (96) show that the approximation on \(E^+\) retains at least half the rank fraction in Equation (88): \[\lim_{j\to\infty}\frac{R_j^+}{N_{js}} =\frac{w^+}{s^{d-1}} \ge\frac{w}{2s^{d-1}}\ge c_n>0.\] Fix \(\delta>0\), depending only on \(n\), so small that, for each fixed \(q\) and its selected data, fewer than \(R_j^+/2\) monomials have transverse degree greater than \((1-\delta)js\) for all sufficiently large divisible \(j\). For each remaining monomial, Equation (99) and Lemma 100 give \[\mathop{\mathrm{ord}}_u(c_{\alpha,\gamma}) \ge j(s-C_n)-(1-\delta)js =j(\delta s-C_n)\ge\frac{\delta js}{2}.\] The last inequality holds once \(q\) is large, since \(s\ge c_nA_q\to\infty\).

Over the same discrete valuation ring \(R\), the torsion-free sheaf \(\mathcal F_j^+\) becomes a free module of rank \(R_j^+\). Write its inclusion into the symmetric power as a matrix with the monomials as rows. Every maximal minor uses at least \(R_j^+/2\) of the rows whose coefficients have order at least \(\delta js/2\). All maximal minors therefore have order at least \(\delta jR_j^+s/4\). By the definition of \(h_j\), this proves Equation (95). Dividing the determinant map by \(\sigma_{J_+}^{h_j}\) gives a holomorphic map at every point of codimension one: the coefficients have the required order along \(J_+\), and its local equation is a unit away from \(J_+\). Hartogs’ theorem extends the divided map across the remaining set of codimension at least two, giving the stated factorization through \(\det\mathcal F_j^+(h_jJ_+)\). ◻

Completion of the proof of Theorem 92. Fix \(q\) sufficiently large for the preceding lemmas, with its chosen \(P,s,T\) and modifications. Choose a very general point \(x\) in the first factor of \(X^2\). The slice of \(Z^+\) above \(x\) is \[S=\mathbb P_Y(\mathcal O_Y\oplus a^*\mathcal{L}),\qquad a:Y=\mathop{\mathrm{Bl}}_xX\longrightarrow X,\] and \(J_+|_S\) is the pullback of \(F\). We choose \(x\) so that Equation (62), Equation (68), the restrictions of the currents below, and all determinant maps for divisible \(j\) can be tested on this slice. These exclude at most countably many proper analytic sets and the measure-zero exceptional sets for current restrictions.