Proof. It suffices to prove \[ \kappa(f\mid D_M)\geq\kappa(S,K_S+D_S), \tag{9}\] because Assumption 4 can then be applied to the original pair \((M,D_M)\). We construct a neat model in the precise equivalence class occurring in that assumption.

Flatten \(f\) by a projective modification of its base and resolve that base, obtaining \(q:S'\to S\) with \(S'\) smooth projective. The main strict transform before resolving the source can be taken equidimensional over \(S'\). Indeed, start with the flat strict transform provided by flattening and then base-change to the smooth resolved base. Flatness persists; the unchanged smooth connected generic fiber is irreducible, and flatness rules out vertical components. The main space consequently has pure fibers of dimension \(\dim M-\dim S\). Resolve it and its boundary to obtain a commuting diagram \[\begin{tikzcd} M' \arrow[r,"p"] \arrow[d,"f'"'] & M \arrow[d,"f"]\\ S' \arrow[r,"q"'] & S . \end{tikzcd}\] Here \(p\) is a projective birational morphism, \(M'\) is smooth projective, and \(f'\) has connected fibers: its general fiber is connected, so Stein factorization over the normal base \(S'\) has trivial finite factor. Set \(D_{M'}\) equal to the strict transform of \(D_M\) plus the full reduced \(p\)-exceptional divisor, and arrange that its support is simple normal crossing.

This is an allowed orbifold modification of the source pair. It pushes \(D_{M'}\) to \(D_M\). If a prime upstairs has positive pullback order over a coefficient-one prime of \(D_M\), it is either that prime’s strict transform or a \(p\)-exceptional prime; in both cases its coefficient upstairs is one. Thus the infinite-multiplicity condition in the definition of an orbifold morphism is satisfied.

Moreover, \(p\) witnesses neatness. Let \(E\subset M'\) be a divisor whose image in \(S'\) has codimension \(c\geq2\). In the equidimensional main space its image has dimension at most \[(\dim S-c)+(\dim M-\dim S)=\dim M-c.\] Its image in \(M\) therefore has codimension at least two, so \(E\) is \(p\)-exceptional. Subsequent resolutions preserve this argument.

Let \(B_{f'}\) be the orbifold base divisor on \(S'\). Every component of a pullback over a prime of \((q^{-1}\mathop{\mathrm{Supp}}D_S)_{\mathrm{red}}\) belongs to \(D_{M'}\): this follows from (7) if it is not exceptional over \(M\), and from the definition of \(D_{M'}\) otherwise. The multiplicity assigned to each such source component is infinite. Consequently \[ B_{f'}\geq(q^{-1}\mathop{\mathrm{Supp}}D_S)_{\mathrm{red}}. \tag{10}\] There is also an effective \(q\)-exceptional divisor \(E_S\) such that \[ K_{S'}+(q^{-1}\mathop{\mathrm{Supp}}D_S)_{\mathrm{red}} =q^*(K_S+D_S)+E_S \tag{11}\] for compatible canonical divisors. Above the boundary, this is the nonnegativity of log discrepancies of the simple normal crossing pair \((S,D_S)\); away from the boundary, it is the nonnegativity of the ordinary discrepancies of the smooth space \(S\). Equivalently, a blowup of a smooth center of codimension \(c\) contained in \(s\) boundary components contributes \(c-s\) if the center lies in the boundary, and \(c-1\) otherwise, both nonnegative.

The neat-model formula in Assumption 4, followed by (10) and (11), now gives \[\kappa(f\mid D_M) =\kappa(S',K_{S'}+B_{f'}) \geq\kappa(S,K_S+D_S).\] This proves (9). In particular, the infimum in the definition of the invariant has been computed on an allowed neat model, rather than replaced by the divisor of an arbitrary base model. ◻

Proof. Lemma 7 is exactly the logarithmic-Iitaka hypothesis of Theorem [proj:conditional-log-abundance]. Apply that theorem, whose full proof is included in the appendices. ◻

Proof. Take the almost-holomorphic rational-chain quotient of a smooth resolution \(W\) of \(Y\). Campana’s rational quotient theorem in class \(\mathcal C\) gives a quotient with base in that class and very general fibers rationally connected after resolution (Campana 2004, Theorem 2.6). Resolve the quotient and the pair simultaneously to a fibration \[f:W'\longrightarrow T,\qquad p:W'\longrightarrow Y,\] with \(W',T\) smooth compact Kähler. The very general fibers are projective: rational-chain-connected manifolds in class \(\mathcal C\) are Moishezon by Campana’s algebraic connectedness criterion (Campana 1981), and a smooth Kähler Moishezon manifold is projective. They are then rationally connected.

The quotient base is not uniruled; see also (Ou 2025, Lemma 8.10). For completeness, suppose otherwise and choose a covering rational curve through a very general point of \(T\). Pull the fibration back to its normalization \(\mathbb{P}^1\) and resolve the main component, obtaining a smooth compact Kähler space \(Z\) over \(\mathbb{P}^1\) with rationally connected very general fiber. Such a fiber has no holomorphic one- or two-forms. A global two-form on \(Z\) restricts to zero on those fibers; the relative differential sequence over the smooth locus then places it in the base-one-form times relative-one-form piece. Its restriction there is again zero, so the two-form vanishes on a dense open and hence everywhere. Thus \(H^0(Z,\Omega_Z^2)=0\). Rational approximation of a Kähler class, followed by Kodaira’s criterion, makes \(Z\) projective. The Graber–Harris–Starr theorem supplies a section of \(Z\to\mathbb{P}^1\) (Graber et al. 2003).

The chosen base point can be taken so that its original quotient fiber is not contained in any of the countably many analytic exceptional sets in the very-general quotient property. This follows from the proper fiber-dimension theorem on the resolved quotient. Consequently the rational curve is contained in none of the corresponding bad parameter sets. Choose two distinct good parameters on it. In each smooth projective rationally connected fiber, rational chains join a very general point to the point of the section. The section joins these two chains. Their projection to \(W\) is a rational chain joining points of different very general rational-quotient fibers, where the original almost-holomorphic quotient is defined. This contradicts the quotient property. Hence \(T\) is not uniruled. Its dimension is positive, since a point quotient would make \(W\) Moishezon, and is at most three; the fiber dimension is likewise between one and three.

On the simultaneous log resolution write \[ K_{W'}+\Gamma\sim_{\mathbb{Q}}p^*J+E, \qquad 0\leq\Gamma<1,\quad E\geq0, \tag{12}\] where \(\Gamma\) is simple normal crossing and \(E\) is \(p\)-exceptional. Concretely, take the nonnegative part of the crepant log pullback boundary; its negative part becomes \(E\). The adjoint in (12) is pseudo-effective. At a very general smooth fiber \(F\) we can impose the klt simple normal crossing restriction and also restrict its positive current, by the parameter observation above. Projective nonvanishing in dimension at most three gives \(\kappa(F,K_F+\Gamma|_F)\geq0\). The non-uniruled base similarly satisfies \(\kappa(T,K_T)\geq0\).

For a smooth base, the invariant orbifold-base dimension is at least its ordinary canonical Kodaira dimension: on every equivalent smooth model the orbifold divisor is effective, and the smooth canonical Kodaira dimension is birationally invariant. Assumption 4 therefore gives \(\kappa(W',K_{W'}+\Gamma)\geq0\). A resulting section pushes through (12) to a section of a positive multiple of \(J\). The exceptional pole allowance disappears away from a codimension-two subset of the normal target, and reflexive extension fills that subset. ◻

Proof. Resolve the Stein-factor base of the Albanese map and its main pullback. This gives a fibration on smooth compact Kähler models, with a positive-dimensional smooth base \(T\) mapping generically with full rank to the Albanese torus. A suitable wedge of the invariant one-forms on that torus pulls back to a nonzero section of \(K_T\). The very general fiber has dimension at most three; if its dimension is zero, it is a point. The pseudo-effective canonical class of the total space restricts to the canonical class of almost every smooth fiber. Lower-dimensional canonical nonvanishing and the parameter observation supply \(\kappa(K_F)\geq0\) on very general fibers. Assumption 4, with zero boundary, now gives canonical nonvanishing on the resolved total space and hence on \(W\). ◻

Proof. Let \((Y,\Delta)\) be the nef klt endpoint in Theorem 2, and put \(J=K_Y+\Delta\). The Moishezon case was settled by Corollary 8. Let \(W\) be a smooth compact Kähler resolution in the remaining case. If \(W\) is uniruled, Lemma 9 applies. Otherwise Ou’s theorem makes \(K_W\) pseudo-effective. A section of a divisible canonical power on \(W\) pushes to the corresponding reflexive canonical power on \(Y\), and multiplication by the effective boundary section, at a common multiple, gives a section of \(J\). Thus canonical nonvanishing on \(W\) suffices. Lemma 10 handles \(q(W)>0\), leaving exactly the stated case.

Once nonvanishing on this original endpoint is established, Assumption 6 makes its adjoint semiample. Its other good-minimal-model and discrepancy properties have already been supplied by Assumption 5. None of the auxiliary models constructed later needs to replace this endpoint. ◻

Proof. Every curve \(C\) in \(F\) has nonzero class, since a Kähler class has positive degree on it, and its class belongs to \(R\). Therefore \[r_F=\frac{L\cdot C}{-J\cdot C}>0\] is independent of \(C\). The restriction of \(L\) to the projective fiber is numerically a positive multiple of the relatively ample rational line \(-J\). It is thus ample on \(F\), and the openness of fiberwise ampleness makes \(L\) relatively ample after shrinking around the image point of \(F\).

Work over a small Stein neighborhood of that point, with compactum equal to the point. At \(r_F\), the restriction of \(L+r_FJ\) is numerically trivial, hence nef but not ample on the nontrivial fiber. Fujino’s relative rationality theorem applies to this finite positive threshold: in lowest terms its denominator is at most \(k(\dim F+1)\) (Fujino 2023, Theorem 4.3.1). The non-lc-locus condition is empty for a klt pair, and the singleton compactum satisfies the required local condition on the Stein base. The relative numerical space over it is finite dimensional, as is also seen by restriction to the projective fiber. The theorem consequently gives \(r_F\geq1/(k(\dim F+1))\), which implies (13).

The line-bundle formulation is expressly permitted by (Fujino 2023, Remark 4.3.4); the nearby ampleness assertion is (Fujino 2023, Lemma 2.2.4). Canonical divisors in that theorem may be handled locally, in its formal canonical-class convention. Alternatively, over the Stein neighborhood the proper direct image of a rank-one canonical sheaf is coherent of rank one, since the contraction is bimeromorphic. Cartan’s theorem supplies a generically nonzero section, hence a meromorphic canonical representative there. This requires no global meromorphic canonical frame on the compact source and no local \(\mathbb{Q}\)-factoriality. The Cartier index used in the bound belongs to this stage alone. ◻

Proof. At a stage \(V_i\) of the empty-boundary program that still maps to \(S_0\), write \(b_i:V_i\to S_0\) and \(H_i=b_i^*H\). All these stages are canonical. Indeed, a common projective resolution of a negative canonical step gives the comparison \[ p^*K_{\mathrm{before}}\sim_{\mathbb{Q}}q^*K_{\mathrm{after}}+G, \qquad G\geq0, \tag{16}\] with \(G\) exceptional over the after model. The comparison follows from the projective analytic negativity lemma over the contraction base: there is no extraction, and canonical negativity supplies the relative sign. For flips both sides and their common resolution are projective over that base. Compatible local canonical comparisons glue to the intrinsic discrepancy divisor in (16). Discrepancies therefore do not decrease. Starting from smooth \(W\), this preserves canonicity. The remaining category and pseudo-effectivity properties are part of Assumption 5.

Choose a Cartier index \(k_i\) for \(K_{V_i}\), and choose an integer \[ N\geq N_0,\qquad N>5k_i. \tag{17}\] This choice is made anew at the current stage. The actual line \(K_{V_i}+NH_i\) has an effective rational klt boundary representative: take a general divisor in a sufficiently high free multiple of \(NH_i\) and divide by that multiple. On a fixed log resolution the system has no fixed exceptional component, and Bertini makes the chosen member transverse to the exceptional strata; its small coefficient preserves klt. Its adjoint is pseudo-effective because \(K_{V_i}\) is pseudo-effective and \(H_i\) is semipositive.

If this adjoint is not analytically nef, Assumption 5 supplies an extremal ray on which \(K_{V_i}+NH_i\) is negative. Since \(H_i\) is the pullback of a semipositive form, the ray is also \(K_{V_i}\)-negative. Take its contraction and its negative step for the empty-boundary program. In fact \[ H_i\cdot R_i=0. \tag{18}\] Otherwise Lemma 13, with \(L=H_i\) and \(J=K_{V_i}\), gives \[\frac{1}{5k_i} \leq\frac{H_i\cdot C}{-K_{V_i}\cdot C} <\frac1N\] on any curve of a nontrivial contraction fiber, contrary to (17).

Every connected projective fiber of the contraction maps to a point under \(b_i\). If its image had positive dimension, some curve in that fiber would map nontrivially to the projective base and have positive \(H_i\)-degree. Proper descent over the normal contraction target therefore factors \(b_i\) through it. In a flip, composition with the positive-side morphism gives \(b_{i+1}\). Thus the actual pullback line \(H_i\), not merely its numerical class, continues to be the pullback of the fixed \(H\).

Repeat this procedure as long as the selected twist is not nef. Every step actually taken is a step of the single empty-boundary \(K\)-program starting from \(W\). The changing integers \(N\) select rays; they do not change that program’s boundary. If the procedure were infinite, it would contradict the arbitrary termination assertion in Assumption 5. It therefore reaches a stage \(V_i\) at which its selected \(K_{V_i}+NH_i\) is nef. In particular, if the canonical program reaches a nef canonical class, the selected semipositive twist is nef there as well.

For this final value of \(N\), (14) already gave a section on the original \(W\). Clear the finitely many Cartier indices of the chosen program. Since no step extracts a prime and all maps agree to \(S_0\), that section pushes through every step as a section of the corresponding actual twisted canonical power. Reflexive extension across codimension two on each normal target justifies the pushforward. Hence \(\kappa(K_{V_i}+NH_i)\geq0\) before abundance is invoked. Assumption 6, applied to the klt representative above, makes this actual line semiample.

Set \(V=V_i\), choose a free integral multiple \(m(K_V+NH_i)\), and let \(h:V\to\mathbb{P}^r\) be its morphism. Take the Stein factorization \(g:V\to T\) of \((b_i,h)\). The variety \(T\) is normal and finite over the projective image, hence projective. Let \(a:T\to S_0\) and \(c:T\to\mathbb{P}^r\) be the induced maps. We have \(\dim T\geq d\) because \(a\) is surjective, and \(\dim T\leq a(V)=d\) because \(T\) is projective. Define \[ A_T=\frac1m c^*\mathcal{O}_{\mathbb{P}^r}(1)-N a^*H \quad\text{in }\mathop{\mathrm{Pic}}(T)\otimes\mathbb{Q}. \tag{19}\] The defining evaluation isomorphism for \(h\) and the actual identity \(H_i=g^*a^*H\) prove (15). ◻

Proof. Choose a dense open set of \(S\) over which \(f\) is smooth and the generator supplied by (20) is not identically zero on any fiber. This requires deleting a proper analytic subset: a horizontal divisor cannot contain a whole fiber over a divisor in the base without being vertical, by the fiber-dimension theorem; the remaining degeneracies are proper analytic images or jumping loci. Include all \(\mu\)-exceptional curves among the finitely many complementary curves.

For each such curve \(P\), fix a component \(D\) of \(f^*P\) dominating \(P\), write \[e=\mathop{\mathrm{ord}}_D(f^*P),\qquad h=\operatorname{coeff}_D R,\] and make these choices before choosing any ample test curve or fiber power. If \(P\) is not exceptional over \(T\), take \(D\) with \(h=0\). Indeed, over a general point of its image divisor in the normal \(T\), a local equation pulls back under the holomorphic \(g\) to a divisor on \(V\) dominating that base divisor. The strict transform of one such component has zero coefficient in the \(p\)-exceptional \(R\). For an exceptional \(P\), set \[ c_P=h/e. \tag{23}\]

Replace \(H'\) by a sufficiently large very ample multiple and choose a smooth member \(C\) of genus at least one. It can be chosen transverse to all complementary curves at their general points, avoiding their finitely many excluded points and any isolated bad base points. Moreover, \(M_C=f^{-1}(C)\) is smooth by Bertini for the pulled-back free system, and connected because \(f\) has connected fibers. To retain (21), choose \(C\) through one parameter where all those countably many equalities hold. A high enough linear system through that point is free away from it; on its fiber \(f\) is a submersion, so the same Bertini conclusion holds there. Each jumping locus then cuts a proper analytic subset of \(C\). Thus very general fibers of \(f_C:M_C\to C\) satisfy all the equalities in (21).

By adjunction, \[ K_{M_C/C}\sim_{\mathbb{Q}} f_C^*((A-K_S)|_C)+R_C, \qquad R_C=R|_{M_C}\geq0. \tag{24}\] At each \(z\in C\cap P\), a general point of the chosen component \(D\) over \(z\) admits coordinates \((x,y_1,y_2)\) on \(M_C\) and a parameter \(t\) on \(C\) in which \[ t=x^e,\qquad R_C=h\{x=0\} \tag{25}\] in that chart. A unit in the pullback equation is absorbed into \(x\). Such charts exist over general points of \(P\): the map \(D\to P\) is generically submersive, and its intersections with the other divisorial supports have smaller generic fiber dimension. Properness and the fiber-dimension theorem allow the exceptional base points to be discarded before \(C\) is chosen. These choices are independent of the next integer \(r\).

For \(r\geq1\), take the main component of the \(r\)-fold fiber product of \(M_C\) over \(C\), and a smooth compact Kähler resolution \(Z_r\) of it. Over the good open set this fiber product is smooth with connected fibers \(F^r\), hence has a unique main component. It is an analytic subspace of a product of compact Kähler spaces; the required Kähler resolution and connected-fiber morphism to \(C\) follow as above. The general fiber has a canonical section by (21), and \(\kappa(C)\geq0\). Assumption 4, with zero boundary and in dimension \(2r+1\), gives a nonzero section of \(mK_{Z_r}\) for some \(m>0\). By taking a power, make \(m\) divisible by \(m_0\) and all indices in use. It may depend on \(r\) and \(C\).

Over good parameters, the product formula and (21) make the fiberwise pluriform space one-dimensional. The chosen section is therefore the product of the \(r\) relative generators in (24), times a base pluriform and a scalar base coefficient. This coefficient is a section on the good open of the line \[ \mathcal{O}_C\bigl(m(K_C+r(A-K_S)|_C)\bigr). \tag{26}\] Indeed, its quotient by the displayed product is constant on very general fibers; local submersion sections and analytic continuation give meromorphic descent to that open. Evaluation at points where the product does not vanish identically on the fiber makes the descended coefficient holomorphic there.

We compute its order at a missing point \(z\). In (25), a frame of \(\mathcal{O}_C(m(A-K_S)|_C)\) maps, up to a unit, to \[x^{mh}\left(\frac{dx\wedge dy_1\wedge dy_2}{dt}\right)^{\!m}.\] On a normalization branch of the local \(r\)-fold product we have \(x_1=x\), \(x_j=\zeta_jx\), where \(\zeta_j^e=1\). Multiplying the \(r\) relative factors by one base factor \((dt)^m\), and using \(dt=ex^{e-1}dx\), gives order \[ m\bigl(rh-(r-1)(e-1)\bigr) \tag{27}\] along \(x=0\). This smooth normalization branch belongs to the main component, being a closure of points with \(t\ne0\). The global pluriform on \(Z_r\) has no pole at its generic divisor, by birational comparison. Its scalar coefficient \(u\), pulled back by \(t=x^e\), therefore extends meromorphically, with \[e\,\mathop{\mathrm{ord}}_z(u)+m\bigl(rh-(r-1)(e-1)\bigr)\geq0.\] In particular, \[ \mathop{\mathrm{ord}}_z(u)\geq-\frac{mrh}{e}. \tag{28}\] There is no pole allowance at a nonexceptional base curve, where \(h=0\). The exceptional allowances are exactly \(mr c_P\), with the fixed \(c_P\) in (23).

The nonzero meromorphic coefficient in (26), with these pole bounds, implies \[\deg K_C+r\left(A-K_S+\sum_Pc_PP\right)\cdot C\geq0.\] Divide by \(r\) and let \(r\to\infty\), keeping \(C\) fixed. Since \(C\) is a positive multiple of \(H'\), this proves (22). ◻

Proof. First consider meromorphic functions and tensors. A proper generically finite map of normal complex spaces factors, by Stein factorization, as a proper modification followed by a finite map. Meromorphic functions on a modification descend to the normal target, and finite maps have local meromorphic traces and norms. These constructions use local meromorphic function fields and therefore do not require a global meromorphic frame.

If \(u\) is a meromorphic function on \(N\), its characteristic polynomial over \(M\), computed on the finite part of the factorization, has meromorphic coefficients on \(M\). These coefficients are constant because \(a(M)=0\). Thus \(u\) satisfies a polynomial with constant coefficients. Connectedness then makes \(u\) constant, and \(a(N)=0\).

Suppose that \(s\) is a nonzero meromorphic section of \(mK_N\). Over a coordinate neighborhood \(U\subset M\), choose a nowhere vanishing local frame \(\tau\) of \(mK_M\). The differential pullback \(f^*\tau\) is a holomorphic pluricanonical tensor, generically nonzero, so \(s/(f^*\tau)\) is a meromorphic function over \(f^{-1}U\); zeros of the Jacobian merely contribute a meromorphic divisor to this quotient. If \(d\) is the generic degree of \(f\), its norm is the coefficient of a meromorphic section of \(mdK_M\): replacing \(\tau\) by \(g\tau\) divides the norm by \(g^d\). The local sections consequently glue and are nonzero. This contradicts (94). The ordinary Jacobian formula \[K_N=f^*K_M+R_f,\qquad R_f\geq0,\] proves pseudo-effectivity of \(K_N\).

It remains to prove \(q(N)=0\). If \(q(N)>0\), the Albanese map has a positive-dimensional image. Let \(h\colon N\to B\) be its Stein factorization onto its normal image. The space \(B\) is finite over a subvariety of a torus. On a smooth compact Kähler model \(B'\) of \(B\), the pullbacks of suitable ambient one-forms have a nonzero wedge of degree \(\dim B\). Hence \(\kappa(B')\geq0\). On every neat smooth model used to compute the invariant base in Assumption 4, the orbifold boundary is effective; birational invariance of ordinary Kodaira dimension therefore gives \[\kappa(h\mid0)\geq\kappa(B')\geq0.\] This comparison concerns the invariant base term, not a boundary on an arbitrarily chosen initial model.

Choose a positive singular metric on \(K_N\). Its local potentials restrict to almost every smooth fiber of \(h\), by local integrability and Fubini’s theorem. We may choose such a fiber outside the countably many proper analytic subsets excluded by the very-general-fiber convention in Assumption 4. If \(F\) is that fiber, then \(K_F\) is pseudo-effective and \(\dim F\leq3\). Ordinary canonical nonvanishing for smooth compact Kähler manifolds of dimension at most three gives \(\kappa(F)\geq0\). In dimension three this follows from the Kähler threefold minimal model theorem and abundance for nef normal compact Kähler threefold pairs; see (Höring and Peternell 2016; Das and Ou 2025). In dimensions at most two it is the classical curve and surface statement. Applying Assumption 4 to \((N,0)\) and \(h\) gives \[\kappa(N)\geq\kappa(F)+\kappa(h\mid0)\geq0,\] contrary to the absence of a meromorphic pluricanonical section on \(N\). The same proof applies to any further morphism of the stated kind. ◻

Proof. Choose an integer \(r>0\) such that \(A=rL\) is a line bundle. A section of \(\mathcal V\otimes A^k\) is a morphism \(A^{-k}\to\mathcal V\). Among all nonzero sections with \(k>0\), choose a tuple \(s_1,\ldots,s_t\), of twists \(k_1,\ldots,k_t\), which is generically independent and has maximal possible cardinality. If there are no such sections the assertion is immediate. Otherwise \(t\leq\mathop{\mathrm{rk}}\mathcal V\). Let \(\mathcal W\subset\mathcal V\) be the saturation of the image of the corresponding direct sum of line bundles. Every further section lies in \(\mathcal W\) generically, by maximality, and hence everywhere as a map into this saturated subsheaf.

A further nonzero section \(s\) of twist \(k\) replaces at least one \(s_i\) while preserving generic independence. The two determinants are nonzero sections, respectively, of \[\det\mathcal W\otimes A^{\sum k_j} \quad\text{and}\quad \det\mathcal W\otimes A^{\sum k_j+k-k_i},\] where determinants are reflexive. Their ratio is a nonzero meromorphic section of \(A^{k-k_i}\). For \(k>\max_i k_i\), this is a positive multiple of \(L\). After clearing denominators in \(C\), multiplication by its canonical meromorphic divisor section converts it into a meromorphic section of a positive multiple of \(K_M\), contradicting (94). This proves the eventual vanishing without choosing a meromorphic frame of \(A\).

By Lemma 17, \(L\) has a semipositive metric with zero Lelong numbers everywhere. The multiplier ideal of every positive integral multiple of this metric is trivial by Skoda integrability (Skoda 1972). The hard Lefschetz theorem with multiplier ideals (Demailly et al. 2001, Theorem 0.1) therefore gives, for every \(0\leq j\leq4\), a surjection \[H^0(M,\Omega_M^{4-j}\otimes\mathcal{O}_M(mL)) \longrightarrow H^j(M,\mathcal{O}_M(K_M+mL))\] for divisible \(m>0\). The source is zero for all sufficiently large such \(m\). Riemann–Roch makes \(\chi(M,\mathcal{O}_M(K_M+mL))\) a polynomial in \(m\); its vanishing on all these multiples implies that its constant term \(\chi(M,\mathcal{O}_M(K_M))\) is zero. Serre duality in dimension four yields \(\chi(M,\mathcal{O}_M)=0\).

Equation (94) gives \(h^{4,0}=0\), and Lemma 19 gives \(h^{1,0}=0\). If \(h^{2,0}=0\), then all of \(H^2(M,\mathbb{R})\) is of type \((1,1)\). Approximating a Kähler class by a rational one and applying the Kodaira embedding theorem would make \(M\) projective, contrary to \(a(M)=0\). Thus \(h^{2,0}\geq1\). Finally, Hodge symmetry and the Euler characteristic identity give \(h^{3,0}=1+h^{2,0}\geq2\). ◻

Proof. Work first on the complement of a codimension-at-least-two set where \(F,G,E\) and the line factors above are bundles. Let \(D\subset T_M\) be the integrable distribution annihilated by \(F\). Lie derivative along \(D\) defines the partial Bott connection on \(F\). Its second fundamental form for \(E\) is a morphism \[ B_E\colon E\longrightarrow(F/E)\otimes D^* =(F/E)\otimes G. \tag{101}\] For clarity, this is tensorial in both variables: for \(w\in D\) and a local section \(s\) of \(E\), the terms introduced by replacing \(s\) by \(fs\) are multiples of \(s\), while the additional term in \(\mathcal L_{fw}s\) is \(s(w)\,\mathrm{d}f=0\).

If the composite of \(B_E\) with the second line factor of \(G\) is nonzero, it is a nonzero section of the line bundle of class \[(H-E)+(K-j)-E=3K+H-j-2e.\] If that composite vanishes and \(B_E\ne0\), it factors through the first line factor and gives a nonzero section of class \[(H-E)+(j-H)-E=2K+j-2e.\] These sections extend across the omitted set by reflexivity and Hartogs’ theorem. Removing their divisor factors gives a nonzero meromorphic section of \(3K\) or \(2K\), respectively, contradicting (94). Thus \(B_E=0\).

In local Frobenius coordinates for \(D\), write its transverse coordinates as \((z_1,z_2)\) and a frame of \(E\) as \(a\,\mathrm{d}z_1+b\,\mathrm{d}z_2\). Vanishing of \(B_E\) says that the ratio \(a/b\), where defined, is constant along the plaques of \(D\). After multiplication by an invertible local function, the frame is therefore a one-form on the two-dimensional transversal. Every rank-one conormal on a smooth surface is integrable. The resulting Frobenius identity extends to the entire regular locus of \(E\) by holomorphic equality, including general points of divisorial zeros of the original forms.

The wedge annihilator of \(\sigma\) in degree one is \(F\). The annihilator of \(\beta_2\) is the hyperplane of one-forms vanishing on \(v_2\). Since \(v_2\) is not tangent to \(D\), their intersection is a line. It contains \(\iota_{v_2}\sigma\), which both wedges to zero with \(\sigma\) and evaluates to zero on \(v_2\). Hence that line is \(E\). ◻

Proof. Off the singular set of the saturated codimension-one foliation, whose codimension is at least two, holomorphic first integrals give frames \(\mathrm{d}z\) for \(E\). If \(\mathrm{d}z_a=g_{ab}\mathrm{d}z_b\) on an overlap, then \(\mathrm{d}\log g_{ab}\) belongs to \(E\): differentiating the transition identity gives \(\mathrm{d}g_{ab}\wedge\mathrm{d}z_b=0\). Consequently the first Chern cocycle wedges to zero with both \(\sigma\) and \(\beta_2\) there.

These vanishings extend in Dolbeault cohomology to all of \(M\). Indeed, for a locally free sheaf \(\mathcal V\) on a smooth manifold and an analytic set \(A\) of codimension at least two, \(\mathcal H^j_A(\mathcal V)=0\) for \(j=0,1\), by depth and Hartogs’ theorem. The local-to-global sequence for cohomology with support therefore gives an injection \(H^1(M,\mathcal V)\to H^1(M\setminus A,\mathcal V)\). Apply it to \(\Omega_M^3\) and \(\Omega_M^4\). We obtain \[ c_1(E)\wedge[\sigma]=0,\qquad c_1(E)\wedge[\beta_2]=0. \tag{104}\]

We next dispose of the exceptional divisor terms. Fix a Kähler form \(\omega_Y\) on \(Y\). For every exceptional prime \(P\) on \(M\), \[ \int_P \sigma\wedge\bar\sigma\wedge p^*\omega_Y=0,\qquad \int_P \sqrt{-1}\beta_2\wedge\bar\beta_2=0. \tag{105}\] Here and below integration on a prime divisor can be computed on a smooth resolution. To see the assertion, rationality of klt singularities gives \(Rp_*\mathcal{O}_M=\mathcal{O}_Y\), hence \(H^k(Y,\mathcal{O}_Y)\simeq H^k(M,\mathcal{O}_M)\). Resolve \(P\) and the image \(p(P)\), and resolve the graph so that the resulting smooth compact Kähler space \(\widehat P\) maps to a smooth model \(B\) of \(p(P)\). Functoriality of restriction makes the classes of \(\bar\sigma|_{\widehat P}\) and \(\bar\beta_2|_{\widehat P}\) pullbacks from \(H^2(B,\mathcal{O}_B)\) and \(H^3(B,\mathcal{O}_B)\), respectively. Conjugation and compact Kähler Hodge theory give the corresponding pullback assertions for their de Rham classes. The class of \(p^*\omega_Y\) restricts from \(B\) as well. Since \(\dim B\leq2\), the degree-six products in (105) vanish.

By (100) and (95), \(c_1(E)=-c_1(L)\) modulo exceptional divisor classes. Combining (104) and (105) gives \[\int_M T\wedge\sigma\wedge\bar\sigma\wedge p^*\omega_Y=0, \qquad \int_M T\wedge\sqrt{-1}\beta_2\wedge\bar\beta_2=0.\] The integrands are nonnegative: \(\sigma\) is decomposable wherever nonzero, and every three-form on a four-dimensional vector space is decomposable. The form \(p^*\omega_Y\) is strictly positive on the isomorphism locus. Thus the zero-mass equalities imply (102) there.

This positivity argument applies to currents as well as smooth forms. Locally, write the coefficient matrix of \(T\) as a positive semidefinite matrix of Radon–Nikodym derivatives with respect to its trace measure. The two wedge equalities imply, almost everywhere for that measure, that each one-form direction of the matrix annihilates both \(\sigma\) and \(\beta_2\) by wedge product. Lemma 21 identifies their intersection with \(E\). This proves (103). ◻

Proof. Take relatively compact coordinate charts upstairs and downstairs so that \(f\) maps the former into the latter. Let \(J\) be the local holomorphic Jacobian determinant; it is not identically zero. A sufficiently small negative power of \(|J|\) is locally integrable. Choose \(P>1\) large enough that \(|J|^{-2/(P-1)}\) is integrable on the chosen compact set. Hölder’s inequality gives \[ \int e^{-s\phi\circ f} \leq \left(\int e^{-sP\phi\circ f}|J|^2\right)^{1/P} \left(\int |J|^{-2/(P-1)}\right)^{(P-1)/P}. \tag{106}\] Change of variables bounds the first factor by the degree of \(f\) times the corresponding exponential integral downstairs, up to fixed coordinate-volume constants. Covering compact sets by finitely many charts proves both assertions. ◻

Proof. We first specify the extension through the singular model. Cover \(Y\) by neighborhoods \(U\) on which a Cartier pluricanonical power has a nowhere vanishing frame. Take the ordinary local index cover \(\widehat U\to U\) defined by a root of that frame, and normalize. The cover is quasi-étale and klt, by the index-cover discrepancy formula; in particular it has rational singularities. On its smooth locus the root is a canonical frame.

Resolve a main component of \(\widehat U\times_U p^{-1}(U)\), obtaining a diagram \[\begin{tikzcd} \widetilde U \arrow[r] \arrow[d,"f"'] & \widehat U \arrow[d]\\ p^{-1}(U) \arrow[r] & U . \end{tikzcd}\] Here \(f\) is proper and generically finite. The resolution is chosen isomorphic over the dense smooth open where the index cover is unramified and \(p\) is an isomorphism. The coefficient \(\eta\) in the root frame is an ordinary holomorphic one-form on that open. It extends reflexively to \(\widehat U\) by Hartogs’ theorem, since the complement of the relevant smooth locus downstairs has codimension at least two. The extension theorem for rational complex spaces then extends it holomorphically to \(\widetilde U\); we use (Kebekus and Schnell 2021, Corollary 1.8). We continue to write \(\eta\) for this holomorphic form and \(\phi\) for the pulled metric weight in the root frame.

Zero Lelong numbers downstairs give \(e^{-s\phi}\in L^1_{\mathrm{loc}}\) for every \(s>0\). By Lemma 23, the same is true on \(\widetilde U\). In particular the pulled weight has zero Lelong numbers, and the pulled curvature has no analytic-subset mass. The upstairs expression in (107) has \(L^r_{\mathrm{loc}}\) coefficients for every finite \(r\). This holds also after multiplication by \(\phi\): on a compact set \(\phi\) is bounded above, while \(|\phi|e^{-\phi}\) is bounded by a constant times \(1+e^{-2\phi}\).

Define \(\Theta_T\) on \(p^{-1}(U)\) as \(f_*\) of this expression divided by \(\deg f\). On the good open this is the original formula. It is independent of the root frame: if \(\tau'=g\tau\), then \(\eta'=g^{-1}\eta\) and \(\phi'=\phi-\log|g|^2\). A pushforward of the indicated \(L^r\) coefficient forms has no mass on the analytic complement, because its inverse image is a proper analytic subset and hence has zero smooth volume upstairs. Consequently these local pushforwards agree on overlaps and define a global positive current. It is nonzero since \(\boldsymbol\eta\) is generically nonzero.

We now prove closedness, rather than assuming that a transverse-looking density is closed. On \(\widetilde U\), the saturation of the line defined by \(\eta\) gives an integrable codimension-one conormal. Integrability holds first on the good open by Lemma 21 and then everywhere by holomorphic equality. Its singular set \(A\) has codimension at least two. On a regular foliated chart, including one through a divisorial zero of \(\eta\), write \[ \eta=f_0\,\mathrm{d}z \tag{109}\] with \(z\) a holomorphic submersion and \(f_0\) holomorphic. Lemma 22 gives \(\mathrm{d}\mathrm{d}^{c}\phi\wedge\mathrm{d}z=0\) on the good part where \(f_0\ne0\). It holds on the whole chart because the positive closed curvature has no mass on the remaining analytic set.

In coordinates \((z,w_1,w_2,w_3)\), positivity shows that the only coefficient of \(\mathrm{d}\mathrm{d}^{c}\phi\) is its \(\sqrt{-1}\,\mathrm{d}z\wedge\mathrm{d}\bar z\) coefficient. Closedness of that current makes this coefficient independent, as a distribution, of every \(w_j\) and \(\bar w_j\). Thus it is the pullback of a positive measure on the \(z\)-disc. Taking a one-variable subharmonic potential and then solving the pluriharmonic difference on a smaller polydisc gives \[ \phi=\phi_0(z)+g+\bar g,\qquad g\text{ holomorphic}. \tag{110}\] With \(h=f_0e^{-g}\), the upstairs current becomes \[\sqrt{-1}e^{-\phi_0(z)}|h(z,w)|^2\,\mathrm{d}z\wedge\mathrm{d}\bar z.\] Its \(\mathrm{d}\mathrm{d}^{c}\) is positive. More explicitly, only differentiation in the \(w\) directions survives wedging with \(\mathrm{d}z\wedge\mathrm{d}\bar z\), and \[ \mathrm{d}\mathrm{d}^{c}\Theta_T = e^{-\phi_0(z)} \sqrt{-1}\partial_w h\wedge\bar\partial_w\bar h \wedge\sqrt{-1}\,\mathrm{d}z\wedge\mathrm{d}\bar z \ \geq\ 0 \tag{111}\] on the regular foliated chart. This is a distributional identity; the one-variable coefficient is locally integrable, and no transverse derivative of it contributes to the displayed wedge.

We give the extension estimate across \(A\). On a relatively compact coordinate ball, choose smooth cutoffs \(\chi_\epsilon\) equal to zero on an \(\epsilon\)-tube around \(A\) and equal to one outside a \(3\epsilon\)-tube, with \[|\nabla\chi_\epsilon|\leq C\epsilon^{-1}, \qquad |\nabla^2\chi_\epsilon|\leq C\epsilon^{-2}.\] They may be constructed by smoothing the distance cutoffs. The tubes have volume \(O(\epsilon^4)\). To justify the bound also when \(A\) is singular, use locally finite area for its analytic components and monotonicity to bound the number of disjoint balls of radius \(\epsilon/4\) centered on each component by \(C\epsilon^{-2d}\), where \(d\leq2\) is its complex dimension. Enlarging this cover to the tube gives volume \(O(\epsilon^{8-2d})=O(\epsilon^4)\); a compact subset meets finitely many local components.

If \(B\) denotes either \(\Theta_T\) or \(\phi\Theta_T\) upstairs and \(r>2\), then for any fixed smooth test form the second-derivative cutoff terms are bounded by \[ C\epsilon^{-2}\|B\|_{L^r} \operatorname{vol}(N_{3\epsilon}(A))^{1-1/r} \leq C_r\|B\|_{L^r}\epsilon^{\,2-4/r} \longrightarrow0. \tag{112}\] Terms with only one derivative are smaller. Testing (111) against \(\chi_\epsilon\) times a positive test form and passing to the limit therefore proves \(\mathrm{d}\mathrm{d}^{c}\Theta_T\geq0\) on all of \(\widetilde U\).

Proper pushforward commutes with \(\mathrm{d}\mathrm{d}^{c}\), so \(\mathrm{d}\mathrm{d}^{c}\Theta_T\) is a global positive exact \((2,2)\)-current on \(M\). Pairing with the square of a Kähler form gives zero; positivity then makes it the zero current. On the good open the chart maps are local biholomorphisms. All branch contributions to the pushforward are positive, so each upstairs Hessian in (111) vanishes there separately. Since \(e^{-\phi_0}>0\) almost everywhere, the holomorphic derivatives \(\partial_w h\) vanish there, and hence on every regular foliated chart by holomorphic continuation. Thus \[ h=f_0e^{-g}\text{ depends only on }z. \tag{113}\]

Equations (110) and (113) imply, on these charts, \[\mathrm{d}\Theta_T=0,\qquad \mathrm{d}\mathrm{d}^{c}(\phi\Theta_T)=0.\] For the second identity, the coefficient depending only on \(z\) causes no derivative after wedging with \(\mathrm{d}z\wedge\mathrm{d}\bar z\), while the remaining \(g+\bar g\) is pluriharmonic in the leaf directions. The first- and second-derivative versions of (112) extend both identities across \(A\).

Finally choose a smooth reference metric on \(L\), with local weights \(\psi\) and curvature \(\theta\). The difference \(u=\phi-\psi\) is a global function downstairs. The local pushforwards defining \(u\Theta_T\) agree: upstairs they are locally integrable by the exponential estimates, they agree on the good open, and they have no analytic-subset mass. Closedness and the last displayed identities give the global current identity \[ \mathrm{d}\mathrm{d}^{c}(u\Theta_T)=-\theta\wedge\Theta_T. \tag{114}\] It proves (108). ◻

Proof. Regularization with analytic singularities gives currents \(R_k\in[R]\) and numbers \(\epsilon_k\downarrow0\) such that \(R_k\geq-\epsilon_k\omega\) and \(\nu(R_k,x)\leq\nu(R,x)\) at every point; see (Boucksom 2004, Theorem 2.1(ii)). In particular the analytic pole sets of \(R_k\) have codimension at least two. Resolve these pole sets by a projective modification \(f_k\colon X_k\to X\). For \(\beta_k=[R]+\epsilon_k[\omega]\), the pulled positive current has a decomposition \[f_k^*\beta_k=P_k+[D_k],\] where \(D_k\) is an effective exceptional real divisor and \(P_k\) is nef. With the usual smooth-remainder version of analytic regularization, \(P_k\) is represented by a smooth semipositive form. The bounded-remainder version gives a positive residual current with bounded potentials, which is nef by a further regularization.

Since \(f_{k*}[D_k]=0\), the projection formula gives \[\int_{X_k} f_k^*\beta_k\,[D_k]\,(f_k^*[\omega])^2=0.\] Consequently \[ \int_X\beta_k^2[\omega]^2 = \int_{X_k} f_k^*\beta_k\,P_k\,(f_k^*[\omega])^2 \geq0. \tag{115}\] The inequality pairs a pseudo-effective class with three nef classes. It follows by approximating the nef factors by Kähler classes and pairing with a positive current. Passing to the limit proves the lemma. In particular, no positivity of the self-intersection of the exceptional divisor has been used. ◻

Proof. Suppose \(T\in\alpha\) has a positive Lelong number at a point, and let \(f\colon X\to M\) be its blowup. The pullback has a positive generic Lelong number on the exceptional divisor. Write its Siu decomposition as \[ f^*T=R+\sum_j a_j[D_j], \qquad a_j>0, \tag{118}\] where \(R\) has no divisorial mass and the sum is not zero. Fix a Kähler form \(\omega_X\) on \(X\), and set \(\alpha_X=f^*\alpha\). The class \(\alpha_X\) is nef, nonzero, and \(\alpha_X^2=0\). Pairing (118) with \(\alpha_X[\omega_X]^2\) gives zero. All terms are nonnegative, since a nef class paired with a positive current and two Kähler classes has nonnegative intersection. Therefore \[Q_{\omega_X}(\alpha_X,[R])=0,\qquad Q_{\omega_X}(\alpha_X,[D_j])=0 \quad\text{for every }j.\] Lemma 25 and Hodge index give \([R]=b\alpha_X\) with \(b\geq0\); this includes \(b=0\) when \(R=0\). Comparing masses in (118) gives \(b<1\), because the divisorial sum has positive mass. Hence \[ (1-b)\alpha_X=\sum_j a_j\,c_1(\mathcal{O}_X(D_j)). \tag{119}\]

The real span of the integral classes \(c_1(\mathcal{O}_X(D_j))\) is a finite-dimensional, hence closed, subspace. The convergent series in (119) therefore places the rational class \(c_1(f^*L)=\alpha_X/(2\pi)\) in that real span. Choose a finite subset of these integral classes forming a basis of the span. Solving the corresponding rational linear system shows that a rational vector in their real span is in their rational span. Thus, for some rational divisor \(D\) with finite support, \[c_1(f^*L)=c_1(\mathcal{O}_X(D)) \quad\text{in }H^2(X,\mathbb{Q}).\] After clearing denominators and integral torsion, the difference line bundle has zero integral first Chern class. By Lemma 19, \(q(X)=0\), so the exponential sequence makes this difference line bundle trivial. A positive multiple of \(f^*L\) consequently has a nonzero meromorphic section. The discrepancy identity for \(X\to M\to Y\) converts it to a meromorphic section of a positive multiple of \(K_X\), contradicting Lemma 19. ◻

Proof. Fix a smooth reference weight for \(L\), with curvature \(\theta\), and a smooth probability volume on \(M\). The set \[\mathcal C_\alpha= \{T\geq0:\ T\text{ closed of type }(1,1),\ [T]=\alpha\}\] is nonempty, compact, and convex in the weak topology of currents. Its mass against \(\omega^3\) is fixed. Write \(T=\theta+\mathrm{d}\mathrm{d}^{c}u_T\), uniquely normalized by \(\int_Mu_T\,\mathrm{d}V=0\). Thus all local weights used for \(T\) have a fixed common additive normalization.

By Lemma 26, the construction of Lemma 24 applies to every \(T\in\mathcal C_\alpha\). Applying Hodge index as in (116) shows that \([\Theta_T]\) is a positive multiple of \(\alpha\). Define \[ \mathcal F(T)= \frac{\int_M\alpha[\omega]^3}{\int_M\Theta_T\wedge\omega^3}\,\Theta_T. \tag{122}\] This is a selfmap of \(\mathcal C_\alpha\).

We verify continuity, including on the higher charts. If \(T_j\to T\) weakly, compactness for quasi-plurisubharmonic functions with fixed lower curvature bound and the chosen normalization gives \(u_{T_j}\to u_T\) in \(L^1\). Indeed every subsequential limit has the same \(\mathrm{d}\mathrm{d}^{c}\) and normalization as \(u_T\). On a coordinate chart the metric weights are plurisubharmonic and converge in \(L^1\). All complex singularity exponents of the limit are infinite, by Lemma 26 and Skoda integrability. The effective semicontinuity theorem (Demailly and Kollár 2001, Theorem 0.2(2)) consequently gives locally uniform bounds for \(e^{-s\phi_j}\) for every fixed \(s>0\), and in fact convergence of these exponentials in \(L^1\) on smaller neighborhoods.

Apply (106) on the fixed proper chart diagrams. It gives locally uniform upstairs bounds for every inverse exponential power as well. After extraction, the weights converge almost everywhere off the exceptional and critical analytic sets. Pullback preserves almost-everywhere convergence there, because the chart maps are local biholomorphisms; the omitted analytic sets have measure zero. Uniform \(L^r\) bounds for some \(r>1\) then give convergence in \(L^1\) of the densities defining \(\Theta_{T_j}\) upstairs. Proper pushforward gives \(\Theta_{T_j}\to\Theta_T\) downstairs. The argument applies to each subsequence and so proves continuity for the whole sequence. Their masses also converge, and the limiting mass is positive. Thus \(\mathcal F\) is continuous. The Schauder–Tychonoff fixed-point theorem yields a fixed point.

Let \(b\) be the positive normalization factor in (122) at this fixed point. Its equality \(T=b\Theta_T\) holds upstairs on the good locus, giving (120). Both sides extend with no analytic-subset mass, so the equality holds on each entire smooth higher chart.

Taking a smooth Euclidean trace on a smaller coordinate ball gives a scalar Poisson equation whose right side belongs to every finite \(L^r\), because \(\eta\) is holomorphic and all inverse exponential moments of \(\phi\) are finite. Interior elliptic regularity gives \(\phi\in W^{2,r}_{\mathrm{loc}}\) for every finite \(r\). Choosing \(r>8\) gives continuous first derivatives, after which the semilinear equation and ordinary elliptic bootstrap make \(\phi\) smooth. On a regular foliated chart, (110) and (113) give \(\eta=e^g h(z)\,\mathrm{d}z\). Therefore \(\mathrm{d}\eta=\mathrm{d}g\wedge\eta=\partial\phi\wedge\eta\) there. Both sides are now smooth on the entire higher chart, and the identity extends by density. This proves (121). ◻

Proof. The assertion is local at a general point of a boundary prime \(D\). Choose a small disc \(\Delta\) transverse to \(D\) at a smooth point not on another boundary component, with \(\Delta^*=\Delta\setminus\{0\}\) contained in \(S_X\). Choose an original proper higher chart over a neighborhood of the image point in \(M\), take its fiber product with \(X\), and resolve a main component. The resulting map \(g\colon W\to X\) is proper and generically finite, and is unramified over \(S_X\). Near every point of \(W\) there is an ambient holomorphic map to \(\mathbb{P}^1\): compose its map to the original higher chart with the map constructed in (125). Over the good open these maps agree with the developing germs up to constant projective unitary transformations.

A component of the inverse image of \(\Delta\) dominating \(\Delta\) is a curve, finite over \(\Delta^*\). Its normalization has a point over \(0\), and its local map to \(\Delta\) can be written, after changing a parameter, as \(t\mapsto t^r\) for some positive integer \(r\). For a sufficiently small circle in the \(t\)-disc, the lifted loop stays inside one neighborhood on which an ambient map to \(\mathbb{P}^1\) is defined. Continuing this ambient map around the lifted loop returns the same germ. On the punctured part, \(g\) is locally biholomorphic, so this is the analytic continuation of the original developing germ around the \(r\)-th power of the meridian.

A projective unitary transformation fixing that germ must be the identity, since it is a submersive ambient germ on the good open. Thus the meridian has monodromy whose \(r\)-th power is the identity. It is essential here to use the ambient germ: even if the transverse disc happens to be tangent to the foliation, the argument does not infer identity of holonomy merely from its action on the values of the map along that disc. The same fiber-product construction applies after the further modifications specified in the statement. ◻

Proof. Near a point of \(D\), choose a polydisc on which its complement is \((\Delta^*)^r\times\Delta^{n-r}\). Each connected finite cover is described by a finite-index subgroup of its fundamental group \(\mathbb{Z}^r\). Such a subgroup contains \(N\mathbb{Z}^r\) for some \(N>0\). The cover is consequently dominated by the coordinate power cover \[(t_1,\ldots,t_r,w)\longmapsto(t_1^N,\ldots,t_r^N,w).\] It extends across the coordinate hyperplanes as the normal quotient of the full polydisc power cover by the corresponding finite subgroup of its deck group. The normal finite extensions glue uniquely: an isomorphism on the dense punctured locus extends between the normal finite algebras, equivalently by their integral closures. This gives a finite normal cover \(\overline X\to X\).

For completeness, a finite source over a compact Kähler space is Kähler in the local-embedding sense. Over finitely many small base neighborhoods choose relative embeddings of the finite source into products with affine spaces, with relative coordinates \(w_a\). Choose a smooth partition of unity \(\lambda_a\) on the base and put \[h=\sum_a(\lambda_a\circ f)|w_a|^2\] on the source, extending each summand by zero outside its support. This is a smooth function on the complex space. At a point where \(\lambda_a>0\), use the corresponding relative embedding. The other relative coordinate functions have local holomorphic extensions to its ambient space. On vertical tangent directions the Levi form of \(h\) is \(\sum_a\lambda_a|\mathrm{d}w_a|^2\) and is strictly positive. The terms arising from derivatives of the partition have a base-direction factor. A sufficiently large multiple of the pulled base Kähler form dominates their horizontal and mixed contributions. Compactness allows one such multiple on the finite cover. Thus \[C f^*\omega_X+\mathrm{d}\mathrm{d}^{c}h\] has strictly plurisubharmonic local potentials in the chosen ambient embeddings and defines a Kähler form on the normal source. Finally take a projective resolution and then an embedded resolution of the boundary, both unchanged over the smooth covered open. Projective modifications of compact Kähler spaces remain Kähler, by the usual relative ample curvature construction. This proves the claim. ◻

Proof. Let \(\Gamma\) be the image, and take its complex Zariski closure in \(\mathrm{PSL}_2(\mathbb{C})\), regarded as a linear algebraic group through the adjoint representation.

If this closure is all of \(\mathrm{PSL}_2(\mathbb{C})\), apply the semisimple Shafarevich theorem (Campana et al. 2015, Theorem 1). Its hypotheses are exactly a smooth compact Kähler source and a representation Zariski dense in a semisimple linear group; for torsion-free image the Shafarevich base is normal projective of general type, and the representation factors through a smooth model of that base. Algebraic dimension zero forces this base to be a point: a positive-dimensional projective base would supply a nonconstant meromorphic function on \(X\). Factorization through a point makes \(\Gamma\) trivial, contradicting its supposed Zariski density.

If the Zariski closure is proper, its identity component is solvable, by the classification of proper connected algebraic subgroups of \(\mathrm{PSL}_2(\mathbb{C})\). Thus \(\Gamma\) is virtually solvable. Pass to a finite-index subgroup whose image \(\Gamma_0\) is solvable and let \(X_0\to X\) be the corresponding finite étale cover. For every further finite-index subgroup \(H\) of \(\pi_1(X_0)\), its abelianization is finite: it is the first integral homology of a compact Kähler finite cover with \(q=0\), hence is finitely generated of rank \(2q=0\).

Now induct along the derived series of \(\Gamma_0\). The quotient \(\Gamma_0/\Gamma_0'\) is an image of \(\pi_1(X_0)^{\mathrm{ab}}\), so it is finite. Its preimage has finite index in \(\pi_1(X_0)\); that subgroup again has finite abelianization, making \(\Gamma_0'/\Gamma_0''\) finite. Repeating gives finite successive indices down to the identity, because the derived series has finite length. Thus \(\Gamma_0\), and then \(\Gamma\), is finite. A finite torsion-free group is trivial. ◻

Proof. Choose \(l\) so that the corresponding reflexive power of \((h_*\mathcal P)^{**}\) is invertible. This is \((h_*\mathcal P^{[l]})^{**}\), since the two sheaves agree where \(h\) is an isomorphism, including the general points of all prime divisors of \(T\). Call this line \(M\). Evaluation of local sections of \(h_*\mathcal P^{[l]}\), and their images in \(M\), gives meromorphic comparisons with \(h^*M\). These comparisons are the same on their common domain and hence define a divisorial comparison globally. Its orders vanish away from the exceptional primes. The resulting integral exceptional divisor gives (130) in codimension one and therefore everywhere by reflexivity. Local meromorphic generators are enough for this argument; it does not represent an arbitrary global line bundle by a global divisor. ◻

Proof. The codimension-one comparison gives \(q_*F=0\). The signs of the two adjoints make \(-F\) nef over the contraction base, and hence over the target of \(q\). Negativity gives \(F\geq0\).

In a small diagram the non-isomorphism sets in \(Z\) are the same on both sides. Otherwise, over a region where one side is an isomorphism, smallness identifies the other adjoint with a pullback, contradicting its relative ample sign on a contracted curve. A non-isomorphism fiber is positive-dimensional by normality. Over any such point, lift a negative curve to the common resolution. Its \(F\)-degree is strictly negative, so the fiber meets \(\mathop{\mathrm{Supp}}F\). A connected projective fiber meeting the support of an effective relatively anti-nef Cartier divisor is contained in that support: if a component outside the support met it, a curve section through an intersection point would have positive intersection. Clearing denominators gives the same statement for \(F\). Thus \(\mathop{\mathrm{Supp}}F\) contains the full fibers in question. Pullback to any higher model now proves strict increase at each specified place.

For dlt preservation use the characterization by log canonicity and an SNC open set meeting every lc center. A new discrepancy-zero place was already a zero place, and strictness keeps the general point of its center out of the surgery. The SNC characterization, checked on global log resolutions, is therefore preserved. The klt assertion is immediate. Finally, on a globally strongly \(\mathbb{Q}\)-factorial dlt pair, slightly decreasing the coefficients of the floor gives a klt pair: the floor is effective rational Cartier, and every discrepancy-zero place has center in it and positive order on its pullback. ◻

Proof. The line \(L\) is relatively nef, and the base point free hypothesis holds by fiberwise ampleness. Locally over a smaller base neighborhood, all sufficiently high powers are generated. The induced maps are constant on each connected fiber, since their tautological lines have degree zero on every curve there. They factor through \(Z\): the graph projection is finite bimeromorphic onto the normal base. The pulled-back tautological lines descend two consecutive powers of \(L\), and their quotient descends \(L\) itself. The projection formula identifies the descent with \(f_*L\) and the pullback map with evaluation. These intrinsic identifications agree on overlaps. ◻

Proof. Compactness permits a common divisible integer \(m\) such that the relative ring of \(mL\) is generated in degree one. Resolve its base ideal and graph, and denote the maps to \(X_0\) and \(Y\) by \(p\) and \(q\). The map to Proj lifts to the normalization. There is an actual moving/fixed decomposition \[ p^*(mL)=q^*H+G,\qquad G\geq0, \tag{133}\] where \(H\) is the relatively ample tautological line on \(Y\). Degree-one generation and projection show that every relative section of \(p^*(kmL)\) has vanishing at least \(kG\).

First, \(G\) is \(q\)-exceptional. If a component mapped to a prime on \(Y\), then \(q\) would be an isomorphism at its general point. For large \(k\), relative ample generation of \(q_*\mathcal{O}(G)\otimes H^{\otimes k}\) gives a local-over-base section having a pole at that prime relative to \(H^{\otimes k}\). Multiplying its pullback by the section of \((k-1)G\) gives a section of \(p^*(kmL)\) whose vanishing is less than \(kG\), a contradiction.

Second, every \(p\)-exceptional prime \(P\) is \(q\)-exceptional. Otherwise relative generation of \(q_*\mathcal{O}(P)\otimes H^{\otimes k}\) gives a section with a pole at the image prime. Add \(kG\) to obtain a section of \(p^*(kmL)+P\). Since \(p_*\mathcal{O}(P)=\mathcal{O}_{X_0}\) by normality, it comes from the original ring. As a section of the enlarged line it must vanish once along \(P\), whereas the chosen pole prevents that vanishing. Here the coefficient of \(G\) at \(P\) is zero by the first part. This is again a contradiction. Thus no divisor is extracted. Taking the trace of (133) gives \(L_Y=H/m\) as actual rational lines. ◻

Proof. After a common Veronese and shrinking, choose finitely many homogeneous generators of the multigraded ring. Diagonal rings on rational degree rays are finitely generated by the semigroup argument for monomial degrees. Their Proj charts can be taken with homogeneous monomials as denominators. The subsets of generators that occur as supports of monomials on the specified positive degree ray form a finite pattern. Localization at such a monomial inverts exactly its support, and the multidegree-zero subring is the degree-zero localization of the diagonal ring. Intersections use unions of supports with the canonical localization maps. Thus parameters with the same pattern have the same charts and gluing. Taking the main component and normalization preserves finiteness.

The base identification fixes the marking on the common bimeromorphic open. A finite collection of the smaller neighborhoods covers the compact base. If two marked models agree on each neighborhood, their identifications over the base agree on the common dense open, hence everywhere, and glue uniquely. There are consequently only finitely many global marked possibilities. ◻

Proof. Use the finite-dimensional space of degrees of global rational line bundles on curves contracted over \(V\); finite dimensionality follows from first Chern classes in finite-dimensional cohomology. For a fixed relatively ample \(H\), the closed curve cone has a compact slice of \(H\)-degree one. Indeed, for every global line \(L\), both \(kH+L\) and \(kH-L\) are relatively ample for sufficiently large \(k\), so all coordinates on the normalized slice are bounded.

Choose a negative extremal ray. For dlt input decrease the floor coefficients rationally just enough to obtain a klt adjoint \(J_0\) still negative on this ray; for klt input let \(J_0=K_T+B\). Cover the base by interiors of finitely many Stein compacta as above. Projecting the local cone decompositions to global degrees shows that, for each positive rational \(\delta\), the normalized slice lies in the convex hull of its compact part \(J_0+\delta H\geq0\) and finitely many points of actual contracted curves. A projected remainder stays in the global closed cone, and a remainder of zero \(H\)-degree contributes zero. The stated convex hull is compact.

Choose \(\delta\) so that the selected ray is strictly in the truncated negative region. It is therefore represented by an actual curve. Separating its point on the slice from the compact hull of the remaining generators and remainder gives a supporting nef class vanishing only on this ray. In the open set of such supports one can choose a rational global line \(N\) in the rational annihilator of the ray, so that a positive multiple of \(N\) minus \(J_0\) is relatively ample. Strict positivity on the compact slice and fiberwise ampleness give this last assertion. Local klt base point freeness on the finite cover generates a common multiple of \(N\). Its section morphism and Stein factorization give a projective bimeromorphic contraction \(f:T\to Z\) over \(V\), with normal target and connected fibers, contracting precisely the ray. The line \(-J_0\) is \(f\)-ample by fiberwise ampleness.

If an exceptional prime \(E\) exists, it is rational Cartier and negativity gives \(E\) strictly negative degree on the ray. All contracted curves lie in \(E\); they cover the nontrivial fibers. There can be no second exceptional prime, since its negative ray degree would force the same curves, and hence \(E\), into it. For a global rank-one reflexive sheaf on \(Z\), take its reflexive pullback to \(T\). It is rationally invertible. Add a rational multiple of \(E\) to kill its ray degree and apply Lemma 34 after clearing denominators. The descended rational line agrees with the given sheaf off codimension two on \(Z\) and hence everywhere by reflexivity. Thus \(Z\) has the required global strong property.

If \(f\) is small, apply the preceding finite generation over \(Z\) to \(J_0\). Lemma 35 gives a projective positive model with no extracted divisors; it is therefore small over \(Z\). The transformed adjoint is relatively ample and rationally invertible. For any global line \(L\) on \(T\), killing the ray degree and applying integral descent gives an actual rational identity \[ L=cJ_0+f^*M,\qquad c\in\mathbb{Q}. \tag{134}\] Smallness gives the corresponding identity on the positive model. Any global rank-one reflexive sheaf there first transports to \(T\), where a power is such an \(L\). Transforming back and using (134) proves the global strong property on the positive model. The positive morphism cannot be an isomorphism, since that would make \(J_0\) a pullback from \(Z\).

For dlt input the unperturbed adjoint has negative degree on the ray. Its version of (134) has \(c>0\), so its transform has the positive sign as well. Lemma 33 preserves the required singularities. All resulting spaces are projective bimeromorphic over \(V\) and hence compact Kähler, so the construction can continue whenever relative curve-nefness fails. ◻

Proof. Count the dimension of the span of prime \((n-1)\)-cycle classes in \(H_{2n-2}(-,\mathbb{R})\). This is a finite nonnegative integer, since compact analytic spaces are triangulable and have finite-dimensional homology. Nonextraction provides common isomorphic opens whose complement in the target has dimension at most \(n-2\). The localization sequence identifies the target’s degree-\(2n-2\) homology with the Borel–Moore homology of that open. Restriction from the source maps its prime-cycle span onto the target’s span by strict transforms, killing the classes of lost primes. Each lost prime has nonzero class by its positive Kähler volume, so the count strictly drops.

For completeness, that volume detects a topological class also on a normal singular space. Kähler potentials with pluriharmonic differences determine a class in \(H^2(-,\mathbb{R})\) through the real-part sequence \(0\to\sqrt{-1}\mathbb{R}\to\mathcal{O}\to\mathcal{PH}\to0\), with a fixed normalization. If pluriharmonicity is initially known only on the regular locus, pull the difference to a resolution. Its smooth pullback is pluriharmonic. Near the compact fiber over a point, choose holomorphic real-part primitives and adjust imaginary constants to make their values on that fiber agree. The real part is constant there, and each irreducible fiber germ forces the holomorphic primitive to be constant there. The primitives therefore agree on overlaps near the fiber. A finite cover and smaller neighborhoods glue them on a neighborhood of the whole fiber, and properness and normality descend them. Thus the original difference is locally a holomorphic real part. Resolving a compact prime cycle now evaluates the corresponding class power as the strictly positive integral of the pulled-back Kähler form. Its fundamental class pushes to the stated cycle class. ◻

Proof. On an SNC resolution let the log-pullback coefficients be \(d_j<1\), put \(w_j=1-d_j\), and take \(\epsilon=\min(1,\min_jw_j)\), with value one if the list is empty. For a place still exceptional over this resolution, choose at the general point of its center normal coordinates \(x_1,\ldots,x_s\), the first \(b\) defining the boundary components through the center. The top exterior Jacobian calculation gives \[ a_E\ \geq\ \sum_{j=1}^{b}w_j\mathop{\mathrm{ord}}_E(x_j) +\sum_{j=b+1}^{s}\mathop{\mathrm{ord}}_E(x_j). \tag{135}\] At most one differential saves one order by normal differentiation along the place. This proves the lower bound. A place in the strict cutoff must have center a stratum: an additional normal coordinate would give at least \(1+\epsilon\) (or at least two if no boundary is present).

Blow up that closed stratum. Its new SNC weight is the sum of the passing weights, and the next center lies in the new exceptional component. Until the place is divisorial, only strata of codimension at least two can occur; each next weight increases by at least \(\epsilon\). The fixed strict cutoff bounds the chain length, and there are finitely many strata at each stage. The finitely many exceptionals already on the starting resolution complete the count. All these blowups can be made globally and projectively.

For a terminal pair, exceptional log-pullback coefficients are negative. The same calculation uses the strict cutoff \(2-b_0\); non-stratum centers and centers involving such exceptional components cannot contribute new places below it. For the lc version take the minimum of one and the strictly positive weights. A center not contained in the non-klt locus meets no zero-weight component generically, so the same argument applies. ◻

Proof. Choose local parameters \(t_1,\ldots,t_{n-2}\) along the locus from an embedding. Take a sufficiently general nearby common value, regular on the smooth locus, upstairs on a projective log resolution, and on all relevant smooth strata; discard images of nondominating strata. The cut upstairs is a smooth surface, with the required SNC support, and no component is exceptional. Codimension-two bad loci are cut to isolated points.

The surface downstairs has dimension two and is regular in codimension one. It is Cohen–Macaulay as well. Indeed the non-Cohen–Macaulay locus of a normal analytic space has codimension at least three, by the local depth/coherent Ext criterion and normality at height at most two. Thus at the chosen general point the parameters are a regular sequence. The cut is generically reduced and Cohen–Macaulay, hence reduced; Serre’s criterion gives normality. Read the restricted boundary transversely as a cycle and take its closure. Complete intersection adjunction off the isolated bad set, followed by reflexive extension, identifies the restricted rational adjoint with \(K_S+B_S\), using division by the same parameter volume form upstairs and downstairs. The restriction of the crepant formula has the correct nonexceptional coefficients. Any difference from the surface crepant formula is exceptional and relatively numerically zero, hence zero by both signs of negativity. This also restricts klt formulas, or dlt formulas from resolutions preserving an SNC good open, with their stated discrepancies.

For a coefficient-one prime, first subtract its smooth strict transform in the resolved formula, then take residue and push to its normalization. At general points over the chosen codimension-two locus, the cut of that strict transform is a smooth curve germ, finite and generically an isomorphism onto its branch of the reduced surface curve. It is therefore the curve normalization. Transversality preserves coefficient orders. Restricting first to the surface and then taking residue gives the same result. Only the indicated sums need be rational Cartier on the normalization; the individual canonical and boundary terms need not be. ◻

Proof. The set is finite by Lemma 39. Choose a projective SNC resolution \(h:R\to T\) carrying it. Put an effective SNC klt boundary \(\Theta\) on \(R\), retaining the strict boundary and the crepant coefficients of the specified primes, and choosing every other exceptional coefficient strictly above its crepant value. With \(J=K_T+B\) this gives \[ D(0)=K_R+\Theta=h^*J+P,\qquad P\geq0, \tag{136}\] where \(P\) is supported exactly on the undesired exceptionals.

Fix an \(h\)-ample line \(H_0\) and let \(E_1,\ldots,E_N\) be all exceptional primes of \(R\). On a small full-dimensional rational polytope in formal coefficient space consider \[ D(u)=D(0)+u_0H_0+\sum_{i=1}^{N}u_iE_i, \qquad u_0\geq0,\quad |u_i|\leq\eta u_0. \tag{137}\] Choose \(\eta>0\) small enough that the perturbation is relatively ample at every nonzero parameter, and then make the polytope small enough. On the finitely many Stein neighborhoods choose effective representatives of \(H_0,-H_0\), and \(\pm E_i\). A small common positive multiple of the effective sum representing \(H_0-H_0\sim0\) supplies a common ample part. Taking the other coefficients sufficiently small on a simultaneous resolution gives equivalent ordinary klt adjoints at the vertices. Lemmas 35 and 36 give finitely many marked normal relatively ample models over \(T\) for all rational parameters.

For each model occurring arbitrarily near zero, take the affine spans of its parameter sets in successively smaller punctured neighborhoods. These nested affine spaces eventually stabilize. Discard models not accumulating at zero. If every eventual span were proper, finitely many proper affine subspaces would cover all sufficiently small rational points in the full-dimensional cone, which is impossible. Thus one marked model \(Y\) occurs arbitrarily near zero with full eventual affine span.

Choose affinely independent rational parameters for this model. Their actual ample adjoint traces solve a rational linear system for the traces of \(D(0),H_0,E_1,\ldots,E_N\) in the group of rank-one reflexive sheaves tensored with \(\mathbb{Q}\). All these traces are therefore actual rational line bundles. A sequence of its parameters tending to zero makes \(D(0)_Y\) curve-nef over \(T\). Taking (136) in codimension one gives the actual identity \(D(0)_Y=h_Y^*J+P_Y\), with \(P_Y\) effective exceptional. Negativity forces \(P_Y=0\).

No specified prime is lost. At a nonzero parameter defining \(Y\), the relative system of a divisible power of \(D(u)\) contains the system of the ample perturbation, multiplied by the section of the corresponding power of \(P\) and a base pullback. It embeds relatively at general points outside \(\mathop{\mathrm{Supp}}P\). Every specified prime has generic point there, including a discrepancy-one prime whose crepant boundary coefficient is zero. It cannot be contracted. Thus \(Y\to T\) is crepant with precisely the prescribed exceptional primes and effective klt boundary.

Each surviving exceptional prime is rational Cartier by the traces of the \(E_i\). For an arbitrary global rank-one reflexive sheaf on \(Y\), its trace on \(T\) has an invertible power by the global strong hypothesis. Lemma 32 expresses the corresponding power upstairs as a pullback line with an integral exceptional correction. Clearing the rational Cartier denominators of that correction makes a further power invertible. This proves the global strong property. Projectivity over the compact Kähler base gives the remaining category assertions. ◻

Proof. In dimensions at most two there are no nontrivial small diagrams, and Lemma 38 handles divisorial contractions. In dimension three discard a finite prefix to make all steps small. Terminality persists by Lemma 33. The underlying threefold singularities are ordinary terminal: deleting the effective rational Cartier boundary does not decrease discrepancies. In particular they are smooth at general curve points. This conclusion also holds for local analytic places. On a global resolution the exceptional coefficients for the empty boundary are negative; local places still exceptional over that smooth resolution have ordinary log discrepancy at least two.

Let \(b\) be the largest boundary coefficient, or zero for empty boundary. The coefficient set is fixed under small steps. If \(b>0\), test each step whose positive side has a flipped curve contained in a coefficient-\(b\) component. If \(b=0\), test every step with any flipped curve; a nontrivial positive side exists by the ample-sign argument in Lemma 33. Blowing up the generic point of that curve on the positive side gives an exceptional place with log discrepancy \[ t=2-\sum_j m_jb_j\leq2-b, \tag{138}\] where \(m_j\) are nonnegative integral multiplicities. One can realize this place by the normalized blowup of the whole curve. Positivity and the fixed boundary denominators make the possible \(t\) a finite set. For each such \(t\), the number of exceptional places with discrepancy strictly below \(t\) is finite by Lemma 39. These counts do not increase. At a tested step its tested place had discrepancy strictly below \(t\) before the step and equals \(t\) afterwards, so the corresponding count drops. Only finitely many tested steps occur.

If \(b>0\), consider the normalizations of the finitely many coefficient-\(b\) surfaces on the remaining tail. Both sides map bimeromorphically to the normalization of their common image in the contraction base. The positive map contracts no curve and is therefore an isomorphism. The negative normalization consequently maps projectively bimeromorphically to the next normalization, contracting a curve whenever the corresponding surface contains a flipping curve. These are compact Kähler surfaces. The cycle count leaves a tail on which no maximal-coefficient component contains a contracted curve on either side.

Remove all maximal-coefficient components from the boundary on this tail. The new adjoint is still negative, because an effective rational Cartier divisor has nonnegative degree on a curve not contained in it. Its one-ray identity with the original adjoint, obtained by Lemma 34, has a positive rational coefficient. Smallness then makes its transform relatively positive. Terminality persists after decreasing the boundary. Induct on the number of distinct positive boundary coefficients. The empty-boundary case was handled by testing every step, so the sequence is finite. ◻

Proof. First \(-P\) is \(h\)-ample. Compare any \(h\)-ample line with its rationally invertible trace downstairs. Their difference is a rational multiple of the unique exceptional prime \(P\), by Lemma 32. Negativity makes that multiple strictly negative, proving the assertion.

Divisorial adjunction of \(K_U+P\) to the normalization \(P^\nu\) gives an actual rational line of the form \(K_{P^\nu}+\Delta\), with \(\Delta\geq0\), even though \((U,P)\) need not be lc. We justify the sign, without requiring the individual terms on \(P^\nu\) to be rational Cartier. Take residue along the smooth strict prime on a log resolution and push its restricted crepant boundary to \(P^\nu\). This identifies the actual restricted line in codimension one, then by reflexivity. Lemma 40 computes a tested coefficient on a klt normal surface germ with the reduced curve of all sliced branches of \(P\). Such a surface germ is a quotient of a smooth germ by a small finite group, by the analytic klt surface classification; see (Kollár and Mori 1998, chap. 4).

On the smooth chart, adjunction to a normalized branch of a reduced plane curve has effective different: the conductor contribution from normalization and the intersections with other branches are nonnegative. More explicitly, hypersurface adjunction makes the plane curve dualizing sheaf a line, and finite duality identifies the normalization’s dualizing sheaf with its pullback multiplied by the conductor ideal. The quotient pullback of the reduced divisor is reduced. Frames of a Cartier log-adjoint power pull to the corresponding frames by the codimension-one étale property, and their meromorphic residues are related by pluricanonical pullback. Therefore the different downstairs pulls to the different upstairs plus the ramification divisor on the normalized branch. This proves \(\Delta\geq0\).

On a minimal smooth resolution \(\pi:S\to P^\nu\) the restricted adjoint is \(K_S+\Delta_S\) with \(\Delta_S\geq0\). Indeed \(K_S\) is relatively nef by curve adjunction, negative definiteness, and the absence of exceptional smooth rational \((-1)\)-curves. The correction \(\Delta_S=\pi^*(K_{P^\nu}+\Delta)-K_S\) is relatively anti-nef with effective pushforward; negativity applied to \(-\Delta_S\) gives the sign.

If \(P\) maps to a curve, take a general smooth fiber on \(S\), after Stein factorization. Its \(K_S\)-degree is at least \(-2\). If \(P\) maps to a point, \(P\) and \(S\) are projective. The classification of smooth projective surfaces supplies moving curves covering \(S\) with \(K_S\)-degree at least \(-3\): use ample curves if the minimal canonical class is nef, ruling fibers in the ruled case, or lines on \(\mathbb{P}^2\), and general strict transforms; see (Barth et al. 2004). In either case choose a curve not contained in the correction or in the finitely many excluded curves, including the inverse image of \(\mathop{\mathrm{Supp}}B^{\mathrm{str}}\cap P\). It maps birationally to a curve \(C\) and the effective correction has nonnegative degree there. This proves (139). ◻

Proof. For \(n=0\) the underlying space is terminal with the ordinary exceptional discrepancy gap \(1+\epsilon\). The terminal-point statement above bounds every canonical index by \(\lfloor1/\epsilon\rfloor\). A global rank-one reflexive sheaf has \(\mathbb{Q}\)-Cartier germs by the global strong hypothesis, so its local indices divide those canonical indices. Their bounded common multiple gives \(I(0,\epsilon)\).

Induct on \(n\), seeking a multiple of previous bounds. If there is no low place use the previous case. Otherwise extract only a place \(P\) of discrepancy \(a\in[\epsilon,1]\) by Proposition 41. The pair on \(U\) has effective crepant boundary \(B^{\mathrm{str}}+(1-a)P\). Its exceptional places are exceptional downstairs, except that \(P\) is no longer counted; all hypotheses hold with \(n-1\). Set \(I'=I(n-1,\epsilon)\).

For the curve in Lemma 43, crepancy and effectiveness of \(B^{\mathrm{str}}\) give \[ -3\leq (K_U+P+B^{\mathrm{str}})\cdot C =aP\cdot C<0. \tag{140}\] Let \(\mathcal F\) be a global rank-one reflexive sheaf on \(T\), and let \(\mathcal F_U\) denote its reflexive sheaf pullback. Distinguish this from the rational line pullback defined by an invertible power of \(\mathcal F\). Their meromorphic comparison has the actual rational-line form \[ h^*\mathcal F=\mathcal F_U+sP. \tag{141}\] Both \(I'\mathcal F_U\) and \(I'P\) are integral lines. Degree zero of the left side on \(C\) gives \[ s=\frac{I'\mathcal F_U\cdot C}{-I'P\cdot C}, \qquad 1\leq -I'P\cdot C\leq\frac{3I'}{\epsilon}. \tag{142}\] The numerator and denominator are integers. Put \[ I=I'\operatorname{lcm} \{1,\ldots,\lceil3I'/\epsilon\rceil\}. \tag{143}\] Then \(I\) and \(Is\) are multiples of \(I'\), so \(I\mathcal F_U+(Is)P\) is an integral line of zero degree on all contracted curves. Increase the crepant coefficient of \(P\) by a sufficiently small positive rational number. The pair remains klt, and its adjoint is relatively antiample because \(-P\) is ample. Lemma 34 descends this integral line. Its descent agrees with \(\mathcal F^{[I]}\) away from the codimension-two center of the extraction, hence everywhere by reflexivity. This proves the induction. ◻

Proof. Only dimension three remains. Suppose there is an infinite sequence; after the cycle count all steps are small. The finite sets of exceptional places of discrepancy at most one are nonincreasing, so stabilize to a set \(\mathcal E\). There is a uniform positive \(\epsilon\) of the type in Proposition 44 on this tail. Indeed start with Lemma 39 on its first model. Only finitely many places lie below its strict cutoff \(1+\epsilon_0\); outside \(\mathcal E\) their discrepancies are greater than one, with a positive minimum gap. Take the minimum of this gap, \(\epsilon_0\), and the positive starting discrepancy lower bound. All subsequent comparisons are nondecreasing.

The boundary denominators are fixed. Proposition 44 gives a common denominator for the actual adjoints on all models of the tail, and hence for all discrepancies by the crepant formulas. The nondecreasing values on \(\mathcal E\), bounded above by one, therefore eventually become constant. Over the first model of this stabilized tail extract exactly \(\mathcal E\) crepantly. The resulting pair is terminal with effective boundary and is globally strongly \(\mathbb{Q}\)-factorial.

Lift a flip \(X_-\dashrightarrow X_+\) over \(Z\) from its extracted terminal model \(Y_0\to X_-\). Run relative elementary steps over \(Z\) using Proposition 37. On a common projective resolution of any finite prefix ending at \(Y\), denote the pulled-back adjoints of \(Y_0,Y,X_+\) by \(P_0,P,P_+\) respectively. The comparisons give \[ P_0=P+G=P_++F, \qquad G,F\geq0, \tag{144}\] where \(G\) is exceptional over \(Y\) and \(F\) is exceptional over \(X_+\). The line \(P_+\) is nef over \(Z\). Over \(Y\) the divisor \(G-F=P_+-P\) is nef and has nonpositive pushforward; negativity gives \(G\leq F\), before any termination is known. A lost prime of \(Y_0\) would have positive coefficient in \(G\), but zero coefficient in \(F\): this follows from smallness below for old primes and from constancy of discrepancies on \(\mathcal E\) for the extracted ones. Thus no prime is lost. The run stays small and terminal, so Proposition 42 makes it finite. Its endpoint has \(P\) nef over \(Z\).

At the endpoint negativity over \(X_+\) applied to \(F-G=P-P_+\) gives the opposite inequality, hence \(P=P_+\). Since the adjoint on \(X_+\) is relatively ample, its projection from the common resolution is constant on the fibers over \(Y\): otherwise a curve in a connected projective fiber would have positive pullback degree. Factoring the graph over the normal space \(Y\) gives a projective crepant morphism \(Y\to X_+\). It extracts the same places, by smallness below and the absence of lost primes, so the construction repeats.

Each nontrivial flip below forces a negative step in its lift, by lifting a negative curve. Infinitely many flips would concatenate to an infinite small terminal sequence with effective strictly transformed boundary, contrary to Proposition 42. This proves the theorem. ◻

Proof. The lc centers are the images of the finitely many coefficient-one strata on a log resolution, and each is generically an SNC floor stratum. Order the floor components defining the chosen generic chain. Take a projective log resolution isomorphic over an SNC open meeting all lc centers. Successively restrict its crepant formula to the smooth strict strata by residue, then push the adjunction boundaries in codimension one to their normalizations. At each stage the SNC log pullback has coefficients at most one, and its rational adjoint is the restriction of the ambient actual line. Reflexive extension gives (145); any exceptional difference from the crepant comparison is numerically zero and vanishes by both signs of negativity. A discrepancy-zero stratum of a restricted formula comes from an ambient coefficient-one stratum whose image meets the good open. This gives the sub-dlt discrepancy and good-open properties, and identifies nested lc centers by the generic SNC calculation.

There is also a canonical meromorphic comparison, not merely an abstract equality of line classes. On a smooth strict stratum the residue of the ambient log-pullback frame has exactly the pole and zero orders of the remaining crepant boundary. It therefore gives the frame of the pushed adjunction in codimension one on the normalization. Extending the line identification reflexively and composing with the natural embedding of its divisorial adjoint sheaf into meromorphic pluricanonical tensors gives the claimed map. Different resolutions give the same map on the dense generic SNC stratum, and hence everywhere meromorphically. Permuting the residues changes only signs, removed in even degree. Sequential adjunction through normalized intermediate strata agrees with this strict-chain calculation by the same dense-open test. In particular all codimension-one boundary orders agree.

We prove effectivity and the DCC assertion at one restriction step; iteration then proves the statement. By Lemma 40, a tested different coefficient is a normalized-curve coefficient on a normal dlt surface with a coefficient-one branch. If the tested point is an lc center, the surface pair is SNC there. Indeed on a sliced log resolution preserving the good open, a zero center must lift to the intersection of two strict floor curves, with no exceptional locus through it; the resolution is an isomorphism there.

Otherwise deleting the boundary leaves a numerically klt surface germ. The Mumford numerical pullbacks of effective terms are effective by negative definiteness. We recall why this is ordinary klt even if rational Cartierness of the separate canonical term has not yet been established. On the minimal resolution of the singular germ, write \[M=(E_i\cdot E_j)<0,\qquad K_S-\pi^*_{\mathrm{num}}K=\sum_i a_iE_i.\] The numerical klt condition gives \(a_i>-1\). Relative nefness of \(K_S\), from minimality and curve adjunction, gives \(Ma\geq0\) and hence \(a_i\leq0\). For any nonzero effective integral exceptional cycle \(Y=\sum n_iE_i\), set \(c_i=n_i+a_i>0\) on its support and \(c_i=0\) elsewhere. Off-diagonal entries of \(M\) are nonnegative, so \[ c^{\mathsf t}M(n+a)\leq c^{\mathsf t}Mc<0. \tag{146}\] Thus some supported \(E_i\) has \((K_S+Y)\cdot E_i<0\). Consequently \(\mathcal{O}_{E_i}(-Y+E_i)\) has degree greater than \(2p_a(E_i)-2\) and has zero \(H^1\) by curve duality. Peeling off such components with the cycle exact sequences proves \(H^1(\mathcal{O}_Y)=0\) for every exceptional cycle \(Y\). Grauert formal functions gives rationality; multiples of the full exceptional curve are cofinal with the maximal-ideal thickenings.

An integral multiple of the numerical pullback of any Weil divisor is now a line-bundle divisor on the resolution with all exceptional degrees zero. Its class restricts to zero in \(H^2\) of the exceptional curve, by normalization and the degree description of the curve’s top cohomology. Continuity around the compact fiber, or topological proper base change, makes that class zero after shrinking. On the preimage of a small Stein neighborhood, \(H^1(\mathcal{O})=0\) by rationality. The exponential sequence makes this line actually trivial there. Pushing in codimension one proves that the original Weil divisor is rational Cartier; in particular the canonical divisor is. The numerical klt test is therefore the ordinary one.

The analytic log-terminal surface classification now realizes the germ as a smooth germ modulo a small finite group (Kollár and Mori 1998, chap. 4). Pull up the full boundary by canonical pullback, which is étale in codimension one. No exceptional place over the origin on the chart has nonpositive discrepancy, by finite discrepancy comparison with positive ramification factor. The point blowup therefore tests total boundary multiplicity strictly below two. There is exactly one smooth coefficient-one branch. The finite group preserves it and acts faithfully on its tangent: finite actions linearize, and a nonidentity element with trivial tangential eigenvalue would be a quasi-reflection. Thus the group is cyclic of order \(r\), also the branch ramification index; \(r=1\) is allowed.

Residue and ramification give \(1-1/r\) for the branch alone. Other components of coefficients \(d_j\) contribute intersection multiplicities \(k_j\geq0\) on the chart. The different coefficient is consequently \[ 1-\frac1r+\frac{\sum_jk_jd_j}{r}, \qquad 0\leq\sum_jk_jd_j\leq1. \tag{147}\] The upper bound follows from the sub-lc resolution formula. This proves effectivity and hence the full dlt condition at this step. Positive elements of a nonnegative DCC set have a positive minimum, if any occur. The sums in (147) thus have a bounded number of positive terms, counted with multiplicity, and satisfy DCC. In a decreasing sequence of coefficients below one, \(r\) is bounded as well, since \(1-1/r\) is a lower bound. This proves DCC for each restriction and for its iterations. ◻

Proof. The morphisms are the restrictions of the ambient projective morphisms followed through finite normalizations. The relative ample signs restrict as stated. Choose a common ambient resolution preserving the generic SNC chain. By Lemma 33 its effective pullback difference has support containing the full fibers over the non-isomorphism set. Restrict the two crepant formulas along the common strict chain. The same coefficient-one terms are subtracted, and Lemma 46 identifies the remaining canonical comparisons. Their difference is precisely the restriction of that effective divisor. The stratum itself is not contained in its support, since its generic point is preserved. The full inverse image of any indicated center is contained in the support, so pullback gives strictly positive multiplicity at each such place. This proves both assertions for actual crepant boundaries, rather than only for restricted first Chern classes. ◻

Proof. We induct simultaneously on dimension. The order within a dimension is (i), then (ii), then (iii) when \(d\leq3\). Dimension zero is immediate. In proving (i) in dimension \(d\), discard a prefix so that all steps are small, by Lemma 38. A zero-discrepancy place on a new space was already a zero place. Strictness in Lemma 33 excludes containment of its center in the exceptional locus. The finite lists of lc centers therefore stabilize, and the generic point of every remaining center is preserved at every subsequent step.

Induct increasingly on the dimension \(e\) of a surviving center; the point case is already settled by preservation of its generic point. Normalize its successive transforms and use the same generic adjunction chain. By the smaller-center conclusion, the stratum diagrams and adjunction pairs are isomorphisms near their non-klt loci on a tail. Their discrepancies are nondecreasing by Lemma 47.

If a prime is extracted by a stratum transformation, it lies on the new side over the ambient exceptional locus. Its earlier discrepancy is strictly smaller than its new discrepancy, the latter being at most one by effectivity. At the start of this tail it therefore had discrepancy less than one and center outside the non-klt locus, since the diagrams are unchanged near that locus. Lemma 39 gives finitely many possible exceptional places; the possible nonexceptional positive-boundary primes add only finitely many. For each such place, repeated extraction would give a strictly decreasing sequence of boundary coefficients, by the strict comparisons. The fixed adjunction DCC set of Lemma 46 excludes infinitely many such occurrences. After discarding a prefix there are no stratum extractions.

The cycle count then leaves no prime contractions by stratum transformations either; for normal curves the diagrams are already isomorphisms. The primes now match. Their coefficients do not increase, and finite positive support and DCC stabilize the whole boundary. Neither morphism of a stratum diagram to \(S\) can contract a prime on this tail: its generic point would lie over the ambient exceptional locus, and strict discrepancy increase would contradict the matched coefficients. Thus the stratum diagrams are small, allowing isomorphisms, and their effective pullback differences are exceptional over the positive stratum. If the stratum transformation is an isomorphism, its discrepancies agree, and strictness excludes any ambient exceptional intersection with the stratum. One may test a divisor over a point of such an intersection, or the point itself for a curve.

For each remaining nontrivial stratum diagram, its dimension is \(e<d\). Use the modification assertion already proved in dimension \(e\) to start on a projective globally strong crepant dlt model of the negative stratum. Run relative elementary steps over \(S\) by Proposition 37. The lower-dimensional termination assertion makes this run finite, with a relatively nef endpoint. On a common higher model write \[ P_0=P+G=P_++F, \tag{149}\] where \(P_0,P,P_+\) are the initial, endpoint, and positive-stratum adjoint pullbacks. The divisors \(G,F\) are effective and exceptional over the respective endpoint sides. Both \(P\) and \(P_+\) are relatively nef, so negativity in both directions gives \(G=F\) and \(P=P_+\). Relative ampleness on the positive stratum makes its projection constant on the connected projective fibers over the endpoint, as in the proof of Theorem 45. The graph therefore factors into a projective crepant morphism to that stratum. Remaining exceptional primes came from old exceptional primes, since there was no extraction upstairs and the stratum transformation was small; they retain coefficient one. This endpoint can start the next lift.

Each nontrivial stratum surgery forces at least one negative step in its lift. Its negative morphism is nontrivial as well, by smallness, matched boundary data, and the ample signs; lift a negative curve to see the assertion. Infinitely many surgeries would concatenate to an infinite lower-dimensional dlt elementary sequence across possibly varying bases, contrary to (iii) in dimension \(e\). This proves disjointness for the chosen center. There are finitely many centers, completing (i) in dimension \(d\).

For (ii), take a projective SNC resolution of the lc pair with boundary equal to the strict boundary plus the full reduced exceptional divisor. Its adjoint is the pullback of the original adjoint plus an effective exceptional correction, by log canonicity; the correction is supported in the floor. Run relative elementary steps over the original space. At every stage the same identity holds with the pushed-forward correction. If the run were infinite, (i) and the cycle count would leave a tail disjoint from the floor on both sides. The current adjoint could not have negative degree on a contracted curve there, since its correction is supported in the floor and its other term is a base pullback. Thus the run ends. Relative nefness and negativity kill its effective exceptional correction. The endpoint is crepant dlt and globally strongly \(\mathbb{Q}\)-factorial by continuation. Every remaining exceptional prime came from an exceptional prime on the resolution and still has coefficient one. This proves (ii).

Finally, if \(d\leq3\), an infinite dlt sequence would, by (i) and the cycle count, have a small tail disjoint from the floor on both sides. Delete the floor. The resulting pair is klt, using the global strong condition and the dlt discrepancy characterization. The ample signs are unchanged by disjointness on both sides. This contradicts Theorem 45, proving (iii) and completing the induction. ◻

Proof. Choose a common canonical power on the two spaces. On their isomorphic open, the actual canonical identification transports the metric. Nonextraction makes the complement of this open in the target have codimension at least two. In a local frame of the target line we must extend a psh weight over that complement.

Here is the local upper bound needed for this Hartogs argument on a normal germ. Take a finite local projection to a ball. Off the branch locus and the image of the missing set, the maximum of the weight on the finite fibers is psh. It extends across the branch locus while the fibers still avoid the missing set, by local upper boundedness. The bad image has codimension at least two, so psh Hartogs extension on the smooth ball extends this maximum there as well. It bounds the original weight on a dense analytic complement. The bound holds on its whole original domain by the disc test, using discs generically outside the excluded analytic sets through any tested point. Normality and the psh Riemann extension theorem on locally irreducible spaces now give the psh extension by upper regularization. These extensions respect the transition functions of the actual line. They define the required semipositive metric. For a related actual-line statement under rational singularities, see (Höring et al. 2025, Lemma 3.6). ◻

Proof. We give the two phases separately. In the first phase the current pair \((T_i,G_i)\) stays dlt with reduced boundary, and we make a step whenever \(K_{T_i}+G_i\) has a negative extremal ray of the cone \(\overline{\mathrm{NA}}(T_i)\) specified in Assumption 5. The underlying zero-boundary pair is klt by the dlt floor perturbation argument, and canonical pseudo-effectivity is retained by Lemma 49.

For a chosen negative ray, take a rational \(t<1\) sufficiently close to one that it stays negative for \(K_{T_i}+tG_i\). This is an effective klt pair, and its adjoint is pseudo-effective because \(K_{T_i}\) is. Assumption 5 supplies the projective bimeromorphic ray step with its prescribed face, negative-side Bott–Chern rank, global strong condition, and Kähler category. The full adjoint has negative degree on this ray, hence the negative relative ample sign. Its one-ray degree-zero adjustment and Lemma 34, applied to the klt contraction, give the positive relative ample sign on a flip. Thus it is an elementary dlt step for the full adjoint, and Lemma 33 keeps the transformed pair dlt.

This first phase is finite. Otherwise the cycle count and Theorem 48(i) leave a small tail disjoint from the boundary on both sides. It is then a genuine zero-boundary \(K\)-negative klt program. The same rays are \(K\)-negative, as tested by their actual contracted curves; the contraction faces and negative-side Bott–Chern conditions are the ones already supplied. Both relative canonical signs are unchanged because the floor is disjoint from all fibers involved in the surgery. Moreover these are the actual canonical flips stipulated by the assumption: under a small map, direct images of common Cartier canonical powers agree as reflexive sheaves, and the positive canonical powers are relatively ample. They give the required relative canonical algebra and tautological line. The supporting-class existence for these zero-boundary negative rays is also supplied by the same assumption. This infinite tail would therefore be one of the programs excluded by its arbitrary termination clause, starting from its first zero-boundary klt model. No dlt fourfold termination assertion has been used.

At the end of phase one write \(A_i=K_i+G_i\). It has no negative extremal ray and, in particular, is nonnegative on curves. We justify this conclusion without asserting an absolute curve criterion for nefness. The normalized Kähler-mass slice of the positive closed current cone defining \(\overline{\mathrm{NA}}(T_i)\) is compact for smooth Bott–Chern pairings. Indeed positive closed currents of unit mass have locally bounded order-zero coefficients by positivity and the Kähler mass; weak limits remain positive and closed. Their pairing image is compact in the product of the pairing coordinates, or in the finite-dimensional realization when used. Its positive cone with zero is closed, since the mass coordinate and normalized coordinates control every convergent sequence. Thus this is the full cone in the definition. A negative value of \(A_i\) would give a negative minimum face of the compact convex slice and hence an extreme point, contradicting the absence of a negative extremal ray. Only curve nonnegativity is needed at this stage.

In the second phase we make \(A_i\)-trivial steps of a single zero-boundary \(K\) program. Maintain the properties that \(A_i\) is nonnegative on curves, \((T_i,G_i)\) is lc, \((T_i,0)\) is klt, and \(K_i\) is pseudo-effective. If some rational \(t_0\in[0,1)\) already has \(K_i+t_0G_i\) analytically nef, every rational \(t\in[t_0,1)\) does as well. Indeed these pairs are klt by interpolation between the zero-boundary klt pair and \((T_i,G_i)\), and their adjoints are pseudo-effective. Their degrees are nonnegative by convexity between \(K_i+t_0G_i\) and the curve-nonnegative \(A_i\). A non-nef one would, by Assumption 5, have a negative ray with a projective contraction and an actual negative contracted curve, a contradiction. Closedness of the analytic nef cone then makes \(A_i\) analytically nef as \(t\) tends to one.

Suppose instead that every rational \(K_i+tG_i\), \(0\leq t<1\), is non-nef. Choose current positive integers \(m,r\) with \(mA_i\) and \(rK_i\) Cartier. Choose a rational \(t\) so close to one that \[ \frac{m(1-t)}{t}<\frac{1}{r(\dim T_i+1)}. \tag{151}\] The klt assumption gives a negative extremal ray \(R\) for \[K_i+tG_i=tA_i+(1-t)K_i.\] Its contraction supplies an actual curve class. Since \(A_i\) is nonnegative on curves, \(R\) is \(K_i\)-negative. Choose the prescribed contraction and step for this same ray now from the zero-boundary application of Assumption 5.

We claim \(A_i\cdot R=0\). If this degree were positive, the Cartier line \(mA_i\) would be relatively ample on that contraction, since its fiber-curve degrees are a positive multiple of those of \(-K_i\). Work over a Stein neighborhood with a property-(P) compactum containing a point with nontrivial fiber. The local projective analytic rationality theorem for the ample Cartier line \(mA_i\) and the klt canonical adjoint gives denominator at most \(r(\dim T_i+1)\) for its positive finite relative nef threshold; see (Fujino 2023, Theorem 4.3.1 and Remark 4.3.4). All contracted curve degrees are proportional, so for a curve \(C\) of the ray that threshold is exactly \[ \frac{mA_i\cdot C}{-K_i\cdot C} \geq\frac{1}{r(\dim T_i+1)}. \tag{152}\] The actual line data and canonical representatives needed for the theorem are available locally over this bimeromorphic Stein base, as in Section 7.2. On the other hand, negativity for the test boundary gives \[\frac{mA_i\cdot C}{-K_i\cdot C} <\frac{m(1-t)}{t},\] contradicting (151). This proves the claim using only current-stage indices, with no uniform index assumption on the fourfold sequence.

By Lemma 34, a Cartier multiple of \(A_i\) descends to an actual line on the contraction base. The pushed adjoint on a divisorial step, or the transformed adjoint on a small positive model, is the corresponding pullback by codimension-one identification and reflexivity. On a common projective model any remaining exceptional comparison is relatively numerically zero, and hence zero by negativity. Thus the step is crepant for the actual adjoint \(A_i\), and the pair stays lc. Curve nonnegativity on the next model follows by lifting each curve to a common model and using the equality of pullbacks. The zero-boundary klt and canonical pseudo-effective hypotheses persist by Assumption 5 and Lemma 49.

Repeat this procedure whenever no rational nef parameter exists. Every step is a prescribed \(K_i\)-negative step of the same zero-boundary program. Arbitrary termination in Assumption 5 excludes indefinite repetition. We therefore reach the asserted \(T,G,A\) and nef klt interval. Neither phase extracts divisors, so their composite is nonextracting.

Apply Theorem 48(ii) to \((T,G)\) to obtain (150). The boundary \(D\) is the strict transform of \(G\) plus coefficient-one exceptional primes, hence reduced. If \(G\) is nonzero its strict transform is nonzero. Analytic nefness pulls back, giving that of \(J\).

Finally take the stated simultaneous resolution and write \(\rho=h\circ\mu:M\to T\). For every rational \(t\) in the nef klt interval, Lemma 17 applied to \((T,tG)\) gives a semipositive metric with minimal singularities and zero Lelong numbers on \(\rho^*(K_T+tG)\). Add the metric of the effective rational divisor \((1-t)\rho^*G\). This is a semipositive metric on the fixed actual line \[ \rho^*A=\rho^*(K_T+tG)+(1-t)\rho^*G=\mu^*J. \tag{153}\] A metric with minimal singularities on that line is no more singular than each of these metrics, up to an additive constant in weights. At every \(x\in M\) its Lelong number is therefore at most \((1-t)\nu([\rho^*G],x)\). The latter is finite and tends to zero as rational \(t\) tends to one. All Lelong numbers of the minimal metric vanish. This proves (iv) and the proposition. ◻

Proof. First, both \(\mathcal{O}_V\) and \(\mathcal{O}_V(-D)\) are Cohen–Macaulay. The first assertion follows from klt rationality: decreasing the reduced dlt boundary makes the zero-boundary space klt. For the second, work locally and trivialize a Cartier multiple of the integral Weil divisor \(D\). The resulting normal cyclic index cover is étale in codimension one: divisorial orders in the defining equation are multiples of the index. The cover is klt by the finite canonical pullback and discrepancy formula, hence Cohen–Macaulay. The desired divisorial sheaf is an eigensummand of the pushforward of its structure sheaf, so is Cohen–Macaulay as well. Normality identifies the reduced ideal of \(D\) with \(\mathcal{O}_V(-D)\). The sequence \[0\longrightarrow\mathcal{O}_V(-D)\longrightarrow\mathcal{O}_V \longrightarrow\mathcal{O}_D\longrightarrow0\] now shows that \(D\) is Cohen–Macaulay, in particular \(S_2\).

For the codimension-one description, take general transverse surface germs in the dlt adjunction calculation. At a single coefficient-one branch the plt local model is a cyclic quotient of a smooth pair, and its boundary curve is smooth. At two coefficient-one branches, the different and the sub-lc condition give an ordinary double SNC point; these are precisely the generic deeper lc strata described above. The branches come from distinct primes. To lift the assertion about a single-branch transverse curve to regularity at the corresponding point of \(D\), note that the transverse boundary curve is generically reduced and has no embedded points by Cohen–Macaulayness. Its regularity lifts by lifting generators of the maximal ideal through the transverse parameters. This proves the asserted codimension-one description.

Let \(S\) be a normalized conductor surface on \(Y_i\). Chain adjunction gives \[ J_S=K_S+\Xi_S=J|_S. \tag{156}\] The matching branch on \(Y_j\) is the normalization of the same image, and the two normalizations agree. At a general SNC point the two iterated residue maps differ by the sign from interchanging two differentials. Their even powers are equal. On a common smooth model the two invertible pullback subsheaves of meromorphic pluricanonical forms are therefore equal: they have the same invertible domain and the same meromorphic map on a dense open. Thus this is an equality of the actual residue lines, not only a numerical comparison. The same argument applies to further normalized lc strata. Proper subcenters are generically deeper SNC strata, so the chains on either side identify the same further centers and the same even residue maps.

At an ordinary double point the usual two-branch local calculation glues sections precisely when their values agree on the intersection. Hence a matching tuple is a section of the ambient invertible restriction away from a subset of codimension at least two in \(D\). For a reduced \(S_2\) analytic ring, regularity of a meromorphic element is tested in its height-one localizations. Applied after trivializing \(\mathcal{O}_D(mJ)\), this extends the section across the omitted subset. Equivalently, the class of the normalization-pushforward section modulo the invertible sheaf has support of codimension at least two and vanishes by the \(S_2\) Hartogs criterion. The residue comparisons used here are the comparisons induced by the one ambient line bundle; no arbitrary scalar choices in adjunction enter the descent. ◻

Proof. The normal compact Kähler threefold log-abundance theorem applies to the effective rational adjunction pair (155): its actual adjoint is analytically nef. One may first pass to a projective crepant dlt modification and use the \(\mathbb{Q}\)-factorial dlt form of the theorem. We use here Das–Ou’s normal threefold theorem, together with their dlt-modification theorem (Das and Ou 2025, Corollary 1.3) (Das and Ou 2024, Theorem 5.2). Generation on the modification descends by projection to the normal target. These inputs concern normal pairs; no nonnormal gluing assertion is being invoked.

For completeness, the passage to divisor representatives in these theorems does not impose an additional global-frame hypothesis. One can first produce a section of a power of \(J_i\) as follows. If \(Y_i\) is projective, use projective threefold log abundance. Otherwise resolve it by a smooth compact Kähler space. If that space is non-uniruled, smooth threefold canonical nonvanishing gives a section which pushes to \(J_i\), since \(\Gamma_i\ge0\). If it is uniruled and non-Moishezon, take a resolved rational quotient. Its smooth non-uniruled compact Kähler base has nonnegative Kodaira dimension, and its very general smooth quotient fibers are projective of dimension at most two. On a simultaneous resolution of the pair, use the strict boundary and all reduced exceptional divisors as an effective SNC boundary \(\widetilde\Gamma\). Log canonicity gives \[K_{\widetilde Y}+\widetilde\Gamma =p^*J_i+E,\qquad E\ge0\quad\hbox{$p$-exceptional}.\] This adjoint restricts pseudo-effectively on a very general quotient fiber. Projective log nonvanishing in dimensions one and two supplies fiberwise sections. Assumption 4, including unit coefficients and its invariant-base interpretation, now supplies a section upstairs exactly as in the rational-quotient reduction. It pushes to a section of an actual power of \(J_i\) by normality and reflexivity. The rational-quotient and uniruledness inputs are used in their smooth compact Kähler scopes (Campana 2004; Ou 2025).

Cancelling the boundary contribution in such a section gives a meromorphic pluricanonical tensor. Its divisor divided by its degree is a rational canonical representative, compatible with pullback to resolutions. The discrepancies are the local canonical discrepancies, and the divisible systems are still the actual systems of \(J_i\). Thus the divisor formulation of threefold abundance gives precisely the claimed semiampleness. The Stein factorization of a sufficiently high generated system yields (157). ◻

Proof. Fix one orbit of objects. At an object \(L\), take the set of all sections transported to \(L\) from all chosen initial sections in that orbit, along all paths. The set is finite: there are finitely many starting objects, and two paths from the same object differ by a loop, whose image is finite. Arrows biject these finite sets. Multiply all their distinct members. The resulting section is nonzero because the object is irreducible, and the products are carried to one another because pullback is multiplicative. They have equal degree within the orbit. The initial section at \(L\) occurs among its factors, so its prescribed zeros remain. Taking further powers makes the degrees equal across the finitely many orbits. ◻

Proof. On an initial smooth projective log resolution of \(Y\), restrict the crepant equality \[K_W+\Gamma_W=p^*J_Y\] to a very general smooth connected fiber of the map to \(Z\). Its right side is trivial. Pair with the appropriate power of the pullback of a Kähler class from \(Y\). Exceptional terms have zero pairing, because their images have codimension at least two on this general fiber, or they do not meet it. Strict boundary terms have nonnegative pairing; a dominant coefficient-one branch gives a strictly positive term. Hence the canonical class of this smooth fiber is not pseudo-effective. Ou’s theorem implies that the fiber is uniruled (Ou 2025). For \(r=0\) this argument is applied to the total space.

Take the almost holomorphic rational quotient of the smooth Kähler resolution. Its fiber has positive dimension, since the rational curves through very general points of the fibers just considered are quotient-contracted (Campana 2004). Its base cannot have dimension at most one. Indeed, rational connectedness of the general quotient fibers and the differential sequence would then force every holomorphic two-form on a resolved total space to vanish. The Kähler projectivity criterion would make that total space projective, a contradiction. The quotient base is therefore a surface, and the general quotient fiber is a smooth rational curve. Those curves are vertical over \(Z\), so \(f\) factors meromorphically through the quotient. Resolving the graphs and the base gives (159). Stein factorization and very general connectedness give connected \(g\)-fibers; the fibers of \(u\) are connected as images of the connected fibers of \(fp\).

If \(P\) were projective, the rational-curve fibration would have Moishezon total space. For example, coherence and GAGA give meromorphic sections of \(g_*\mathcal{O}_W(-K_{W/P})\) generating its general fibers. Evaluation embeds the general smooth rational fiber, and these sections together with base functions give full algebraic dimension. Thus \(P\) is nonprojective.

Every exceptional prime of the initial projective resolution of \(Y\) is projective over a compact curve or a point, hence Moishezon. It cannot dominate \(P\): a dominant generically finite map of surfaces preserves algebraic dimension, by the norm or characteristic-polynomial argument after finite Stein factorization. The strict transform of a Moishezon prime remains Moishezon. The further graph resolutions can be made isomorphisms over a general quotient-base open, so their new exceptional primes are vertical too. Consequently all horizontal crepant boundary terms are strict transforms of effective boundary components. Adjunction on the general rational fiber makes their weighted degrees sum to two.

A nonprojective compact Kähler surface has a nonzero holomorphic two-form, again by the Kähler projectivity criterion. Finally, if \(r=2\), then \(P\to Z\) would be generically finite over a projective surface, forcing \(P\) to be Moishezon and projective. Thus \(r\le1\). ◻

Proof. If \(r=1\), a vertical surface which dominates \(Z\) would have image a curve in \(P\) dominating \(Z\). This contradicts the preceding no-multisection observation. It remains to prove horizontality when \(r=0\), so that \(J_Y\sim_\mathbb{Q}0\).

Choose a nonzero \(\eta\in H^0(P,K_P)\). On the general rational fiber write the distinct horizontal marked points and their coefficients as \[b_1,\ldots,b_q\in\mathbb{Q},\qquad 0<b_j\le1,\qquad \sum_{j=1}^q b_j=2.\] Take a smooth compact Kähler generically finite base change \(b:P^+\to P\) splitting and ordering the markings. It can be constructed from a main component of a fiber product of the horizontal prime normalizations over \(P\), on the finite étale locus parametrizing an ordering of the distinct marks, and then resolved. Resolve the main component of the pulled-back rational fibration, obtaining \[h:W^+\longrightarrow W,\qquad g^+:W^+\longrightarrow P^+.\] Let \(H_j\) be the prime closures of the ordered generic sections. The very general rational fibers are unchanged by these resolutions.

For \(j\ne k\) there is a meromorphic top form \[ \beta_{jk}=d\log x_{jk}\wedge(g^+)^*b^*\eta, \tag{160}\] where \(x_{jk}\) has horizontal divisor \(H_j-H_k\). This description is valid meromorphically even across degenerations. To see its local construction on the base, the coherent sheaf \((g^+)_*\mathcal{O}_{W^+}(H_j-H_k)\) has generic rank one, because its restriction to a general rational fiber has degree zero and is trivial. Divide the canonical meromorphic section by the evaluation of a locally chosen generically nonzero direct-image section. Different choices change \(x_{jk}\) by a base function on the general fibers. Its logarithmic differential wedges to zero with the pulled-back top form of \(P^+\), so (160) is independent of the choices. Reversing \(j,k\) changes its sign.

The rational vector \((b_j)\) lies in the convex hull of the vectors \(\mathbf e_j+\mathbf e_k\) for \(j<k\). These are exactly the vertices of the rational polytope \[\bigl\{(x_j):0\le x_j\le1,\ \sum_jx_j=2\bigr\}.\] Average a rational convex decomposition over the permutations which preserve the coefficients. After clearing denominators and multiplying by an even integer, we obtain nonnegative even integers \(l_{jk}\) and an integer \(m>0\) such that \[ \sum_{j<k}l_{jk}=m,\qquad \sum_{k\ne j}l_{\min\{j,k\},\max\{j,k\}}=mb_j. \tag{161}\] We also make \(m\) divisible by all adjunction indices. The tensor \[ \beta=\bigotimes_{j<k}\beta_{jk}^{\otimes l_{jk}} \tag{162}\] is a nonzero meromorphic \(m\)-canonical tensor, invariant under the permutations of the ordered marks. Evenness removes the signs from reversed pairs. It therefore descends to \(W\). More explicitly, relative to the pullback of a local canonical power frame, its coefficient is the same on all generic sheets. Factoring \(h\) through its finite Stein part, normalized trace descends this meromorphic coefficient, and modifications do not change meromorphic functions. Formula (161) shows that the descended tensor has exactly the allowed horizontal poles, of orders \(mb_j\).

We must also control every vertical prime, including a prime whose image on \(P\) is a point. Let \(E\) be a vertical prime of \(W\), and choose a prime \(E^+\) above it, with ramification index \(e\) under \(h\). At its general point the tangential map is generically finite and separable, so the canonical Jacobian has order \(e-1\). Near a general point of the image in \(P\), take coordinates \(t,s\) with \(t\) vanishing on that image. Whether the image is a curve or a point, \[\mathop{\mathrm{ord}}_{E^+}(h^*g^*t)\ge e.\] In logarithmic differential coordinates at \(E^+\), the differential of this function retains that order. Both \(d\log x_{jk}\) and the pullback of \(ds\) are at most logarithmic. A logarithmic top form has at most one simple pole. Writing \(\eta\) in the \(dt\wedge ds\) frame, with its holomorphic coefficient, gives \[ \mathop{\mathrm{ord}}_{E^+}(\beta_{jk})\ge e-1. \tag{163}\] Thus (162) has order at least \(m(e-1)\), exactly paying the canonical ramification in descending an \(m\)-canonical tensor. The descended tensor on \(W\) is regular at \(E\).

Push it to \(Y\). The horizontal pole orders are allowed by the effective boundary, and at vertical primes it is regular; hence normality gives a nonzero section of \(mJ_Y\). Since \(J_Y\) is torsion, this section has no zeros: after trivializing a torsion power, a nonzero section is a nonzero holomorphic function on a connected compact normal space. A vertical floor prime would give a positive zero in the adjoint section, since its coefficient is one and the canonical tensor is regular there. This is impossible. Horizontality follows also for \(r=0\).

A horizontal conductor surface is generically finite over the nonprojective surface \(P\), so is non-Moishezon. A Moishezon neighbor would be projective, and its normalized conductor surfaces would be projective. Thus every vertex reached across a dominant edge remains non-Moishezon. Finally, each dominant branch has coefficient one and contributes its generic degree to the horizontal degree sum two. This proves the last claim. ◻

Proof. Let \(C\) be the normal connected-fiber system base of \(J_S\). It is a polarized projective curve. A sufficiently divisible system embeds \(C\), and the loop action induces automorphisms of \(C\) preserving a fixed ample power. Take the smooth minimal compact Kähler surface \(Q\) in the bimeromorphic class of \(S\). Classical compact-surface theory will be used in its Kähler and elliptic-surface forms (Barth et al. 2004). Here \(a(Q)=1\) and \(H^0(Q,K_Q)\ne0\); bimeromorphic self-maps of this minimal surface of nonnegative Kodaira dimension are automorphisms. The map to \(C\) is holomorphic on \(Q\). Indeed, on a common resolved model no curve can dominate \(C\) by the nonpositive Néron–Severi test, so the blowdowns to \(Q\) factor the map. Its general fibers are connected. Every meromorphic function on \(Q\) comes from \(C\): the function field has transcendence degree one and is algebraic over that of \(C\), so the function is constant on connected general smooth fibers and descends meromorphically.

Let \(F\) be the general fiber class. It is nonzero, rational and isotropic. Since the intersection form on \(\mathop{\mathrm{NS}}(Q)_\mathbb{R}\) is nonpositive, \(K_Q\cdot F=0\). Adjunction gives genus one for a general smooth fiber. The induced cohomology group preserves the rational line \(\mathbb{Q}F\). On \[ F^\perp/\mathbb{Q}F \tag{164}\] it preserves an integral lattice and the Hodge summands. The form is positive definite on real \((2,0)+(0,2)\) and negative definite on the remaining \((1,1)\) quotient by the Hodge index theorem. Reversing the latter sign gives an invariant positive definite norm. Its integral isometry group is finite. In particular the action on \(H^{2,0}(Q)\) is finite.

The action on the base is also finite. For genus at least two this follows from finiteness of the curve automorphism group; for genus one the subgroup preserving a fixed ample line bundle is finite. Suppose \(C=\mathbb{P}^1\). We claim \(q(Q)=0\). Restriction of holomorphic one-forms to a general smooth elliptic fiber is injective. Indeed, a form vanishing on one general fiber vanishes on all general fibers by constancy of periods, since holomorphic forms on a compact Kähler space are closed. It is then pulled back from a one-form on the smooth base open. This base form extends at a critical value: at a general point of a fiber divisor the local parameter is \(t=w^d\), and a pole or a nonremovable singularity cannot pull back to a regular form. There is no nonzero holomorphic one-form on \(\mathbb{P}^1\). Thus \(q(Q)\le1\). If equality held, the elliptic Albanese map would be nonconstant on a general elliptic fiber. A general Albanese fiber would then contain a curve dominating \(\mathbb{P}^1\), contradicting the absence of multisections. This proves the claim.

It follows that \[ \chi(\mathcal{O}_Q)=1+h^{2,0}(Q)\ge2, \qquad e(Q)=12\chi(\mathcal{O}_Q)-K_Q^2\ge24. \tag{165}\] There are at least three critical values of the relatively minimal elliptic fibration. Otherwise the smooth base is \(\mathbb{P}^1\) with at most two points removed. Its universal cover is compact or parabolic, so the genus-one period map to the upper half-plane is constant. The integral linear monodromy is then finite. Kodaira’s singular-fiber and monodromy classification, including multiple fibers, bounds the Euler number of a fiber with finite linear monodromy by ten. The types \(I_n\) and their multiples for \(n>0\), and \(I_n^*\) for \(n>0\), have infinite monodromy. Euler addition with at most two critical values would therefore give \(e(Q)\le20\), contrary to (165). The finite set of critical values is preserved by the base action, and an automorphism of \(\mathbb{P}^1\) fixing three points is the identity. The base image is finite in this final case too.

Pass to the finite-index kernel fixing both \(C\) and all holomorphic top forms. Choose a nonzero holomorphic top form \(\eta\) on \(Q\). Any meromorphic \(m\)-canonical tensor, divided by \(\eta^{\otimes m}\), is a meromorphic function and hence comes from \(C\). The kernel fixes it. It therefore fixes all the adjoint section spaces under their meromorphic canonical realizations. The original loop action has finite image. ◻

Proof. Use the dominant surfaces as objects, with conductor matchings and the internal arrows just constructed. Lemma 57 supplies loop finiteness, so Lemma 54 gives invariant nonzero sections in a common degree \(m'\).

We verify their descent at each node. A degree-one branch \(S\) is bimeromorphic to \(P\), so its general fibers over \(Z\) are connected. Its section of \(m'J_S\) is therefore pulled back from \(m'H\) on \(Z\). When there are two such branches, the internal arrow makes these base sections identical. For a degree-two branch, resolve \(S\dashrightarrow P\). Over a general \(z\in Z\), every component of the fiber dominates the connected smooth curve \(P_z\), with total degree two. Indeed, an exceptional or bad curve on \(P\) cannot dominate \(Z\), by the no-multisection observation. Divide the section by the pullback of a local frame of \(m'H\). This holomorphic function is constant on each compact connected component of the fiber. Invariance under the sheet-exchange arrow equates the two generic values. It descends meromorphically to \(Z\), and then holomorphically: a meromorphic function on a normal base whose pullback by a proper surjection is regular has no pole, as can also be checked through finite Stein factorization.

Pull these base sections back to \(Y\). Their restrictions match along dominant edges because the object sections do. They vanish on the prescribed nondominant images because the invariant object sections retain the initial vanishing factor and dominate \(Z\). This gives the desired tuple. ◻

Proof. All node adjoints are torsion. Choose a common sufficiently divisible even degree and a trivializing section at each node. Conductor comparison across an edge is multiplication by a nonzero constant on these sections. A graph with no cycles permits all comparisons to be solved by rescaling. In general it is enough to prove that the oriented product of these constants around each cycle is a root of unity, and then to take one further common power.

The graph has degree at most two by Lemma 56. At a node on a cycle, including a cycle formed by parallel edges, the two distinct branches must each have degree one over its \(P_i\). The section from the proof of Lemma 56 is now a power of \(d\log x\wedge\eta_i\): the two unit points exhaust the horizontal weighted degree. Its even residues on the two branches are the corresponding powers of the base top form \(\eta_i\). Following the cycle gives bimeromorphic identifications of the \(P_i\) and hence isomorphisms of their smooth minimal models \(Q_i\), since they have nonnegative Kodaira dimension. The cycle constant is the scalar on \(\eta^{\otimes m}\), or its inverse, for the resulting automorphism \(\phi\) of one minimal model \(Q\).

The existence of a single ambient Kähler class restricts this automorphism. Fix a Kähler form \(\omega\) on \(V\). At the two branches of a cycle node, choose smooth branch resolutions and maps \[r_k:B_k\longrightarrow W_i,\qquad \alpha_k:B_k\longrightarrow Q_i,\qquad k=1,2,\] where \(\alpha_k\) is bimeromorphic. Put \[v_k=(\alpha_k)_*r_k^*p_i^*[\omega] \in H^{1,1}(Q_i,\mathbb{R}).\] Each \(v_k\) has positive square. In fact, the pulled-back ambient class \(\beta_k\) on \(B_k\) has positive square because the branch maps generically birationally onto its surface image in \(V\). Write \(\beta_k=\alpha_k^*v_k+e_k\) with \(e_k\) exceptional. Orthogonality and negative definiteness of exceptional surface classes give \[ v_k^2=\beta_k^2-e_k^2>0. \tag{166}\]

Furthermore, \[ v_1-v_2\in\mathop{\mathrm{NS}}(Q_i)_\mathbb{R}. \tag{167}\] Indeed, the rational Hodge morphism \[(\alpha_1)_*r_1^*-(\alpha_2)_*r_2^* :H^2(W_i,\mathbb{Q})\longrightarrow H^2(Q_i,\mathbb{Q})\] kills \(H^{2,0}\) and \(H^{0,2}\). Holomorphic two-forms are basic on the smooth rational-fiber open, by the differential sequence and the absence of fiberwise one-forms; their restrictions to the two birational sections agree. Hence the rational image of this Hodge morphism has type \((1,1)\), and is contained in the rational Néron–Severi space. Apply it to the ambient real class to obtain (167). Across a conductor edge the corresponding branch maps to \(V\) agree on a common higher model. The transported \(v_k\) therefore agree across that edge. Going around the cycle yields \[ \phi^*v-v\in N:=\mathop{\mathrm{NS}}(Q)_\mathbb{R},\qquad v^2>0. \tag{168}\]

The form on \(N\) is nonpositive. If it is negative definite (including \(N=0\)), project \(v\) orthogonally to \(N^\perp\). The result \(w\) has positive square and is fixed by \(\phi^*\), by (168). The Hodge index theorem makes the \((1,1)\) complement of \(w\) negative definite; the real \((2,0)+(0,2)\) summand is positive definite. Together these give a positive definite real norm on \(H^2(Q,\mathbb{R})\) preserved by the integral action. The cyclic image is finite, so in particular its eigenvalues on \(H^{2,0}(Q)\) are roots of unity. If \(N\) has a radical, the Hodge index theorem makes it a rational invariant isotropic line. Its orthogonal quotient has the definite Hodge summands and integral lattice used in (164); the image on \(H^{2,0}\) is again finite.

These exhaust the possibilities for \(N\). Since \(\phi^*\eta^{\otimes m}\) is proportional to \(\eta^{\otimes m}\), the nonzero form \(\eta\) itself is an eigenform: its meromorphic ratio to its pullback has an \(m\)th power which is constant, and hence is constant. Its eigenvalue, and thus the cycle constant, is a root of unity. Taking a common power over the finitely many cycles and then rescaling the node sections solves all conductor comparisons. ◻

Proof. Fix either projection \(d\). Nonexceptional coefficients are those of an effective subboundary, so the negative part contributing to \(N\) is exceptional. The divisor \[N-T_\Theta=-\lfloor\Theta\rfloor\] differs from the smooth klt adjoint \(K+\{\Theta\}\) by \(-(K+\Theta)\), a pullback from the base. It is thus relatively nef and big; relative bigness here uses that \(d\) is birational. Relative Kawamata–Viehweg vanishing gives \[ R^1d_*\mathcal{O}(N-T_\Theta)=0 \tag{169}\] (Kawamata 1982). Normality gives \(d_*\mathcal{O}(N)=\mathcal{O}\). The divisor sequence and (169) therefore give a surjection \[\mathcal{O}\longrightarrow d_*\mathcal{O}_{T_\Theta}(N).\] It factors through \(d_*\mathcal{O}_{T_\Theta}\), which injects into \(d_*\mathcal{O}_{T_\Theta}(N)\) because \(N\) has no component in \(T_\Theta\). Thus \(d_*\mathcal{O}_{T_\Theta}\) is the structure sheaf of the reduced image of \(T_\Theta\). This image is the non-klt locus. Stein factorization, or the absence of nontrivial fiberwise idempotents in this equality, shows that the fibers are connected. A proper surjection with connected fibers induces a bijection on connected components. The common \(T_\Theta\) gives the assertion for both projections. Since the pairs become klt after decreasing the floor, their non-klt loci are exactly their floor supports. ◻

Proof. Take a projective small \(\mathbb{Q}\)-factorialization for the underlying klt pair (Birkar et al. 2010). It is crepant for the full adjunction pair and isomorphic at general SNC stratum points. The full pair remains dlt, its centers correspond bimeromorphically, and the chain adjunctions and even residue maps correspond on common resolutions. We may therefore work on this \(\mathbb{Q}\)-factorial model.

Proceed by dimension. In dimension one, a unit marking on a log-trivial curve forces a rational curve; there are at most two unit markings. For two markings the even residues of the logarithmic generator agree by (158). A single marking requires no comparison.

Suppose first that \(\lfloor\Gamma\rfloor\) is connected. Each normalized floor component with full adjunction again has a dlt log-trivial pair, and it admits a klt pair by adjunction keeping only that component in the ambient boundary. Within it use induction if there are proper lc subcenters; otherwise that component is itself the minimal lc center. For two successive intersecting floor components choose a minimal lc center in their intersection. Such centers exist by ordinary dlt stratum adjunction; intersecting distinct unit primes meet in lc strata, as is also seen on a general transverse surface. The two even chain restrictions through this shared center agree. Connectedness of the floor then links all the minimal centers.

Suppose instead that the floor is disconnected. Decrease all its coefficients by one fixed small positive rational \(\epsilon\). The resulting pair is klt, and its adjoint is \(\mathbb{Q}\)-linearly equivalent to \(-\epsilon\lfloor\Gamma\rfloor\), so is not pseudo-effective. The projective klt MMP in dimension at most three ends in a \(\mathbb{Q}\)-factorial elementary Mori fiber space \[Y'\longrightarrow P\] (Kollár 1992; Birkar et al. 2010). These steps are crepant for the full log-trivial pair. One may see this by transporting a trivializing pluricanonical tensor: there is no extraction, its divisor remains the negative of the corresponding multiple of the pushed full boundary, and the resulting pullback formulas agree. Equivalently, the exceptional comparison of the full, relatively numerically trivial adjoints is zero by negativity.

The pushed floor is relatively ample and supports the entire non-klt locus, because its decrease is klt. Lemma 60 shows that it is still disconnected. Some floor component is horizontal by relative ampleness. A vertical floor prime has degree zero on the Mori ray, since it misses a general fiber. By the whole-fiber property it contains every fiber it meets, and so meets every horizontal floor component. Every other vertical prime is likewise connected to the horizontal components. The presence of any vertical floor would consequently connect the entire floor. There are therefore no vertical floor primes.

Disconnectedness is now visible on a general fiber, since every component is horizontal. That fiber must be a curve. Indeed, on a normal projective Cohen–Macaulay fiber of dimension at least two an effective ample divisor has connected support. To check this, take a large Cartier multiple \(A\) of it. Serre duality and Serre vanishing give \(H^1(\mathcal{O}(-kA))=0\) for \(k\gg0\); the divisor sequence then gives \(H^0(\mathcal{O}_{kA})=\mathbb{C}\), implying connected support. The general fiber here is of klt type and hence Cohen–Macaulay, and the floor restricts to an effective ample divisor. Its restriction is lc and becomes klt after decreasing the floor, as can be checked on the restricted common log resolution.

The general fiber is consequently a rational curve with exactly two degree-one unit markings: it has a nonempty boundary and log-trivial adjoint, the total weighted degree is two, and disconnectedness gives two different horizontal components. Over a smooth projective model \(R\) of \(P\) we obtain a bimeromorphic model \(\mathbb{P}^1\times R\) with these marks as zero and infinity. The trivializing tensor is, over the function field and hence meromorphically, a power of the two-pointed \(d\log\) generator times a meromorphic pluricanonical tensor on \(R\). Resolving its divisor on \(R\), the crepant subboundary on the product is therefore \[ \{0\}\times R+\{\infty\}\times R +\operatorname{pr}_R^*B_R, \tag{170}\] where \(B_R\) may be signed, is log smooth, and has all coefficients at most one.

No vertical coefficient in (170) can equal one. Such a prime would connect the two sections by an SNC path of unit components. This path remains connected through unit components on a common resolution mapping to \(Y'\). Explicitly, when a smooth blowup separates two adjacent unit components at their general codimension-two crossing, its exceptional divisor has coefficient one and joins their strict transforms. Blowups not separating that crossing leave the connection. The image of the path would lie in the non-klt locus of \(Y'\) and meet both disconnected floor components, a contradiction.

All vertical coefficients are therefore strictly below one. The log-smooth discrepancy formula now says that the only zero-log-discrepancy places are the two horizontal sections themselves. In fact zero places of a log-smooth subboundary with coefficients at most one are generated by its unit strata; here the two unit divisors are disjoint and there are no deeper unit strata. Both section valuations occur as divisors on the original pair, since its floor is disconnected. They are exactly its minimal lc centers. Pair them bimeromorphically through \(R\). Their residue restrictions from the common generator agree in even powers by (158). Since these restrictions trivialize their adjunction lines, equality as meromorphic tensors on a common resolution gives \(B\)-crepancy. This finishes the disconnected case and the induction. ◻

Proof. Use as objects the normalized minimal lc centers of the nodes, with their iterated full adjunction pairs. Their adjoints are torsion. Lemma 61 supplies internal arrows connecting all objects at a node and preserving even restrictions from a trivializing node section.

For a matched conductor surface choose a minimal lc center inside it. More precisely, choose a minimal ambient lc stratum contained in the double stratum. It is a minimal center for the chains on both sides: any further subcenter would again be a nested ambient stratum, by the generic SNC description. Its shared normalization gives an arrow between the two node objects, with the symmetric even residue comparison from \(V\).

The projective representation theorem gives finite loop images. Lemma 54 therefore provides nonzero invariant sections in a common degree. They are trivializing sections on the objects. At a fixed node, their scalar ratios to restrictions of one trivializing node section agree by the internal arrows, so one node section induces all of them. Across a conductor surface the two node restrictions are trivializations of the same torsion line. Their ratio is constant on that connected normal surface. It is one on the chosen minimal center by the even residue comparison, so it is one everywhere. The node sections thus match on every conductor surface. ◻

Proof. At a node with a dominant branch, the general \(f_i\)-fiber is a smooth connected rational curve. Indeed it has \(K_F+\Gamma_F\sim_\mathbb{Q}0\) and a unit marking, so adjunction forces genus zero and horizontal weighted boundary degree two. Use dominant surface branches as objects. If there are two degree-one unit branches, or one degree-two unit branch, the two-mark calculation (158) gives their internal comparison or sheet-exchange involution over \(Z_i\). These are \(B\)-crepant and preserve restrictions of base sections. A single degree-one unit branch needs no internal comparison, even if other horizontal markings have fractional coefficients.

Combine these arrows with conductor matchings and apply the projective representation theorem and Lemma 54. At a degree-one object, a section descends to the base by the birational comparison. At a degree-two object, divide by a local base frame. Invariance under the involution equates the two generic values, so the ratio is a meromorphic base function; normality and properness give holomorphic descent. For two degree-one objects the internal arrow equates their descended sections. Pulling back gives the desired node sections. Conductor matchings and the retained initial vanishing factors give exactly the required matching and nondominant vanishing. ◻

Proof. Start on a projective crepant \(\mathbb{Q}\)-factorial dlt model of \(Y\). Decrease all floor coefficients by a fixed small rational \(\epsilon>0\) and run the projective klt threefold MMP over \(Z\). The decreased adjoint restricts to a negative multiple of the nonzero effective floor on a general fiber, so it is not relatively pseudo-effective. The program ends in a relatively elementary Mori fiber contraction \[ Y'\longrightarrow P\longrightarrow Z, \tag{172}\] with \(Y'\) \(\mathbb{Q}\)-factorial. The full adjoint remains the actual pullback of \(H\) at every step: all steps are over \(Z\) and extract no divisors, and negativity kills the exceptional relatively numerically trivial comparison. Thus the full lc data are crepant throughout. The transformed floor \(\Sigma'\) is relatively ample over \(P\) and supports the full non-klt locus, since its decrease is klt. Both maps in (172) have connected fibers; for the second this also follows by pushing the structure sheaf of the connected system map through the first.

On a common smooth projective resolution of the general fibers \(F\) and \(F'=Y'_z\), the crepant subboundary is the same. Lemma 60, applied to these surfaces, shows that the floor support on \(F'\) remains disconnected. This application does not require that the transformed full pair be dlt: it is lc, and its decrease is klt, which are the hypotheses used in that lemma.

If \(\dim P=1\), the floor on \(F'\) would be ample and connected. One may use the connected-support argument in the proof of Lemma 61, or the surface Hodge index theorem: a disjoint splitting of an ample rational divisor would give orthogonal classes with positive square. Consequently \(\dim P=2\). A general Mori fiber over \(P\) is rational, with total horizontal weighted boundary degree two and at least one unit marking. A vertical floor prime has degree zero on the Mori ray. If it dominates \(Z\), the whole-fiber property gives, on a general \(F'\), a full fiber over a point of \(P_z\), meeting every horizontal floor curve. Every other vertical floor contribution likewise joins the horizontal ones. All components restricted from a horizontal surface dominate \(P_z\) for general \(z\), after discarding the finitely many nongeneral base values. Thus such a vertical floor prime would connect the entire floor of \(F'\), which is impossible.

There are therefore no vertical floor contributions on general \(F'\). Disconnectedness and the degree sum two force exactly two disjoint degree-one floor curves over \(P_z\). The curve \(P_z\) is smooth and connected for general \(z\): the normal surface base has only isolated singularities, and generic smoothness and connected fibers apply.

We also need to exclude hidden deeper zero-discrepancy places. Choose the rational fiber coordinate \(x\) over \(P_z\) with these two curves as zero and infinity. This gives a bimeromorphic product model \(\mathbb{P}^1\times P_z\) of \(F'\). A divisible pluricanonical tensor trivializing its full log adjoint is, in relative notation, \[ (d\log x)^{\otimes m}\otimes\gamma, \tag{173}\] where \(\gamma\) is a meromorphic \(m\)-canonical tensor on \(P_z\). There are no other horizontal boundary terms because the two unit marks exhaust the degree. The crepant subboundary on the product is the two unit sections and vertical fibers with coefficients \(-\mathop{\mathrm{ord}}_p(\gamma)/m\le1\).

A coefficient-one vertical fiber would join the two sections by an SNC unit path. Successive point blowups preserve this path through unit components: a blowup at a crossing of two unit components inserts a unit exceptional curve. On a common resolution its image in \(F'\) would join the two disconnected floor curves inside the non-klt locus. Thus every vertical coefficient is strictly below one. The log-smooth discrepancy calculation now leaves only the two horizontal sections as zero-log-discrepancy places. They must both appear as divisors on \(F\), since its floor is disconnected. Hence its floor consists of those two curves and has no deeper lc center.

In particular, no zero-discrepancy place horizontal over \(Z\) is extracted or lost between \(Y\) and \(Y'\): restricting such a place to a general surface fiber would give another zero-discrepancy place there. The two Mori marks consequently correspond bimeromorphically to the original dominant floor data. Pair them over \(P\), using normalization of the degree-two cover when the two marks belong globally to one surface. The full crepant equalities over \(Z\) transport local base frames, and the even \(d\log\)-residue calculation gives the asserted internal \(B\)-comparison. ◻

Proof. Use dominant surface branches as objects. For a single smooth floor curve on a general fiber, the corresponding surface has connected general fibers over \(Z\), so its sections come from \(Z\). In the disconnected case use the internal arrows of Lemma 64. Over a general \(z\), the two curves are bimeromorphic to the connected curve \(P_z\). The values of a section relative to a local base frame are constant on each of them, and internal invariance equates the constants. This remains true when the two curves are in one global surface and the arrow is an involution. Thus invariant sections descend meromorphically and then holomorphically to \(Z\), using normality and properness.

The projective representation theorem and Lemma 54 provide the invariant nonzero object sections. Their descended base sections pull back to node sections. Matching follows from the conductor arrows, and vanishing from the retained initial base factors. ◻

Proof. The objects now are normalized curve strata, flagged by the node \(Y_i\), a dominant surface branch \(S\) at that node, and a further normalized curve branch on \(S\) dominating \(Z_i\). The flag records the actual chain of residue maps. Give each curve its full effective adjunction boundary and the actual restricted line \(J_L\).

There are three kinds of arrows. First, identify the corresponding flags across a shared conductor surface. Second, identify the flags through the two surface branches at a deeper curve center within one node. The generic SNC calculation and even residues identify these chains. Third, consider a fixed \(S\) and the curve Stein factor of \(S\to Z_i\). If a general connected fiber has two floor crossings, pair these two markings over that Stein curve. They give a bimeromorphic map between curve objects or an involution of one object. By the surface-fiber description preceding Lemma 64, this fiber is log rational with exactly these two unit points. Thus (158), now applied to the full adjunction on \(S\), gives a \(B\)-crepant arrow preserving restrictions of base sections. A floor fiber curve with only one crossing requires no internal pairing of points.

These arrows connect all flagged crossing points over a general \(z\in Z_i\) within a node. Indeed, the whole floor graph of \(F\) is connected, each floor curve meets another at one or two points, the two crossings on a curve are paired by the third kind of arrow, and the two flags at a crossing are paired by the second. This connects the flags along every path in the connected graph.

All curve objects are projective lc pairs with semiample adjoints, so the projective representation theorem gives finite loop images. Apply Lemma 54, with initial sections pulled from the respective node bases and vanishing on the required nondominant images. In the common final degree, divide the invariant curve sections by a local pulled-back base frame. The preceding connectivity says that their values coincide on all flag sheets over a general \(z\). They therefore define one meromorphic base section. It is holomorphic, since its pullbacks to the finite dominating curve covers are holomorphic and the base is normal. Pulling it back gives a section on the node.

Across a dominant conductor surface \(S\), equality on a chosen dominant further curve implies equality of the two node restrictions on all of \(S\). To see this precisely, both restrictions are sections of the semiample actual line \(mJ_S\), hence are pulled from its connected-fiber Stein system base. That base is a curve, and the chosen curve stratum dominates it, since it dominates \(Z_i\). Pullback of sections to that stratum is injective. The canonical even adjunction compares these pullbacks, which agree by the first kind of arrow; hence the original restrictions agree. Finally, the initial vanishing factors and dominance of the objects show that the node sections vanish on every nondominant branch. This proves the claim. ◻

Proof of Proposition 51. Take a cluster of vertices of maximal system dimension \(r\). If it is isolated, choose a nonzero high base section vanishing on all nondominant branch images, as explained above. Otherwise \(r\le2\). A cluster containing a non-Moishezon vertex is entirely non-Moishezon and has \(r\le1\); Lemmas 58 and 59 handle its two possibilities. Every other cluster is projective. Lemmas 62, 63, 65, and 66 cover all its possibilities.

In each case we have, in one sufficiently divisible even degree, a nonzero tuple of node sections matching on dominant edges and vanishing on all nondominant branches. Put zero on components outside the chosen cluster and take a further common power if necessary so that \(mJ\) is Cartier on the ambient \(V\). At every conductor surface either the two sections match by construction or both restrictions are zero. Lemma 52 therefore descends the tuple to an actual section of \(\mathcal{O}_D(mJ)\). It is nonzero because its restriction to a node in the chosen cluster is nonzero. This proves (154) on the whole reduced boundary. ◻

Proof. Suppose a finite collection of distinct prime divisors satisfies a nonzero rational relation in \(H^2(M,\mathbb{Q})\). Clearing denominators and then torsion gives a nonzero integral divisor \(E\) with \(c_1(\mathcal{O}_M(E))=0\) in integral cohomology. The exponential sequence and \(H^1(M,\mathcal{O}_M)=0\) imply \(\mathcal{O}_M(E)\simeq\mathcal{O}_M\). Its canonical meromorphic section is then a meromorphic function with divisor \(E\). Since \(a(M)=0\), every meromorphic function is constant, so \(E=0\), a contradiction. The classes are rational, so a real dependence would give a rational dependence by linear algebra. They are therefore linearly independent over \(\mathbb{R}\); their number is bounded by \(\dim H^2(M,\mathbb{R})\). ◻

Proof. Assume the contrary. Interpret a section as a morphism \(\mathcal{O}_M(-mL)\to\mathcal E\); this requires no meromorphic frame of \(L\). Among all such morphisms choose a tuple \(s_1,\ldots,s_k\), with respective indices \(m_1,\ldots,m_k\), whose generic rank is maximal. Let \(\mathcal H\subset\mathcal E\) be the saturation of the image of their direct sum. Every further such morphism has image in \(\mathcal H\): otherwise adding it would increase generic rank. A nonzero further section can replace at least one of the \(s_j\) while preserving generic rank.

There are infinitely many increasing indices \(m\) with a nonzero section, so one fixed replacement index \(j\) works for infinitely many of them. Taking the nonzero determinant gives effective integral divisors in the actual line bundles \[ \det\mathcal H\otimes\mathcal{O}_M((c+m)L), \qquad c=\sum_{i\ne j}m_i. \tag{175}\] The determinant is interpreted reflexively; on the smooth space its double dual is a line bundle, and the wedge morphism extends to it.

By Lemma 67, all the effective divisors in (175) have the same finite set of possible prime components. Their coefficient vectors lie in \(\mathbb{Z}_{\ge0}^{N}\). Such an infinite sequence has an infinite componentwise nondecreasing subsequence: successively pass to a constant or increasing subsequence in each coordinate. Choose two members with indices \(m'<m''\). Subtracting their divisors gives an effective divisor in the actual line \((m''-m')L\), contrary to \(\kappa(M,L)=-\infty\). This determinant argument is related to the nonvanishing method in (Höring et al. 2025). ◻

Proof. If \(G\ne0\), then \(D\ne0\). By Proposition 51, a positive Cartier multiple of \(J|_D\) has a nonzero section on the whole reduced divisor \(D\). Increase this multiple to clear the ambient index. We show that, under the contrary assumption \(\kappa(T,A)=-\infty\), high powers of this section extend to \(V\).

Take a simultaneous projective compact Kähler log resolution \(p:M\to V\) also resolving the map to \((T,G)\), and put \[ L=p^*J,\qquad K_M+F\sim_{\mathbb{Q}}L. \tag{176}\] The support of \(F\) is SNC and every coefficient is at most one; negative coefficients are allowed. All comparisons here are actual rational line-bundle comparisons. Normality and projection give \(\kappa(M,L)=\kappa(V,J)=\kappa(T,A)\). The space \(M\) still has \(a=q=0\).

For every sufficiently divisible \(m\), \[ p_*\mathcal{O}_M(mL-\lfloor F\rfloor) =\mathcal I_D\otimes\mathcal{O}_V(mJ). \tag{177}\] Indeed, the nonexceptional coefficient-one components of \(F\) are the strict transforms of the primes of \(D\). Every other coefficient-one component has center in \(D\), since \((V,D)\) is dlt and is klt away from \(D\). A holomorphic function vanishing on the reduced \(D\) pulls back with order at least one along all of these components. Conversely, any poles allowed by negative coefficients of \(\lfloor F\rfloor\) are exceptional. Normality and Hartogs extension remove them downstairs, while vanishing is required along each prime of \(D\). This proves (177), after the actual pullback line is removed by projection.

The low-degree Leray sequence therefore gives an injection \[ H^1\bigl(V,\mathcal I_D\otimes\mathcal{O}_V(mJ)\bigr) \hookrightarrow H^1\bigl(M,\mathcal{O}_M(mL-\lfloor F\rfloor)\bigr). \tag{178}\] No vanishing of a higher direct image is needed for this injection. Write the line on the right as \(K_M+\mathcal L_m\), where \[ \mathcal L_m= \mathcal{O}_M(-K_M-\lfloor F\rfloor+mL) \sim_{\mathbb{Q}}(m-1)L+\{F\}. \tag{179}\] For \(m\ge1\), give \(\mathcal L_m\) the metric obtained from the zero-Lelong semipositive metric on \(L\) and the divisor metric of \(\{F\}\), taking roots of identified powers if necessary. Its curvature is semipositive and its multiplier ideal is trivial. To see the latter directly, the zero-Lelong weight has every finite local exponential integrability exponent by Skoda’s theorem. The SNC coefficients of \(\{F\}\) are strictly below one, and hence their divisor weight has an integrability margin. Hölder’s inequality combines that margin with a sufficiently high finite exponent for the zero-Lelong weight.

Hard Lefschetz with multiplier ideals (Demailly et al. 2001, Theorem 0.1) gives a surjection \[H^0(M,\Omega_M^3\otimes\mathcal L_m) \longrightarrow H^1(M,K_M+\mathcal L_m).\] Its source vanishes for all large divisible \(m\) by Lemma 68, applied to the fixed bundle \(\Omega_M^3\otimes\mathcal{O}_M(-K_M-\lfloor F\rfloor)\). It follows from (178) that \[H^1\bigl(V,\mathcal I_D\otimes\mathcal{O}_V(mJ)\bigr)=0.\] The exact sequence for the reduced divisor \(D\) now makes restriction of sections of \(\mathcal{O}_V(mJ)\) onto \(D\) surjective. High divisible powers of the nonzero floor section remain nonzero because \(D\) is reduced, and therefore extend. Their extensions push to \(T\), since \(J=h^*A\) and \(h_*\mathcal{O}_V=\mathcal{O}_T\). This contradicts \(\kappa(T,A)=-\infty\). ◻

Proof. Suppose \(\kappa(W,K_W)=-\infty\). The Albanese argument of Lemma 10 gives \(q(W)=0\). Both \(a=q=0\) persist on every smooth compact Kähler model used below. Their canonical classes are pseudo-effective by non-uniruledness (Ou 2025). We distinguish the existence of a nonzero meromorphic section of a positive canonical power. That property is birationally invariant on smooth spaces: Jacobian comparison gives it on a common resolution, and meromorphic tensors extend over codimension two.

Proof of Theorems 2 and 1. The reduction in Proposition 11 treats the projective, uniruled and irregular cases and leaves smooth non-uniruled non-Moishezon fourfolds with \(q=0\). Positive algebraic dimension is handled by Proposition 12; algebraic dimension zero is handled by Theorem 70. The sections push to the actual adjoint of the nef pair exactly as in the reduction. This proves Theorem 2.

Finally start Assumption 5 from the original pair \((X,B)\). Its endpoint \((Y,B_Y)\) is in the required global strong category, with nef actual adjoint, no extraction, and all the stated discrepancy inequalities. Theorem 2 gives a first section there. Assumption 6 then makes a positive Cartier multiple of that same endpoint adjoint globally generated. These are all the assertions of Theorem 1. ◻

Proof. Put \(F=u^*(K_X+\Delta)-v^*(K_{X'}+\Delta')\). Discrepancy improvement on divisors of \(X\) gives \(u_*F\geq 0\), while \(-F\) is \(u\)-nef because \(K_{X'}+\Delta'\) is nef. The negativity lemma therefore gives \(F\geq 0\). At a prime of \(W\) that is not \(v\)-exceptional, the coefficient of \(F\) is zero unless that prime is extracted on \(X'\). In the latter case it equals the negative of its log discrepancy over \((X,\Delta)\). It is therefore nonpositive, and hence zero. Thus \(F\) is \(v\)-exceptional. If the source adjoint is nef, then \(F\) is also \(v\)-nef, so the negativity lemma gives \(F=0\).

Equality of the pullbacks transfers semiampleness, since a proper birational morphism onto a normal variety has direct image of its structure sheaf equal to the structure sheaf. More explicitly, global sections of a Cartier multiple and its pullback coincide; generation upstairs implies generation downstairs. Finally, for a klt source the coefficient at an extracted prime would be strictly negative, which is impossible. The remaining discrepancy inequalities show that the model is klt. ◻

Proof. The ambient variety \(V\) is klt. The reduced floor of a \(\mathbb{Q}\)-factorial dlt pair is \(S_2\), and it has ordinary double crossings in codimension one; its irreducible components are normal. For the \(S_2\) assertion one can work locally with the cyclic class cover of the \(\mathbb{Q}\)-Cartier divisor \(D\). This cover is quasi-étale and klt, hence Cohen–Macaulay. The eigensheaf \(\mathcal{O}_V(-D)\), being a direct summand of its finite direct image, is Cohen–Macaulay. The divisor sequence then shows that \(D\) is Cohen–Macaulay.

Divisorial adjunction on the normalization \(\coprod D_i\to D\) gives lc pairs \((D_i,\operatorname{Diff}_{D_i}(D-D_i))\). The different includes the conductor with coefficient one. Removing that conductor contribution defines an effective boundary on \(D\); together with \(D\) it is an slc pair whose adjoint line bundle is \(M|_D\). The divisible residue identifications hold in codimension one, with even powers eliminating residue signs at double crossings, and extend by \(S_2\).

Each normalized adjoint is nef and semiample by Assumption 73 and Lemma 74. Semiampleness descends from the normalization by the slc gluing theorem (Fujino and Gongyo 2014a, Theorem 1.5, equivalently Theorem 4.3). This gives the required statement on the whole reduced scheme, not merely on its individual components. ◻

Proof. In an infinite ample-scaling program the scaling limit is zero: a positive limit is covered by the termination theorem for a dlt pair with an ample summand in the scaling divisor (Birkar 2012, Theorem 1.9(ii), equivalently Theorem 4.1(ii)). Discard the finitely many divisorial contractions and write the flips as \(V_i\dashrightarrow V_{i+1}/Z_i\), with scaling numbers \(\lambda_i>0\) tending to zero. Fix a component \(S_1\) of \(\lfloor\Delta_1\rfloor\), write \(S_i\) for its normal strict transform, and let \(T_i\) be the normalization of its image in \(Z_i\). Since the ambient contraction is small, \(S_i\to T_i\) is projective birational. The standard discrepancy argument makes \(S_i\dashrightarrow S_{i+1}\) an isomorphism in codimension one after finitely many steps (Birkar 2010, proof of Lemma 3.6).

Here are the precise auxiliary relative programs in that argument. Take a small projective \(\mathbb{Q}\)-factorialization \(p_i:S'_i\to S_i\), and put \[D_i=K_{S'_i}+B_i=p_i^*((K_{V_i}+\Delta_i)|_{S_i}), \qquad C_i=p_i^*(A_i|_{S_i}).\] Adjunction gives a dlt pair \((S'_i,B_i)\), with \(B_i,C_i\geq0\), and \((S'_i,B_i+\lambda_i C_i)\) is lc. The scaling divisor has no floor component, so its restriction is effective. The number \(\lambda_i\) is rational, being the ratio of intersections of rational divisors on the contracted rational curve. The globally nef divisor \(L_i=K_{V_i}+\Delta_i+\lambda_i A_i\) descends rationally linearly across \(V_i\to Z_i\). Indeed, for sufficiently small rational \(\delta>0\), the pair \((V_i,(1-\delta)\Delta_i)\) is klt and its negative adjoint is ample over \(Z_i\). Relative basepoint freeness applies to a Cartier multiple of \(L_i\); its numerical triviality and the connected fibers give descent. Restricting and pulling back yields \[D_i+\lambda_i C_i\sim_{\mathbb{Q}}0/T_i.\]

By (Birkar 2012, Theorem 1.1(3)), a \(D_i\)-LMMP over \(T_i\) with scaling of a fresh ample divisor terminates. The theorem applies to the effective rational lc boundary \(B_i+\lambda_i C_i\), with \(\lambda_i C_i\) rational Cartier and the driving pair \(\mathbb{Q}\)-factorial dlt. Since the base map is birational, the endpoint is a log minimal model, not a Mori fiber space. Comparison with the relatively ample model \(S_{i+1}\) identifies this endpoint with a small \(\mathbb{Q}\)-factorialization \(S'_{i+1}\), as in (Birkar 2012, Remarks 2.9–2.10).

These finite relative programs are genuine pieces of an absolute program with scaling of the transforms of \(C_1\). Throughout the \(i\)-th piece, \(D_i+\lambda_i C_i\) remains the pullback of a nef divisor on \(T_i\), hence is globally nef. Every contracted negative ray has positive \(C_i\)-degree, so its global scaling threshold is exactly \(\lambda_i\). The relative cone is a face of the absolute cone, cut out by the pullback of an ample divisor on the projective base \(T_i\); its extremal rays are therefore absolute extremal rays. Thus the pieces concatenate as asserted in the cited remarks. Rescaling the initial scaling divisor by \(\lambda_1\), if necessary, makes its sum with the initial boundary lc and nef.

If the concatenation were infinite, its positive scaling numbers would tend to zero. Its initial adjoint is pseudo-effective: for each late scaling value, pull back the corresponding nef scaled adjoint and use the effective comparison divisor for the preceding nonpositive steps, then take the limit. Assumption 73 therefore gives an absolute log minimal model for that initial pair. The concatenation terminates by (Birkar 2012, Theorem 1.9(iii)), since its zero limit is never attained. The floor-component argument of (Birkar 2010, Lemma 3.6) now gives special termination. No effective representative in dimension \(n\), nor any arbitrary relative good-model assertion, has been used.

For the final assertion, a curve disjoint from the round-down has degree zero against any signed divisor supported there. It therefore cannot generate an adjoint-negative flipping ray. Special termination rules out all sufficiently late flips. There are only finitely many divisorial contractions, since each lowers the Picard number, and pseudo-effectivity excludes a Mori fiber space as the endpoint. Thus the endpoint is nef. ◻

Proof. Both \(\Delta-\epsilon P\) and \(\Delta-\tfrac12P\) are effective klt boundaries. If \(R\) is a ray negative for \(L-\epsilon P\), nefness of \(L\) shows that \(R\) is also negative for \(L-\tfrac12P\). Choose a rational curve \(C\) spanning this ray and satisfying the length bound \[-\bigl(L-\tfrac12P\bigr)\cdot C\leq 2n.\] Writing \(a=L\cdot C\geq 0\) and \(p=P\cdot C\), we have \[p\leq 2a+4n,\qquad a<\epsilon p, \qquad (1-2\epsilon)a<4n\epsilon.\] Our choice of \(\epsilon\) gives \(a<1/m\). Since \(ma\) is a nonnegative integer, \(a=0\).

For the driving klt contraction, the line bundle \(\mathcal{O}_Z(mL)\) is numerically trivial over the contraction base. The line-bundle clause of the contraction theorem gives its descent as a line bundle, with no change of \(m\) (Fujino 2009, Theorem 3.74(3)). One may also see the unchanged index directly from relative basepoint freeness: two consecutive sufficiently large powers descend, so their quotient descends. For a flip, pull this descended line bundle back on the other side. The transformed \(L\) is nef, and \(mL\) remains Cartier.

The driving boundary remains klt throughout its LMMP. The transformed \(\Delta-\tfrac12P\) remains effective and is smaller than the driving boundary, so it too is klt. The same estimate and the same integer \(m\) apply inductively at every subsequent step. ◻

Proof. Hashizume’s reduction gives nonvanishing and log minimal models for projective lc pairs with real boundary in dimensions at most \(n\) (Hashizume 2018, Theorem 1.4). Thus we may pass to a \(\mathbb{Q}\)-factorial dlt log minimal model. Fix the support of its boundary, impose the rational affine constraints that its coefficient-one components remain equal to one, and take a sufficiently small rational polytope around the given boundary within those constraints. On a fixed log resolution witnessing dlt, the strict discrepancy inequalities remain strict in this neighborhood; thus its boundaries remain dlt. The rational-polytope theorem for nef adjoints (Birkar 2011, Remark 3.1 and Proposition 3.2(3)), applied to all extremal rays and intersected with this neighborhood, expresses the given real boundary in a finite rational simplex of dlt boundaries with nef adjoints. It is enough to prove semiampleness for these rational adjoints. Their positive real combinations are semiample, using the product of the associated contractions. For a rational adjoint, real semiampleness can also be rationalized: the finite linear equations expressing its divisor and principal-divisor coefficients have rational data, so a real solution with positive coefficients has a rational solution with the same positivity.

We therefore work with a rational dlt pair \((Z,\Delta)\) and \(L=K_Z+\Delta\) nef. Real nonvanishing gives rational nonvanishing in this case: retain the finitely many divisors and principal divisors in an effective real representative and solve the resulting rational linear system with nonnegative rational coefficients. If \(\kappa(Z,L)\geq 1\), lower-dimensional good models give a good minimal model by (Fujino and Gongyo 2017, Lemma 3.5). Lemma 74 transfers semiampleness back to \(Z\). It remains to treat \[\kappa(Z,L)=0.\]

Proof. The assertion for \(v=n\) is the numerical criterion for bigness of a nef divisor. Suppose \(v<n\). Intersecting the pseudo-effective class \(L-cD\) with a product of nef classes gives \[0\le (L-cD)L^vH^{n-v-1} =-c\sum_iD_iL^vH^{n-v-1}.\] Every summand on the right is nonnegative. Thus each is zero. For \(v=0\), ampleness gives \(D=0\), and the signed representative gives \(L\sim_{\mathbb{Q}}0\).

Now suppose \(v>0\). On a component of \(D\), the numerical dimension of \(L|_D\) equals its image dimension under \(\phi\). The preceding vanishing gives the upper bound \(r\). On the other hand, \[0<L^vH^{n-v}=\sum_i a_iD_iL^rH^{n-v},\] so a positive-coefficient component attains that bound.

Consider the symmetric matrix \[Q_{ij}=D_iD_jL^rH^{n-r-2}.\] Its off-diagonal entries are nonnegative: distinct effective \(\mathbb{Q}\)-Cartier divisors have effective intersection cycles. The mixed Hodge index theorem, obtained by approximation with ample classes and the ordinary Hodge index theorem on complete-intersection surfaces, says that the ambient intersection form has at most one positive direction. In this form, \(L\) is isotropic, is orthogonal to every \(D_i\), and has strictly positive pairing with \(H\). Its orthogonal space therefore has negative semidefinite form. The same applies after pullback to a resolution, so this argument does not require \(X\) to be smooth.

The signed relation implies \(Q(a_i)=0\). Write \(a=a^+-a^-\) temporarily for the positive and negative coefficient vectors. Then \[Q(a^+,a^+)=Q(a^+,a^-)\ge0.\] Negative semidefiniteness makes both sides zero and gives \(Qa^+=0\). The vanishing rows belonging to negative or zero coefficients show that each intersection of a positive component with a negative or zero component has zero \(L^r\)-degree against \(H^{n-r-2}\). Semiampleness on \(D\) gives image dimension less than \(r\). If \(r=0\), an effective nonzero intersection has positive ample degree, proving emptiness. ◻

Proof. Start with the normalized simultaneous root stack of the Cartier divisors \(mG_+\), \(mG_-\), and \(mD_0\), taking an \(\ell\)-th root of each divisor with its section. Empty divisors cause no difficulty. Write \(S_+,S_-,S_0\) for the tautological root divisors. To check their reducedness, work at a generic prime where the downstairs multiplicity is \(b\). A normalized chart of \(y^\ell=x^b\) has ramification index \(\ell/b\), and \(y\) has order one. This applies to \(D_0\) because \(m\) divides \(\ell\). The divisors are Cartier; normality and their generic reducedness imply that they are reduced.

Log ramification gives a log canonical pair with this full reduced boundary and \[K+S_++S_-+S_0\sim_{\mathbb{Q}}a(S_+-S_-).\] Its ambient charts are klt. Indeed they are klt off the boundary, and decreasing all boundary coefficients slightly removes all zero-discrepancy places, as can be checked on a log resolution. The integral Weil class given by the difference of the two sides is torsion. Take the finite representable cover formed from its reflexive powers, with a chosen periodicity isomorphism, and normalize. On codimension-one charts this is the usual cover of a torsion line bundle, hence is etale there. The adjoint relation becomes a fixed linear equivalence; log canonicity, klt ambient singularities, and reduced Cartier boundary divisors persist. More explicitly, before taking this cover write \(B=S_++S_-+S_0\). The torsion reflexive sheaf is \[\mathcal F= \bigl(\omega(B)\otimes\mathcal{O}(-a(S_+-S_-))\bigr)^{**}.\] The relative spectrum of its reflexive-power algebra, with a chosen periodicity \(\mathcal F^{[q]}\simeq\mathcal{O}\), has a tautological evaluation trivializing the pulled-back \(\mathcal F\) in codimension one. This is a morphism of sheaves on the cover stack itself. The resulting adjoint isomorphism is therefore equivariant on every atlas, rather than a separately chosen nonequivariant trivialization.

Blow up the ideal of \(S_+\cap S_-\) and normalize. Locally the ideal has two generators, so the ordinary blowup embeds in a relative projective line. Its fibers, and those of its finite normalization, have dimension at most one. Every exceptional divisor therefore lies over a codimension-two component of the intersection. Such a component is a dlt stratum downstairs and is generically SNC. Separate normalized Kummer charts, followed by purity for the index cover at the smooth generic locus, give the same description upstairs.

It follows that the tautological exceptional divisor \(E\) is reduced Cartier and \[S'_+=p_1^*S_+-E,\qquad S'_-=p_1^*S_--E\] are disjoint reduced Cartier divisors, where \(p_1\) denotes this normalized blowup. No codimension-two two-branch stratum is contained in \(S_0\), so \(p_1^*S_0\) is reduced Cartier without an exceptional component. The boundary \[S'_++S'_-+E+p_1^*S_0\] is crepant and log canonical. Write \(B'\) for this total boundary on a chart \(V'\). The crepant equality establishes log canonicity for every valuation. Outside \(B'\), the blowup is an isomorphism to the klt complement of the old boundary. Thus a divisor \(F\) with \(a(F;V',B')=0\) has center in \(\mathop{\mathrm{Supp}}B'\). As \(B'\) is effective Cartier, \(\mathop{\mathrm{ord}}_F(B')>0\), and \[a(F;V',0)=a(F;V',B')+\mathop{\mathrm{ord}}_F(B')>0.\] Divisors with positive pair discrepancy remain positive after dropping the boundary. This proves that \(V'\) is klt, including at higher-codimension singular loci. Set \(S=S'_+\) and \(T=S'_-+E+p_1^*S_0\) for the moment. Near \(S\), the disjoint divisor \(S'_-\) contributes its unit section, and the fixed linear equivalence becomes \[\omega(S+T)\simeq\mathcal{O}(aS).\] The ambient charts remain klt and hence Cohen–Macaulay. Since all displayed boundary divisors are Cartier, this isomorphism also makes those charts Gorenstein.

The strict divisor \(S\) is finite over \(S_+\). Before normalization it lies in the zero section of the projective ratio coordinate: its fibers over \(S_+\) have at most one point. The morphism is proper, and normalization is finite. In particular a codimension-one point of \(S\cap T\) lies over a codimension-two intersection downstairs. The generic SNC description above shows that \(T|_S\) is generically reduced. It is Cartier on the Cohen–Macaulay scheme \(S\), so it is reduced everywhere.

We next remove the root characters that will not be used. The line \(\mathcal{O}(S-S'_-)\) has \(\ell\)-th power equal to the pullback of \(N_0\). It defines a morphism to the gerbe of \(\ell\)-th roots of \(N_0\) on \(X\). Take the relative coarse space over this gerbe and call it \(\mathcal X\). On a trivializing chart this means quotienting by the kernel of the finite group character acting on the indicated root line. These are ordinary quasi-projective quotient schemes: the preceding covers are finite, the blowup is projective, and a finite quotient preserves quasi-projectivity. The chartwise construction patches.

Retain \(S,T\) for their reduced images. Near \(S\), the pole section is a unit, so both the line of \(S\) and its section descend through this kernel quotient. Thus \(S\) remains Cartier. A finite-group norm of a local equation for the upstairs \(T\) shows that its image is set-theoretically principal; we do not need reduced \(T\) to be Cartier. The only divisorial ramification is on the boundary. Log ramification therefore preserves the lc pair and descends the equivariant adjoint isomorphism to [proj:sg:ambient-adjunction]. Cohen–Macaulayness of the ambient chart and of \(S\) follows from the finite CM covers and their invariant direct summands. For clarity, the coarse kernel acts trivially on the retained root line, and hence on \(\mathcal{O}(aS)\). Equivariance of the fixed adjoint isomorphism gives the same kernel character on \(\omega(S+T)\). Taking invariants therefore descends the isomorphism, with log ramification identifying the invariant log dualizing sheaf. Off \(T\) the descended adjoint isomorphism makes the canonical sheaf invertible, giving the claimed Gorenstein property there.

The reduced supports \(S\) and \(J\) are unions of log canonical centers: intersections of lc centers are again unions of lc centers. Hence both are Du Bois by (Kollár and Kovács 2010, Theorems 1.4 and 1.7). For the more precise coherent statement, on one quotient chart write \(S^{\mathrm{up}}\to S\) for the finite cover used above. Since \(T^{\mathrm{up}}|_{S^{\mathrm{up}}}\) is reduced, \[\mathcal I_{J/S} = \bigl(f_*\mathcal{O}_{S^{\mathrm{up}}} (-T^{\mathrm{up}}|_{S^{\mathrm{up}}})\bigr)^{\mathrm{inv}}.\] Indeed an invariant function vanishes on the reduced quotient image precisely when its pullback vanishes on this reduced preimage. The sheaf on the right is maximal Cohen–Macaulay: upstairs it is invertible on a CM scheme, finite pushforward has the same depth over the quotient, and invariants are a direct summand in characteristic zero.

Finite-map duality with trace, followed by invariants, gives \[\mathcal Hom_S(\mathcal I_{J/S},\omega_S) = \bigl(f_*\omega_{S^{\mathrm{up}}} (T^{\mathrm{up}}|_{S^{\mathrm{up}}})\bigr)^{\mathrm{inv}} \simeq\mathcal{O}_S(aS).\] This use of duality allows ramification and does not assert that \(\omega_S\) itself is invertible. The last isomorphism is upstairs Cartier adjunction followed by descent of the \(S\) line. Off \(J\) it is the ordinary residue determined by the fixed ambient isomorphism. Normalized trace preserves that convention, proving compatibility on overlaps.

Restricted to \(S\), the root gerbe of \(N_0\) is \(D\times_P\mathcal P\). The strict-divisor finiteness already proved, together with removal of the relative inertia kernel, makes \(S\to D\times_P\mathcal P\) finite and representable. Its image covers the positive boundary, and its root line is \(\mathcal{O}_S(S)=C\). Lemma 80 gives all the asserted image-dimension bounds, including the bound for \(J\). The resulting \(h:S\to\mathcal P\) is representable projective.

Its Stein algebra defines a finite stack \(\mathcal T=\mathop{\mathrm{Spec}}_{\mathcal P}h_*\mathcal{O}_S\) over \(\mathcal P\). There are enough vector bundles on \(\mathcal P\) to generate this coherent algebra as a module. Namely, decompose by the finitely many inertia characters, twist each summand by a power of \(C\) to descend it to \(P\), and apply Serre generation. A vector-bundle surjection gives an embedding of \(\mathcal T\) in a vector bundle over \(\mathcal P\). Its closure in the projective completion is still \(\mathcal T\), since it is proper over the base. This gives \(\mathcal Y\) and \(g\). The Stein construction supplies \(g_*\mathcal{O}_S=\mathcal{O}_{\mathcal T}\) and the required line identities.

Finally, etale morphisms extend uniquely over nilpotent thickenings, compatibly as \(q\) varies. The reduction \(S_U\) is a scheme. The extension \(qS_U\) has trivial inertia and is an algebraic space. It is separated over \(X\): the chosen chart is separated, the morphism to the root gerbe is finite, and that gerbe has finite diagonal. Its finite-type geometric fibers over \(X\) are zero-dimensional, and this is unchanged by nilpotents. Thus \(qS_U\to X\) is separated and quasi-finite. Zariski’s main theorem embeds it in a scheme finite over \(X\), proving that it is a quasi-projective scheme. This completes the construction. ◻

Proof. Because \(p\) is finite on the support, its direct image has only perverse cohomology in degree zero. Denote that system on \(\mathcal P\) by \(M'\). Projective strictness on the ordinary scheme charts gives \[ Rp_*\mathop{\mathrm{Gr}}^F_k\mathop{\mathrm{DR}}_{\mathcal Y}(M) \simeq \mathop{\mathrm{Gr}}^F_k\mathop{\mathrm{DR}}_{\mathcal P}(M'). \tag{184}\] We first explain why this is a global identity of descended complexes, rather than a claim of local vanishing.

For right differential modules, the direct image is formed from the filtered transfer bimodule \[\mathcal D_{\mathcal Y\to\mathcal P} = \mathcal{O}_{\mathcal Y}\otimes_{p^{-1}\mathcal{O}_{\mathcal P}} p^{-1}\mathcal D_{\mathcal P},\] with its order filtration, by derived tensor over \(\mathcal D_{\mathcal Y}\) and derived direct image. These operations may equivalently be performed on Rees modules. Applying de Rham on the base means further derived tensor with \(\mathcal{O}_{\mathcal P}\), whose filtration begins in degree zero. The filtered Spencer resolution, associativity of derived tensor, and projection formula identify the result with the direct image of de Rham upstairs: the transfer module tensored over the base differential operators with its structure sheaf is \(\mathcal{O}_{\mathcal Y}\). These are canonical sheaf constructions on the etale site; the Spencer resolutions are locally free over differential operators. They therefore respect chart changes before passing to associated gradeds. Strictness and the concentration in perverse degree zero can be checked on the base charts by the ordinary projective direct-image theorem. Coherent cohomology on those charts is unchanged by using their etale sites. This proves Equation (184) as an identity that computes global coherent hypercohomology. The same argument applies to the representable finite morphism used next.

There is a representable finite surjective morphism \[v:\mathbb{P}^s_z\longrightarrow\mathcal P\] defined by the power map \([z_0:\cdots:z_s]\mapsto[z_0^\ell:\cdots:z_s^\ell]\) and the root \(\mathcal{O}_{\mathbb{P}^s_z}(1)\). In particular, \(v^*C=\mathcal{O}_{\mathbb{P}^s_z}(1)\). On the standard affine opens of \(\mathbb{P}^s\) choose the trivial root, obtaining scheme etale charts \(\check U_i\to\mathcal P\). On \(z_i\ne0\), the morphism \(v\) factors through \(\check U_i\). Pull back \(M'\) on these charts and take perverse cohomology in degree zero. Functoriality and the chosen root identify these objects on overlaps, giving a system \(H\) on the Zariski opens of \(\mathbb{P}^s_z\).

For completeness, this is a global graded-polarizable mixed Hodge module on \(\mathbb{P}^s_z\). The perverse, filtered, and weight data glue. The pure weight gradeds have strict-support decompositions, which glue by uniqueness. On a smooth connected dense stratum of an irreducible support, a polarization from a nonempty Zariski open extends as a flat pairing: the fundamental group of that open surjects onto the fundamental group of the stratum. Its Hodge compatibility extends by continuity; nondegeneracy and positivity persist for the extended flat pairing. Quasi-unipotence at the boundary is already supplied by the local Hodge modules. Saito’s extension theorem for polarizable variations (Saito 1990, sec. 3.b) and uniqueness of strict-support extension identify this global pure object with the glued one. There is no additional boundary at infinity, since \(\mathbb{P}^s_z\) is projective. The weight extensions then give the stated mixed object.

The system \(M'\) is a retract of \(v_+H\). To see this on a chart, use the whole base-changed finite cover. Over \(\check U_i\) it is a disjoint union of copies of the corresponding affine coordinate chart of \(\mathbb{P}^s_z\). The unit on unshifted constants and its dual trace, using smooth duality in equal dimensions, have composition multiplication by the degree. On the finite etale locus the trace is summation over sheets. On each connected smooth base chart, the shifted constant Hodge module is the rank-one intersection complex and has endomorphism ring \(\mathbb{Q}\); its endomorphisms are determined on a dense open. Thus the composition equals the degree globally, with no additional term supported on the branch locus. Ramification is therefore allowed. Tensoring with \(M'\), applying proper projection formula, and taking perverse cohomology in degree zero gives the claimed retraction, since finite direct image is perverse exact. Divide the trace by the degree. All maps are canonical under etale base change, so the retraction descends also on filtered differential modules.

Now apply the finite version of Equation (184) and projection formula. The desired hypercohomology is a direct summand of \[\mathbb H^j\!\left( \mathbb{P}^s_z,\mathop{\mathrm{Gr}}^F_k\mathop{\mathrm{DR}}(H)\otimes\mathcal{O}_{\mathbb{P}^s_z}(-b) \right).\] This is zero for \(j<0\) by ordinary negative-ample Kodaira–Saito vanishing. One may apply it to the pure weight gradeds and then use the exact sequences of the weight filtration; the Hodge filtrations of those sequences are strict. ◻

Proof. The difference triangle for the constant objects of \(S_U\) and \(J_U\), together with Du Bois comparison, identifies its degree-zero graded de Rham complex with \(\mathcal I_{J_U/S_U}\). Coherent duality and the maximal-Cohen–Macaulay property therefore give \[ \mathop{\mathrm{Gr}}^F_k\mathop{\mathrm{DR}}(A^\bullet_U)=0\quad(k<0), \qquad \mathop{\mathrm{Gr}}^F_0\mathop{\mathrm{DR}}(A^\bullet_U)=\mathcal B|_{S_U}[0]. \tag{185}\] The shift \([N]\) cancels the dimension shift in the dualizing complex. In the smooth case this convention says \(\mathbf D(\mathbb{Q}^H[N])=\mathbb{Q}^H[N](N)\); its right differential module has lowest piece \(\omega\) at index zero.

We spell out the passage from the complex to perverse cohomology. Let \(q\) be the smallest filtration index occurring in any perverse cohomology object of \(A^\bullet_U\), locally on a fixed chart. Such a lower bound exists because the complex is bounded and its filtrations are good. At index \(q\), the graded Spencer complex of every perverse cohomology object has only its degree-zero term. The spectral sequence from perverse truncation therefore has a single nonzero row and gives \[\mathcal H^i\mathop{\mathrm{Gr}}^F_q\mathop{\mathrm{DR}}(A^\bullet_U) =F_q\,{}^p\!H^iA^\bullet_U.\] If \(q<0\), this contradicts Equation (185). Thus all these objects have \(F_{<0}=0\). Applying the same argument at index zero gives \(F_0A_{0,U}=\mathcal B|_{S_U}\), and gives zero for the lowest piece of the other perverse cohomology objects. No perversity assertion about the shifted constant complex is required.

For the unfiltered description, duality identifies the dual of the reduced constant complex, in the ambient \(V\), with \[R\Gamma_{[S_U]}(\omega_V)[c].\] Its degree-zero module is the first nonzero support cohomology \(\mathcal H^c_{S_U}(\omega_V)\). The support of \(J_U\) is locally set-theoretically principal in \(S_U\), by the construction. Invert a local defining function, lifted to \(V\); this exact meromorphic localization realizes \(j_*\) on the underlying module. It yields the asserted description of \(A_{0,U}\).

We also need compatibility of the two descriptions of its lowest piece. On the smooth locus of \(S_U\setminus J_U\), filtered duality and smooth graph pushforward identify it with the ordinary dualizing sheaf included by residue. This is the Ext-to-support inclusion, with the normalization fixed by the ambient adjunction isomorphism. Naturality in smooth etale coordinates fixes the same scalar on every chart. Agreement extends over \(S_U\setminus J_U\): the first support-cohomology module has no sections supported on a smaller-dimensional subset. For example, this follows from the Cousin resolution of the smooth dualizing sheaf, whose first term supported on \(S_U\) is a sum of injective modules at its codimension-\(c\) generic points. It extends across \(J_U\) by localization. Hence the inclusion is the one stated in the lemma, not merely an abstract isomorphism of coherent sheaves.

Finally use a graph embedding followed by a smooth projection to compute \(g_+\). At filtration zero the relative Spencer complex has only its top term, namely the direct image of \(\mathcal B|_{S_U}\) on the graph. Projective strictness gives \[F_{<0}M_U=0,\qquad F_0M_U=R^1g_*(\mathcal B|_{S_U}),\] and identifies this sheaf with its actual submodule in the unfiltered degree-one direct image. The constructions used here are canonical on etale changes of charts. ◻

Proof. Define \[E_{j,k}=g_*(I^j/I^{j+k}),\qquad j\geq0,\quad k\geq1.\] Here \(g_*\) uses the underlying topological map. We do not assume that a thickening already has a ringed-space map to \(U\). The first graded layer is \[E_{j,1}=\mathcal{O}_{\mathcal T_U}\otimes C^{-j}.\] The exact sequence for adjacent lengths has kernel \(g_*\mathcal{O}_{S_U}\otimes C^{-j-k}\), and its connecting homomorphism takes values in \[R^1g_*\mathcal{O}_{S_U}\otimes C^{-j-k}.\]

Induct on \(k\), proving the assertion simultaneously for all \(j\). Assume all shorter transitions are surjective. A section of \(E_{j,k}\) with zero length-one term comes from \(E_{j+1,k-1}\) when \(k>1\). By induction it locally lifts from \(E_{j+1,k}\), so its connecting class is zero. Also \(E_{j,k}\to E_{j,1}\) is surjective. For \(k=1\) the factorization is immediate. The obstruction therefore factors through a map \(D_k\) on the graded algebra of length-one layers, with graded shift \(k\). Choose local lifts and take their Cech differences. The difference of a product is the sum of the two first-order differences; products of the errors vanish in the layer at issue. Thus \(D_k\) is a \(\mathbb{C}\)-derivation from this graded algebra to the graded \(R^1g_*\mathcal{O}_{S_U}\)-module.

In degree zero, restrict it along \(\mathcal{O}_U\to\mathcal{O}_{\mathcal T_U}\). The resulting derivation factors through \(\Omega_U^1\), hence defines a section of \(T_U\otimes R^1g_*\mathcal{O}_{S_U}\otimes C^{-k}\). These local sections descend to \[ e\in H^0\!\left( \mathcal Y,T_{\mathcal Y}\otimes R^1g_*\mathcal{O}_S\otimes C^{-k} \right) = H^0\!\left( \mathcal Y,T_{\mathcal Y}\otimes F_0M\otimes C^{-(a+k)} \right). \tag{186}\] The equality uses Equation (183), Lemma 83, and projection formula. For descent, lifts of functions pull back on the unique etale thickenings. Their Cech classes pull back by ordinary flat base change for the coherent obstruction sheaves on the reductions. Differentials pull back and generate under an etale map, proving the asserted compatibility.

Shrink \(U\) and take etale coordinates \(t_1,\ldots,t_d\), where \(d=\dim U\), and a frame of \(C^{-1}\). Its pullback is a conormal frame, denoted by \(y\). By induction lift the coordinates along \(g\) modulo \(I^k\), and lift \(y\) to \(I/I^{k+1}\). On an open cover of \(S_U\), lift these one step further, obtaining functions \(h_{i,\lambda}\) modulo \(I^{k+1}\) and generators \(y_i\) modulo \(I^{k+2}\). Formal etaleness of the coordinate map makes each \(h_i\) a local map from \((k+1)S_U\) to \(U\). On overlaps write \[ h_{j,\lambda}-h_{i,\lambda} =e_{ij,\lambda}y_i^k, \qquad y_j-y_i=\eta_{ij}y_i^{k+1}. \tag{187}\] The reduced coefficients are Cech cocycles representing \(D_k(t_\lambda)\) and \(D_k(y)\) in the chosen frames. Let \(\sigma=y^{-a}\) denote the induced frame of \(\mathcal B|_{S_U}\).

Off \(J_U\), the fixed adjunction isomorphism gives \[\omega_{(k+1)S_U} \simeq\mathcal{O}_{\mathcal X}((a+k)S)|_{(k+1)S_U}.\] It therefore defines local dualizing sections \(b_i=y_i^{-(a+k)}\). Under the trace inclusion \(\omega_{S_U}\hookrightarrow\omega_{(k+1)S_U}\), the identities \[ b_j-b_i=-(a+k)\eta_{ij}\sigma,\qquad y_i^kb_i=\sigma,\qquad y_i^{k+1}b_i=0 \tag{188}\] hold when the dualizing sheaves are realized in support cohomology. Indeed the first identity is the binomial expansion of \((1+\eta_{ij}y_i^k)^{-(a+k)}\); its quadratic terms vanish because \(2k\geq k+1\). The other two identities describe Cartier trace and the annihilator of the thickening.

Embed \((k+1)S_U\) as a closed subscheme of a smooth open \(Z\subset\overline Z=\mathbb{P}^m\). The graph of the reduced map \(g\) is closed in \(\overline Z\times U\), by its properness over \(U\). Use this reduced graph for an ambient realization of \(A_{0,U}\). Each local graph lift \(h_i\) maps \(b_i\), by Ext-to-support cohomology, to a section \(\delta_i(b_i)\) of the underlying graph module. We claim that \[ \delta_j(b_j)-\delta_i(b_i) = -(a+k)\eta_{ij}\sigma +\sum_{\lambda=1}^d (e_{ij,\lambda}\sigma)\cdot\partial_{t_\lambda}. \tag{189}\] The reduced dualizing sections on the right use the inclusion of Lemma 83.

Here is the local calculation, including the non-smooth case. Put \(c_0=\dim Z-N\). The Ext-to-support map identifies the dualizing sheaf of the thickening with its annihilator submodule in \(\mathcal H^{c_0}_{S_U}(\omega_Z)\). One can obtain this from the support-cohomology spectral sequence: there is no support cohomology below \(c_0\), so in total degree \(c_0\) the Ext group is the homomorphisms from the thickening’s structure sheaf into that first support-cohomology module. Thus the calculation does not require \(S_U\) to be a local complete intersection in \(Z\).

Lift \(h_{i,\lambda}\) locally to functions on \(Z\). In \(Z\times U\), the graph inclusion on a thickness-dualizing section \(b\) is the generalized fraction \[\frac{b\,dt_1\wedge\cdots\wedge dt_d} {\prod_{\lambda=1}^d(t_\lambda-h_{i,\lambda})}.\] This is successive support cohomology in the additional coordinate equations, or equivalently the Koszul residue for the graph. It may be computed etale-locally near the graph branch. After first taking support cohomology from \(Z\), the translated new coordinates form a regular sequence, so their support cohomology is concentrated in their top number. Localization in either set of translated coordinates gives the same localization on this support-torsion module: the two translations differ nilpotently on every section.

Changing from \(h_i\) to \(h_j\) therefore gives a finite Taylor expansion on \(b_j\). Products of two differences in Equation (187) annihilate \(b_j\), again because \(2k\geq k+1\). Its linear term uses \[(h_{j,\lambda}-h_{i,\lambda})b_j =e_{ij,\lambda}\sigma.\] For a top differential form the right action satisfies \[\left(\frac{dt_\lambda}{t_\lambda-h}\right) \cdot\partial_{t_\lambda} =\frac{dt_\lambda}{(t_\lambda-h)^2}.\] Consequently the doubled pole has the positive sign displayed in Equation (189). Together with Equation (188), this proves that identity. The calculation initially takes place off \(J_U\). The sections and identities extend meromorphically across \(J_U\) in the unfiltered localized module; arbitrary finite pole orders there are allowed.

Now push by \(\overline Z\times U\to U\), using relative Spencer. If \(i\) denotes the closed reduced graph embedding and \(\pi\) the projection, closed direct image is perverse exact and \(\pi_+i_+A_{0,U}=g_+A_{0,U}\). Thus degree-one holonomic differential-module cohomology of this Spencer direct image is precisely the underlying module of \(M_U={}^{p}\!H^1g_*A_{0,U}\). The cochain \((\delta_i(b_i))\) lies in its top, degree-zero term, which has no outgoing relative Spencer differential. Its Cech difference is therefore a boundary in total degree one. A sufficiently refined ambient cover around the closed graph computes this assertion; all sheaves involved are supported there. The reduced cocycles in Equation (189) represent their \(R^1g_*\mathcal B\) classes inside \(F_0M\), by Lemma 83. Right differentiation in the \(U\)-coordinates acts on the pushforward complex. Hence the right side of Equation (189) represents zero in the unfiltered module \(M\).

That right side lies in \(F_1M\); the term involving \(\eta\) lies in \(F_0M\). Since \(F_1M\) is an actual subsheaf of \(M\), its vanishing in \(M\) is vanishing in \(F_1M\). Taking the order-one symbol thus says that the global section \(e\) of Equation (186) is killed by \[ T_{\mathcal Y}\otimes F_0M \longrightarrow \mathop{\mathrm{Gr}}^F_1M, \tag{190}\] after twisting by \(C^{-(a+k)}\). This reasoning places no Hodge-filtration bound on the auxiliary cochain \((b_i)\).

There is no term preceding degree \(-1\) in \(\mathop{\mathrm{Gr}}^F_1\mathop{\mathrm{DR}}(M)\), because \(F_{<0}M=0\). Its degree-\(-1\) hypercohomology after the indicated twist is therefore precisely the space of global sections of the kernel of Equation (190). Since \(a+k>0\), Lemma 82 makes that space zero. We conclude that \(e=0\).

In a local coordinate and line trivialization, this says that every class \([e_{ij,\lambda}\sigma]\) is already zero in \(F_0M\), hence in \(M\). Its actual differential-operator image is consequently zero, before taking symbols. The unfiltered relation now gives \[(a+k)[\eta_{ij}\sigma]=0.\] Lowest-piece injectivity and \(a+k>0\) imply \(D_k(y)=0\). The degree-zero derivation is zero as well: it vanishes on \(\mathcal{O}_U\), which surjects onto \(\mathcal{O}_{\mathcal T_U}\). Locally the whole graded length-one algebra is generated by that coefficient algebra and the conormal frame \(y\). Thus \(D_k=0\) in every degree, proving the induction and all sheaf transitions.

Their kernels are the coherent sheaves \(\mathcal{O}_{\mathcal T_U}\otimes C^{-j-k}\). On an affine \(U\) these have no first cohomology. The exact sequences of sheaves of abelian groups therefore also give surjectivity on global sections. Choosing lifts successively gives compatible formal lifts of any chosen functions on \(\mathcal T_U\), and, when \(C\) is trivialized, of its transverse conormal frame. ◻

Proof. If \(B\) is a point, a vertical divisor is zero. If the relative dimension is zero, the fiber bound makes \(f\) finite; its geometrically integral connected generic fiber makes it birational, and normality of \(B\) makes it an isomorphism. Both cases are immediate. Assume henceforth that the base and the relative dimension are positive.

The fiber bound implies that every vertical prime divisor maps onto a prime divisor of \(B\). For each such base prime \(P\), write the full pullback as \(f^*P=\sum_jm_jE_j\) and let \(b_j\) be the coefficient of \(G_Y\) along \(E_j\), including zero coefficients. Give \(P\) coefficient \(\min_j(b_j/m_j)\) in \(G_B\). Only finitely many coefficients are nonzero. Then \[R'=G_Y-f^*G_B\] is effective, vertical, \(\mathbb{Q}\)-Cartier, relatively nef, and misses at least one component over each base prime. We prove that \(R'=0\).

If \(R'\ne0\), choose a base prime below its support. Take a general complete-intersection curve in \(B\) meeting that prime at a general point; when \(\dim B=1\) use \(B\) itself. Its inverse image in \(Y\) is normal by normal Bertini. It is integral: generic geometric integrality gives the dominating component, and the fiber bound excludes additional vertical components. Next take \(\dim Y-\dim B-1\) general very ample hyperplanes upstairs. This produces a normal integral surface over the base curve with geometrically connected generic fiber. At the chosen general base point, the cuts retain curves in the residual support and in a missing component of the full pullback; these curves are distinct because the original components were distinct at the generic point of the base prime.

Resolve the surface. The pulled-back residual divisor is effective, vertical, and relatively nef. Its intersection with the full fiber is zero, whereas its intersection with each fiber component is nonnegative. Every one of the latter intersections is therefore zero. The intersection matrix of a connected surface fiber is negative semidefinite with kernel generated by the full fiber. Thus the residual part at the chosen point is a multiple of that full fiber. A missing component forces this multiple to be zero, contradicting the component in its support. Connectedness of the fiber follows from generic connectedness and Stein factorization over the normal base curve. ◻

Completion of the proof of Theorem 79. Lemma 80 handles \(v=0\) and \(v=n\). In the remaining case, combine Propositions 81, 84, and 85.

Apply the nef-reduction theorem to a Cartier multiple of \(L\) (Bauer et al. 2002, Theorem 2.1). It gives an almost holomorphic rational map with connected fibers, with \(L\) numerically trivial on its compact general fibers, and with positive \(L\)-degree on every noncontracted curve through a very general point of \(X\). Let its base dimension be \(b\). A compact \(L\)-trivial \(d\)-fold through such a point is contracted: otherwise general ample slices through the point yield a noncontracted curve of \(L\)-degree zero. Hence \[b\le n-d=v.\]

Resolve the graph, flatten the main component over a modification of the base, resolve that base, and normalize the main transform. We obtain projective morphisms \[\begin{tikzcd} Y \arrow[r,"\pi"] \arrow[d,"f"'] & X\\ B & \end{tikzcd}\] with \(\pi\) birational, \(B\) smooth, \(Y\) normal, and geometrically connected integral generic fiber of \(f\). All fibers have dimension at most \(n-b\). Indeed the flat main transform remains integral after the base modification by flatness and generic integrality, and finite normalization preserves the fiber-dimension bound. Normalization need not preserve flatness; only this bound is used.

The pullback \(\pi^*D\) has no horizontal component. To see this, restrict to a very general fiber of \(f\). The class of \(\pi^*L\) is numerically trivial there, and \(\pi^*(L-cD)\) restricts to a pseudo-effective class. The latter assertion follows by restricting effective approximants, avoiding the countably many fibers contained in their supports. Its restricted class is \(-c\pi^*D\). A negative nonzero effective divisor on a projective variety cannot be pseudo-effective, as intersection with an ample power shows. Thus the restriction of \(\pi^*D\) is zero. In particular \(b=0\) would force \(D=0\), contrary to \(v>0\).

Consequently \(G_Y=\pi^*G\) is vertical and is relatively nef, since \(G_Y\sim_{\mathbb{Q}}\pi^*L\). Lemma 86 gives \[\pi^*L\sim_{\mathbb{Q}}f^*G_B\] for a \(\mathbb{Q}\)-divisor \(G_B\) on the smooth base. It is nef: every curve of \(B\) is dominated by a curve of \(Y\), on which the pullback has nonnegative degree. The intersection formula for a pullback gives \(\nu(X,L)\le b\). Combined with \(b\le v\), this yields \(b=v\) and \(G_B^b>0\). Therefore \(G_B\) is big. Pulling back sections gives \(\kappa(X,L)\ge b=v\), and the general inequality \(\kappa(X,L)\le\nu(X,L)\) for a nef divisor proves abundance. ◻

Proof. If \(q(X)>0\), resolve the Stein factorization of the Albanese map to obtain a morphism \(f:W\to B\) with connected fibers, with \(W,B\) smooth projective and \(\dim B>0\). The map from \(B\) to a subvariety of \(\mathop{\mathrm{Alb}}(X)\) is generically finite. Wedges of invariant one-forms on the abelian variety therefore give a nonzero section of \(K_B\): at the generic point, choose \(\dim B\) independent pulled-back one-forms. Thus \(\kappa(B,K_B)\geq0\).

For a very general smooth fiber \(F\), the restriction of the pseudo-effective class \(K_W\) is pseudo-effective. For example, restrict a positive current representing it to almost every fiber, or use effective approximations with arbitrarily small ample error. Adjunction identifies this restriction with \(K_F\). If \(F\) has positive dimension, the induction hypothesis gives \(\kappa(F,K_F)\geq0\); for a point the same assertion holds by convention. Applying Assumption 71 with both boundaries empty gives \(\kappa(W,K_W)\geq0\), contradicting (191).

This is the only use of Assumption 71 in the proof. Irregularity is a smooth birational invariant. Finally, when \(\mathop{\mathrm{Pic}}^0=0\), a numerically trivial line bundle is torsion, since the group of numerically trivial line bundles modulo \(\mathop{\mathrm{Pic}}^0\) is finite. Clear denominators for rational divisors. On a terminal \(\mathbb{Q}\)-factorial model, pull back to a smooth resolution and then descend the rational linear equivalence. ◻

Proof. Shrink to an open \(U\) where the evaluation has finitely many etale sheets, the parameter map is smooth, and the generated involutive distribution \(\mathcal F\) has constant rank. Its construction is algebraic: spans and brackets stabilize at the generic point after finitely many operations, and descend from the etale sheets. The nonempty open parts of the integral parameter fibers are connected. We use only chains of fiber-image steps lying in \(U\).

For \(x\in U\), endpoints of chains of at most \(k\) steps form a constructible set, by the algebraic fiber-product construction and Chevalley’s theorem. Every endpoint lies in the analytic leaf through \(x\). A locally closed smooth variety contained in an analytic leaf has dimension at most \(\mathop{\mathrm{rk}}\mathcal F\). Indeed, in a Frobenius chart the leaf meets at most countably many plaques: use finite plaque chains in a countable foliation atlas. The transverse coordinate map on any connected smooth piece then has countable image and is constant.

There is therefore a maximal dimension among the closures of all such endpoint loci; it is attained for some finite \(k\). The loci are nested because we allow at most \(k\) steps, including the identity step. Choose an irreducible component \(V\) of maximal dimension and a dense open \(O\subset V\) of reachable points. For each etale sheet, restrict its fiber-equivalence relation to first coordinate in \(O\), and take the component through the diagonal section. Its second-image closure contains \(O\), hence \(V\), while its second image consists of endpoints reachable in at most \(k+1\) steps. Maximality forces that closure to equal \(V\). At a general diagonal point, smoothness of the parameter map identifies its vertical tangent with the corresponding web tangent. Every web tangent is consequently tangent to \(V\), and so is every bracket. Hence \(\dim V\geq\mathop{\mathrm{rk}}\mathcal F\), while the reverse inequality was proved above. A plaque is thus open in \(V\). The equations of \(V\), pulled back to the connected immersed leaf, vanish on a nonempty open and hence on the entire leaf. Its closure is exactly \(V\).

These rank-dimensional invariant subvarieties have countably many Hilbert or Chow parameter spaces. Tangency on the regular locus is an algebraic condition after stratification, so one such family dominates. An invariant irreducible variety meeting \(U\) contains the leaf through any of its points in \(U\): the tangent vector fields preserve its reduced ideal, first on its smooth locus and then everywhere by closure. A rank-dimensional leaf closure is therefore unique through a general point. Assigning that closure gives the desired rational map. Resolve its graph and take the Stein factorization. ◻

Proof. Otherwise take a covering family of nongeneral-type subvarieties, with smooth domains \(V\), and slice its parameters as above. Restricting the ramification formula of its generically finite total evaluation shows that \(K_V\) dominates the restriction of the pulled-back \(K_X\) by an effective divisor. Thus \(K_V\) is pseudo-effective. The induction hypothesis gives a good minimal model of \(V\), so \(\kappa(V)\geq0\). Since \(V\) is not of general type, its general Iitaka fibers have positive dimension and Kodaira dimension zero. Spreading these fibers gives another covering family with smooth domains \(G\), of dimension less than \(n\), and \(\kappa(G)=0\). Each \(G\) has a good minimal model and \(\kappa_\sigma(K_G)=0\).

Fix a positive current \(T\) representing \(K_X\). After slicing and resolving the total evaluation of the \(G\)’s, its restriction to almost every parameter fiber, plus the effective restricted ramification divisor, is a positive current in \(K_G\). Every positive current in that class is supported on the fixed canonical divisor of \(G\). To see this, take a common resolution with its good minimal model. The latter has torsion canonical divisor, so the canonical divisor upstairs is effective exceptional. Intersection with a pullback of an ample class to the power \(\dim G-1\) forces any such positive current to vanish off the exceptional locus. The support theorem makes it divisorial, and independence of exceptional divisor classes, by negativity, fixes its coefficients. Pushing down proves the assertion on \(G\). The excluded support is algebraic in the family: it is the fixed divisor of a sufficiently divisible relative pluricanonical system after shrinking the parameter space.

It follows that \(T\) annihilates the web distribution of the \(G\)’s on a dense open. If that distribution has full rank, \(T\) is supported on a proper algebraic set. The support theorem for positive closed \((1,1)\)-currents expresses it as a finite nonnegative real combination of prime divisors (Siu 1974; Demailly 2012a). Its rational cohomology class then has a nonnegative rational representative: solve the finite rational linear system for the coefficients on the same face of the nonnegative orthant. Lemma 87 converts numerical effectivity to \(\mathbb{Q}\)-linear effectivity, a contradiction.

Otherwise Lemma 88 gives a rational quotient with general smooth fiber \(F\) on a smooth resolved graph, where \(0<\dim F<n\). Its canonical divisor is pseudo-effective. The moving \(G\)’s still generate its tangent space at general points. Indeed the quotient is constant on every general web member, hence induces a rational map on their parameter space; restricting to a general quotient fiber preserves dominance of evaluation. A pluricanonical rational map of \(F\) is constant along each such \(G\). Slice the parameters inside \(F\) to make the total evaluation generically finite: ramification injects the restricted pluricanonical sections into sections of a multiple of \(K_G\), and these span a space of dimension at most one. Consequently their ratios are constant on \(G\). Their differentials vanish on the web and its brackets, so the pluricanonical map is constant on \(F\). Lower-dimensional nonvanishing therefore gives \(\kappa(F)=0\), and its good minimal model gives \(\kappa_\sigma(K_F)=0\).

Now apply the numerical-zero-fiber reduction of Gongyo–Lehmann (Gongyo and Lehmann 2013, Theorem 1.3 and Corollary 4.5). For a projective \(\mathbb{Q}\)-factorial klt rational pair with a connected-fiber morphism and numerical dimension zero on the general fiber, that theorem produces a klt pair on a smooth birational base whose good-minimal-model existence is equivalent. Here the total space is the smooth graph, the boundary is zero, and the base has dimension less than \(n\). The required numerical dimension is \(\kappa_\sigma\), following the convention in (Fujino 2020, sec. 3 and Theorem 3.2). It is zero by the good model of \(F\), so the numerical-dimension hypothesis holds. The induction hypothesis supplies the base good model, hence nonvanishing upstairs, contradicting (191). ◻

Proof. Resolve a moving subsystem of a multiple of \(P\). If its map is generically finite, \(P\) is big and the assertion follows from pseudoeffectivity of \(K_Y\). Otherwise its general fiber is a proper positive-dimensional subvariety through very general points and is of general type by Proposition 89. The canonical divisor of the smooth resolution is relatively big, so adding a sufficiently large ample pullback from the image makes it big. Since that pullback is bounded by a multiple of the resolved moving subsystem, pushforward gives \(K_Y+aP\) big for some \(a>0\). Convexity with the pseudo-effective \(K_Y\), and then addition of the pseudo-effective \(P\), gives the assertion for every \(c>0\). A nonzero nonnegative functional on a closed convex cone is strictly positive on its interior. Applying it to \(K_Y+cP\) proves the last assertion. ◻

Proof. We induct on the integer part of \(M\); discrepancies over a smooth variety are positive integers. Suppose that a strictly increasing sequence exists, and discard initial terms so its members exceed a fixed positive number \(c\). Skoda integrability and change of variables give \[ \nu_E(T)\leq A_V(E)\nu_x(T) \tag{192}\] at a general point \(x\) of the center. Indeed every exponent smaller than \(1/\nu_x(T)\) is locally integrable, whereas integrability after pullback imposes the corresponding discrepancy bound. Hence all centers lie in the fixed proper Siu locus \(\{\nu_x(T)\geq c/M\}\), which is algebraic by Siu analyticity and projectivity (Siu 1974).

If, after passage to a subsequence, the centers lie in a fixed codimension-at-least-two subvariety, principalize its ideal. All lifted centers lie in the exceptional locus, where the relative canonical divisor has positive integral order. The discrepancy bound for the new smooth ambient space has therefore decreased by at least one. Pull back \(T\) and apply induction.

Otherwise the centers lie in a fixed prime divisor \(D\) of the Siu locus. Write \(T=c_D[D]+T'\) with \(T'\) positive and with zero generic Lelong number along \(D\). The integers \(\mathop{\mathrm{ord}}_E(D)\) are bounded: the fixed effective divisor \(D\) has positive log canonical threshold, and \(\operatorname{lct}(D)\mathop{\mathrm{ord}}_E(D)\leq A_V(E)\). Pass to a constant order. The residual numbers \(\nu_E(T')\) are still strictly increasing, and after discarding a term have a positive lower bound. Equation (192) places their centers in a fixed Siu locus of \(T'\), which does not contain \(D\). Their intersection with \(D\) has codimension at least two, so the preceding case applies. ◻

Proof. Suppose otherwise, and choose a connected-fiber quotient of minimal base dimension \(b\). If \(b=0\), the current vanishes on a dense open; the divisorial-current argument in Proposition 89 contradicts (191). Thus \(0<b<n\). Resolve the graph and the space carrying the current: \[\begin{tikzcd} W \arrow[r,"p"] \arrow[d,"h"'] & Y \\ B & \end{tikzcd}\] Here \(W,B\) are smooth projective and \(h\) has connected general fiber. Flatten the main component over a modification of the base, resolve that base, normalize the main transform, and then resolve upstairs. Every vertical prime on the flat transform maps to a divisor of the base: a prime over base codimension \(c\) would have generic fiber dimension at least \(n-b+c-1\), whereas all fibers have dimension at most \(n-b\). Consequently every vertical divisor on \(W\) over base codimension at least two is exceptional over \(Y\). This property persists under later resolutions and base modifications: a nonexceptional prime already maps onto a base prime and its generic image is unchanged.

For divisors and numerical classes put \[T_B=p_*h^*.\] This strict pullback is well defined on numerical classes because \(Y\) is \(\mathbb{Q}\)-factorial: numerical classes upstairs decompose into pulled-back classes from \(Y\) and exceptional classes. It preserves pseudoeffectivity. For a higher base model \(r:B'\to B\), its analogue satisfies \(T_{B'}r^*=T_B\), by computing on a common graph. It kills divisors exceptional over \(B\): otherwise a prime in such a pullback that dominates a divisor of \(Y\) would have center of codimension at least two on \(B\), contrary to the property just established.

Descent of the current. Let \(T\) also denote the pulled-back current on \(W\). Shrink to a smooth open \(B^0\) so that \(h\) is smooth proper, the removed set contains no vertical divisor over \(B^0\), and the remaining open in every fiber is nonempty and connected. Away from that fixed removed algebraic set upstairs, \(T\) descends to a positive current \(S^0\). Indeed in submersion coordinates the horizontal coefficient distributions are independent of fiber variables by closedness, and the open parts of general fibers are connected. They therefore glue to \(S^0\) on \(B^0\). The difference \(T-h^*S^0\) is closed of order zero and supported on the removed algebraic set. The support theorem expresses it as a signed divisor sum; a closed order-zero \((1,1)\)-current supported in codimension at least two is zero. Its horizontal coefficients are nonnegative, since a pullback has zero generic Lelong number along a horizontal divisor.

Subtract those finitely many global Siu components from \(T\), obtaining a positive current \(T'\) that equals \(h^*S^0\) on all of \(h^{-1}(B^0)\). Choose a Kähler form \(\omega\) representing an ample algebraic class and put \(d=n-b\). Fiber integration gives \(h_*(\omega^d)=c>0\), constant on \(B^0\), by closedness. The projection formula therefore gives \[c^{-1}h_*(T'\wedge\omega^d)|_{B^0}=S^0.\] The left side defines a global positive closed extension. Its class is real algebraic, since \(\{T'\}\) is the algebraic class \(p^*K_Y\) minus real divisor classes and algebraic intersection and pushforward preserve such classes. Remove its divisorial parts along \(B\setminus B^0\), and call the result \(S\). On a dense open \(T=h^*S\). Their difference is again closed of order zero, so the same support theorem leaves only a signed divisor sum. It has nonnegative coefficients at every prime dominating a base prime. For the latter assertion, the generic local map is a transverse power map times a submersion. A current with zero generic Lelong number along the base prime has zero generic number at such an upstairs prime: local target balls of a radius equal to a fixed power of the source radius give the usual Lelong comparison. The only possible negative coefficients lie over base codimension at least two and are exceptional over \(Y\). It follows that \[ K_Y-T_B\{S\}\quad\text{is pseudo-effective}. \tag{194}\] Horizontal parts subtracted above merely contribute effective divisors to this difference.

A negative canonical direction on the base. Choose a nonzero supporting functional \(\alpha\) of the pseudo-effective cone at \(K_Y\), and put \(\eta=T_B^*\alpha\). The corresponding functionals \(\eta'\) on higher smooth base models are movable classes by pseudo-effective/movable duality (Boucksom et al. 2013, Theorem 0.2). They annihilate base-exceptional divisors. Equation (194) gives \(\eta\cdot\{S\}=0\).

Only finitely many prime divisors on \(B\) have positive divisorial Lelong coefficient for \(S\). Otherwise their nonzero strict pullbacks are effective divisors on \(Y\) killed by \(\alpha\), with pairwise disjoint prime supports. The supports are disjoint because a nonexceptional prime of \(Y\) cannot map into the intersection of two base primes. Only finitely many base primes have zero strict pullback, since their total pullbacks are supported in the finite exceptional locus of \(p\). In the finite-dimensional rational space of divisor classes, infinitely many remaining strict pullbacks have a rational relation. Its positive and negative sides are nonzero effective divisors with disjoint support; Lemma 87 makes the relation \(\mathbb{Q}\)-linear. They define a moving pencil killed by \(\alpha\), contradicting Corollary 90.

We claim that \[ K_B\cdot\eta<0. \tag{195}\] The smooth general fiber \(F\) of \(h\) is of general type by Proposition 89. The relative-positivity theorem of Kovács–Patakfalvi (Kovács and Patakfalvi 2017, Theorem 9.9) applies to a log canonical fiber space with smooth base, SNC total pair, and log-general-type geometric generic fiber; for a rational base divisor \(M\) with \(\kappa(M)\geq0\), it gives relative subadditivity with the term \(h^*M\). Here both spaces are smooth projective, the boundary is zero, and we take \(M=0\). Invariance of plurigenera on the smooth locus identifies the generic-fiber Kodaira dimension with that of \(F\). Thus \[ \kappa(W,K_W-h^*K_B)\geq \kappa(F,K_F)=n-b>0. \tag{196}\] This is an established general-type-fiber theorem, not another use or strengthening of Assumption 71. Choose compatible canonical divisors and set \(P=K_Y-T_B(K_B)=p_*(K_W-h^*K_B)\). For every sufficiently divisible \(m\), pushing forward an effective divisor \(\operatorname{div}_W(\varphi)+m(K_W-h^*K_B)\) gives \(\operatorname{div}_Y(\varphi)+mP\geq0\). The resulting injection of sections preserves their ratios, so \(\kappa(Y,P)\geq1\). Corollary 90 now gives \(0<\alpha(P)=-K_B\cdot\eta\), proving (195).

Rational curves and an exact zero gap. Apply the multiplier approximation to \(S\) on \(B\), and principalize \(\mathcal J(kS)\) by \(r_k:B_k\to B\). If \(F_k\) is its divisor, the class \[P_k=\frac{r_k^*L_k-F_k}{k}\] is nef. Put \(\eta_k=T_{B_k}^*\alpha\). Since it kills exceptional divisors and \(L_k-k\{S\}\) stays bounded, \[0\leq P_k\cdot\eta_k=O(k^{-1}),\qquad K_{B_k}\cdot\eta_k=K_B\cdot\eta<0.\] For a fixed ample \(H\) on \(B\), polarize by \(P_k+k^{-1}r_k^*H\) plus a sufficiently small rational ample class on \(B_k\). Approximate \(\eta_k\) by positive sums of covering-curve classes. Discard terms with nonnegative canonical degree. Some remaining covering curve has the ratio of polarization degree to anticanonical degree \(O(k^{-1})\). Quantitative bend-and-break (Miyaoka and Mori 1986, Theorem 5) supplies rational curves through its general points with polarization degree at most \(2\dim B\) times that ratio. Parameterization and uncountability give a covering family of rational curves \(R_k\) satisfying \[ P_k\cdot R_k=O(k^{-1}),\qquad H\cdot r_{k*}R_k=O(1). \tag{197}\] The general parametrized member is free in characteristic zero. Consequently \(K_{B_k}\cdot R_k\leq-2\), and \[ 0\leq K_{B_k/B}\cdot R_k =K_{B_k}\cdot R_k-K_B\cdot r_{k*}R_k=O(1). \tag{198}\] Here the upper bound follows from the fixed ample-degree bound in (197).

Write \(\bar R_k=r_{k*}R_k\). Every positive contact with an exceptional prime contributes a positive integer discrepancy times a positive integer contact degree to (198). Thus the number of such contacts, their discrepancies, and their contact degrees are uniformly bounded. Contacts with the finitely many strict transforms of divisorial Lelong components of \(S\) are bounded by their fixed degrees against \(\bar R_k\). Subtracting these divisorial parts from \(r_k^*S\) leaves a positive current. Using (193) and (197) gives \[ \begin{split} 0&\leq \{S\}\cdot\bar R_k -\sum_E\nu_E(S)\,E\cdot R_k\\ &\leq \{S\}\cdot\bar R_k-(F_k/k)\cdot R_k \leq O(k^{-1}). \end{split} \tag{199}\] The sum includes exceptional primes and those strict divisorial components; terms with zero contact are irrelevant.

Bounded ample degree gives only finitely many numerical classes of the integral cycles \(\bar R_k\). Pass to one class. By Lemma 91, the sums in (199) belong to an ACC set: there are a bounded number of terms, their nonnegative integral multiplicities are bounded, and all relevant discrepancies are bounded. Finite sums of this kind preserve ACC, by successively taking nonincreasing subsequences of their entries. The nonnegative gaps in (199) must therefore equal zero for some covering families. Otherwise a sequence tending to zero has a strictly decreasing positive subsequence, giving a strictly increasing sequence of the complementary sums.

For such a family the residual positive current restricts to degree zero on almost every \(R_k\), and hence vanishes there. Off its removed divisors it is the original current. The discussion following Lemma 88 and that lemma itself yield a positive-relative-dimensional rational quotient of \(B\) whose vertical directions are annihilated by \(S\) on an open. Composing with \(h\) gives a quotient of smaller base dimension whose vertical directions are annihilated by \(T\), contradicting the choice of \(b\). ◻

Proof. Pass to a common smooth model \(V\) also dominating the fixed resolution carrying \(T\). Choose on the fixed resolution a sufficiently positive line bundle \(H\). For sufficiently divisible \(k\), multiplier approximation gives a globally generated \(\mathcal{O}(kp^*K_Y+H)\otimes\mathcal J(kT)\). Pull this system to \(V\). A general member \(D_k\) of its underlying effective divisor system has, at every specified prime \(E\), multiplicity \(\mathop{\mathrm{ord}}_E\mathcal J(kT)\); global generation after removal of the ideal ensures no additional generic vanishing. Here the notation \(H\) also denotes its pullback to \(V\).

Put \(D'_k=p_{t*}D_k\) and \(H_t=p_{t*}H\). The first is effective, and both are \(\mathbb{Q}\)-Cartier because \(Y_t\) is \(\mathbb{Q}\)-factorial. Pushing the rational linear relation gives \[D'_k\sim_\mathbb{Q}kK_t+H_t.\] Thus the exceptional divisor \(D_k-p_t^*D'_k\) is rationally equivalent to \[kJ_t+H-p_t^*H_t.\] The two exceptional divisors are equal: their difference is numerically trivial and exceptional, so the negativity lemma applied with both signs makes it zero. Since \(p_t^*D'_k\) is effective, \[\operatorname{coeff}_E D_k \geq k\operatorname{coeff}_E J_t +\operatorname{coeff}_E(H-p_t^*H_t).\] The last coefficient is fixed for this model and \(t\). Combining with (193), dividing by \(k\), and letting \(k\to\infty\) proves the assertion. The same proof can be made on each common model, so the comparison applies to all divisorial valuations needed later. No uniformity in \(t\) of the fixed error is required. ◻

Proof. Suppose such families exist. Slice their parameters and resolve total evaluation, the map to the fixed \(Y\), the fixed smooth resolution carrying \(T\), and the common models comparing \(Y\) and \(Y_t\). Denote the resulting generically finite dominant morphism by \(q_t:U_t\to Y\). The source is smooth near its complete general parameter fiber and has dimension \(n\), with parameter space of dimension \(n-1\). Rational maps from this smooth source to each required proper target extend at codimension-one points by the valuative criterion of properness. Their indeterminacy loci consequently have codimension at least two, and their images cannot dominate the parameter space. Shrink that space to avoid these finitely many images, and resolve and compactify preserving the remaining open. Equivalently, nontrivial smooth-center blowups have codimension-at-least-two centers; a codimension-one Cartier-center blowup is the identity. Thus the complete smooth general parameter fiber is still \(C_t\), unchanged as a curve, and its genus is unchanged.

For a prime \(Z\) of \(U_t\) exceptional over \(Y\), the restricted valuation of its function field is \(r_Z\mathop{\mathrm{ord}}_E\) for a divisorial valuation \(E\) over \(Y\). The coefficient of \(Z\) in the ramification difference is \[ \operatorname{coeff}_Z(K_{U_t}-q_t^*K_Y) =r_Za(E;Y,0)-1. \tag{201}\] This follows by factoring through a model extracting \(E\) and calculating the tame ramification of DVRs. It is positive and bounded away from zero: \(Y\) is terminal, so \(a(E;Y,0)>1\), and a fixed Cartier index of \(K_Y\) puts discrepancies in a fixed rational lattice. The remaining ramification coefficients are nonnegative. On the general parameter fiber, \[K_{U_t}\cdot C_t=2g(C_t)-2,\qquad q_t^*K_Y\cdot C_t\geq0.\] The latter inequality follows from pseudoeffectivity and the covering property. Therefore the total ramification cost is bounded. Formula (201) bounds the number of positive exceptional contacts, their discrepancies, their ramification indices, and their integral contact degrees. Moreover \(q_t^*K_Y\cdot C_t\) lies in a fixed rational lattice inside a bounded interval, so takes only finitely many values.

The generic Lelong number of \(q_t^*T\) at \(Z\) is \(r_Z\nu_E(T)\). Extract \(E\), remove its generic divisorial part, and use the transverse-power-map comparison for the zero-generic-number remainder. Lemma 93 shows that these coefficients dominate the coefficients of the pulled-back canonical difference \(J_t\). Its support is exceptional over \(Y\) as well as \(Y_t\), since the map between these models is small. Subtracting all exceptional divisorial parts of \(q_t^*T\) leaves a positive current. Consequently \[ \begin{split} 0&\leq q_t^*K_Y\cdot C_t -\sum_{Z\ {\rm exceptional}} r_Z\nu_E(T)\,Z\cdot C_t\\ &\leq K_t\cdot C_t \leq (K_t+tA_t)\cdot C_t =O(\mu_t). \end{split} \tag{202}\] All divisor contacts used here are nonnegative, since the curves cover the total space.

On the fixed smooth resolution \(W\to Y\), one has \[a(E;W,0)=a(E;Y,0) -\mathop{\mathrm{ord}}_E(K_W-p^*K_Y)\leq a(E;Y,0),\] because the relative canonical divisor is effective. Thus Lemma 91 applies with a fixed bound. The sums in (202) form an ACC set, their number of terms and integral multiplicities being bounded. Pass to a fixed value of \(q_t^*K_Y\cdot C_t\). Since \(\mu_t\to0\), the exact-zero argument of (199) gives a family for which the left gap is zero.

The residual positive current then restricts to zero on almost every curve in that family. On the dense open away from the removed exceptional divisors it agrees with the original current. Its curve tangents therefore generate, by Lemma 88, an annihilated rational fibration of positive relative dimension on \(Y\). This contradicts Proposition 92. ◻

Proof. Put \(P=K_W+D\), and suppose it is not big. It cannot have Iitaka dimension at least one: by Corollary 90, \(K_W+cP\) would be big, and \((1+c)P=(K_W+cP)+D\) would then be big as well. Hence \(\kappa(W,P)\leq0\).

Run the dlt LMMP for \(P\) with ample scaling. The divisor \(P\) has a signed rational representative supported on the floor \(D\), and its transforms retain that property. Every negative flip must meet the floor: on its complement the representing rational section is nowhere vanishing, so the adjoint has zero degree on every complete curve there. By Proposition 76, an infinite sequence would eventually have all flips disjoint from the floor, a contradiction. Divisorial contractions are finite in number. Thus the program terminates at a \(\mathbb{Q}\)-factorial dlt log minimal model \((Z,D_Z)\).

The boundary is reduced, \(K_Z\) is pseudo-effective as the birational pushforward of \(K_W\), and the nef adjoint has a signed rational representative supported on \(D_Z\). Its restriction to \(D_Z\) is semiample by Proposition 75. Theorem 79 applies, with the pseudo-effective perturbation \(K_Z\), and gives abundance. Since the Iitaka dimension is at most zero, the numerical dimension is zero. Intersecting \(K_Z+D_Z\equiv0\) with an ample \((n-1)\)-fold product, and using pseudoeffectivity of \(K_Z\), forces \(D_Z=0\). The signed relation then gives \(K_Z\sim_\mathbb{Q}0\).

On a common resolution, the pullback of \(K_W\) is rationally equivalent to a signed divisor exceptional over \(Z\). It is pseudo-effective. A positive current in that class has zero intersection with a pullback of an ample class on \(Z\) to the power \(n-1\), so is supported on the exceptional locus. The support theorem makes it effective divisorial, and independence of exceptional divisor classes by negativity says that the original signed coefficients are these nonnegative coefficients. Thus \(K_W\) is \(\mathbb{Q}\)-linearly effective, contradicting (191). ◻

Proof. The dlt modification for an arbitrary effective boundary (Fujino and Hashizume 2023, Theorem 2.10), applied after truncating coefficients exceeding one, gives \[K_Q+B+E=p^*(K_Z+\Theta),\] where \(Q\) is \(\mathbb{Q}\)-factorial, \((Q,B)\) is dlt, and \(E\ge0\) is supported above the non-lc locus. Choose, over the generic point of \(V\), a minimal lc stratum, and denote its closure by \(S\). The divisor \(E\) does not contain \(S\). Iterated dlt adjunction, including the effective restriction of \(E\), produces an effective divisor \(B_S\) with \[K_S+B_S\sim_{\mathbb{Q}} (p|_S)^*((K_Z+\Theta)|_V).\] The pair is klt over the generic point of \(V\); singularities elsewhere are not asserted to be klt.

Factor \(S\to V^\nu\) through its Stein base \(V'\). This is a klt-trivial fibration: generic klt, effectivity of the boundary, and connected fibers give the rank-one condition. The canonical bundle formula, in precisely this generically subklt form, gives a discriminant and a b-nef moduli b-divisor (Fujino and Gongyo 2014b, Definition 3.1 and Theorem 3.5). The discriminant trace on \(V'\) is effective. To see the sign, over a generic base prime a divisor in its inverse image has multiplicity at least one and a nonnegative boundary coefficient; the lc threshold of the fiber is therefore at most one. Coefficients exceeding one in the discriminant cause no difficulty for the present inequality.

Intersect the resulting base formula with the pullback of \(P^{f-1}\). The moduli term is nonnegative: on a model where it is nef this is a nef intersection, and exceptional terms vanish on pushing down. The effective discriminant is also nonnegative. Finally the codimension-one ramification formula for the finite map \(V'\to V^\nu\), followed by projection, can only increase the canonical intersection. Discrepancy terms on resolutions have images of codimension at least two and pair trivially with the pulled-back \(P^{f-1}\). Dividing by the finite degree gives Equation (206). ◻

Proof. The base loci increase with the level. Following containing irreducible components through the marked point produces a nested chain of proper positive-dimensional subvarieties. There are only \(d-1\) possible dimensions, so two successive components agree. For fixed degree, jet kernels form vector bundles after shrinking the marked-point parameter space. Components of their base loci spread after a finite parameter extension and further shrinking. We may fix the chosen step, dimension, and component on an irreducible parameter space \(T\). Its incidence \(\Gamma\subset T\times Z\) dominates \(Z\), because it contains the varying marked point.

Here is the multiplicity argument. Regard a local section of the higher kernel bundle as a varying element of the fixed vector space \(H^0(Z,kP)\). A parameter derivative of order \(j\) loses at most \(j\) orders of vanishing along the moving diagonal. Thus, for \(j<k\delta-O(1)\), every such derivative belongs to the lower kernel and vanishes on the selected incidence \(\Gamma\). At a general smooth point of \(\Gamma\), the projection \(\Gamma\to Z\) is smooth. Therefore the pure parameter directions, together with \(T\Gamma\), span \(T(T\times Z)\). In local coordinates over \(Z\), this says that normal coordinates to \(\Gamma\) can be chosen among parameter coordinates. Vanishing of all parameter derivatives of orders below \(h\) is consequently equivalent there to membership in \({\cal I}_\Gamma^h\). Restricting to a parameter fiber proves generic multiplicity at least \(k\delta-O(1)\) along \(V\), hence at least \(k\delta/2\). This applies to a local basis of the higher kernel bundle, and therefore to every section of the higher subseries.

Put \(a=d-f\) and \(h=\lceil k\delta/2\rceil\); the stronger bound \(k\delta-O(1)\) permits this choice for sufficiently large \(k\). At the smooth generic point of \(V\), the base ideal is maximal-ideal primary and lies in the \(h\)-th power of that maximal ideal. Cut by \(a\) general members of the subseries. Their local intersection multiplicity along \(V\) is at least \(h^a\). For completeness, global excess base components do not invalidate this bound: at each cut discard components wholly contained in the base locus before the next intersection. None contains the generic point of \(V\), since \(V\) is a base-locus component. The discarded cycles have nonnegative \(P\)-degree by nefness. The remaining proper intersections therefore have total \(P^f\)-degree at most \(k^aP^d\). Since \(h\ge k\delta/2\), this gives the first inequality.

The local lc threshold of this primary ideal is at most \(a/h\): the exceptional divisor of the blowup of its smooth closed point in the \(a\)-dimensional transverse local scheme gives this bound. On a log resolution, a sufficiently long average of general members realizes the ideal threshold by an effective rational divisor \(\Theta\sim_{\mathbb{Q}}cP\). It is lc at the generic point of \(V\), with \(V\) an lc center, and \(c\le k a/h\le2d/\delta\). Lemma 96 gives the second inequality. ◻

Proof. The volume bounds follow from pseudo-effectivity of \(K\): \[\mathop{\mathrm{vol}}(K+tA)\le\mathop{\mathrm{vol}}(K+2tA)\le2^n\mathop{\mathrm{vol}}(K+tA).\] Suppose the asserted order bound fails for arbitrarily small \(t\). Taking powers of the offending section permits arbitrarily large divisible degrees. Choose \(n+1\) levels strictly between \(\rho\) and \(4\rho\), with equal gaps comparable to \(\rho\). All corresponding subseries are nonzero. The base point at the lowest level cannot be isolated: local Bezout for \(n\) general members would exceed the total intersection \(k^nP^n\). Lemma 97 therefore supplies a moving proper positive-dimensional component \(V\).

The component is of general type by Proposition 89. On the scaling model for \(K+2tA\), the effective transform of \(A\) does not contain a general such component, and \(K_Z|_V\le(2r)^{-1}P|_V\) in effective order. Equations (207) consequently imply Equation (208) with constants uniform in \(t\); the gap is bounded above and away from zero because \(\rho\) is comparable to the fixed \(\epsilon\). The resulting bounded-genus, bounded-normalized-degree covering curves contradict Proposition 94. Exceptional additions preserve sections and their orders at general points, proving the additional assertion. ◻

Proof. The product \(Y\times Y\) is \(\mathbb{Q}\)-factorial. Indeed on a product resolution the Picard group has no cross term because the smooth models have irregularity zero; the two factorwise \(\mathbb{Q}\)-factorial descent statements then apply. Products of canonical singularities are canonical, and the projective bundles are klt. The two disjoint Cartier axes form a plt boundary \(D\), and \[K_{Z_0}+D=K_{\rm sum},\qquad D\sim2\xi-mK_{\rm sum}.\] The notation \(K_{\rm sum}\) means \(K_1+K_2\), or \(K\) on a slice.

For \(0<\sigma\le1\), put \[\gamma=2\sigma+\frac{q(1+\sigma m)}r,\qquad \Xi_t\sim_{\mathbb{Q}}\xi+ \frac{(1+\sigma m)t}{\gamma}A_{\rm sum}.\] Both endpoint weight systems are big, so there are effective rational representatives of \(\Xi_t\) containing neither axis. We first obtain a threshold bound independent of \(\sigma\). Restrict to \(0<t\le\sigma/(1+m)\). Since \(\gamma\ge2\sigma\), the coefficient \[a=\frac{(1+\sigma m)t}{\gamma}\] lies in \((0,1]\). Thus the numerical classes \(\xi+aA_{\rm sum}\), and their restrictions to either fixed axis, range over bounded segments independent of \(\sigma,q,r,t\). On a fixed smooth resolution their effective pullbacks have uniformly bounded degree against a fixed very ample class. That degree bounds their multiplicity at every point, including the coefficients of exceptional components. Rational denominators do not enter this estimate. The smooth lc threshold is at least the reciprocal of maximal multiplicity.

On the fixed klt space, write the crepant boundary on its resolution as an SNC divisor with coefficients below one. The minimum of one and the positive numbers one minus its positive coefficients bounds its log discrepancies below by a fixed positive multiple of smooth log discrepancies. Therefore the preceding smooth bound gives a common threshold \(\eta>0\) on the original space. The same reasoning on the axes gives, after decreasing \(\eta\), the same bound for both restricted divisors.

Now choose rational \(0<\sigma<\min\{1,\eta/4\}\), independently of \(q,t\), and then take \(t\) small enough that \(t\le\sigma/(1+m)\) and \(q(1+\sigma m)/r<\sigma\). It follows that \(\gamma<3\sigma<\eta\). Inversion of adjunction with the disjoint coefficient-one axes gives plt near them; away from them the threshold bound gives klt. Lowering their coefficients to \(1-\sigma\) consequently shows that \[(Z_0,(1-\sigma)D+\gamma\Xi_t)\] is klt. This choice respects the order: first \(\epsilon\), then \(q\), then sufficiently small \(t\) with \(r\) as in Equation (203). A direct calculation gives its adjoint class \[K_{Z_0}+(1-\sigma)D+\gamma\Xi_t \sim_{\mathbb{Q}}\frac{1+\sigma m}{r}M.\] The big klt adjoint has a good minimal model by (Birkar et al. 2010). Pushing the boundary to this model gives the stated \(\Delta\) and \(c_t=(1+\sigma m)/r\).

For the volumes, the weight-\(l\) summand in degree \(k\) has base classes \(kL_{l/k}\) and \(kL_{q-l/k}\); on a slice there is one such factor. The section decomposition and asymptotic Riemann–Roch give Equations (210)–(211) as Riemann sums. One can justify passage to the integral without assuming uniform asymptotic Riemann–Roch: partition \([0,q]\) into rational bins, compare weight classes in each bin by adding and subtracting the bin width times a fixed very ample divisor dominating both \(mK\) and \(-mK\), take divisible-degree limits, and then shrink the bins. Volume continuity gives the formulas. Equation (204) supplies their uniform bounds and, in particular, proves bigness used above. In the range of Equation (205), the bounds needed when choosing \(q\) are explicitly \[\mathop{\mathrm{vol}}(M)\ge \frac{(2n+1)!}{4(n!)^2}\,q\epsilon^{2n}, \qquad \mathop{\mathrm{vol}}(L+q\xi)\le2(n+1)q\epsilon^n.\] Their coefficients are independent of \(q\); the smallness threshold for \(t\) is allowed to depend on \(q\).

Compare a complete system on a common resolution with the subsystem at a fixed rational weight. The latter has fixed divisor consisting of its toric monomial and the base-model fixed differences. Subtracting the fixed divisor of the full system leaves an effective divisor \(E_b\); its moving class is precisely the sum of the pullbacks of the indicated \(N_b\)’s. This proves Equation (212). The same comparison in individual sufficiently divisible degrees shows that every section at that weight, after the full fixed subtraction, contains the corresponding multiple of \(E_b\).

At a generic base-divisor point the small base-model maps are isomorphisms. The two endpoint systems are free there in divisible degree, so together they have no common zero on the projective-line fiber. The full system therefore has no fixed subtraction there. The same holds on a general fiber. Finally, when restricting Equation (212) to a component sweeping the torus open, choose a general component not contained in \(\mathop{\mathrm{Supp}}E_b\). There are only countably many rational weights, so these effective restrictions can be required simultaneously. ◻

Proof. Write \(d=2n+1\). Choose \(q\) sufficiently large that the volume root in Equation (210) admits \(d+1\) levels above \(2n+3\), with equal gap \(\delta\) as large as needed below. This is possible uniformly for small \(t\), since the volume is bounded below by a positive constant times \(q\). Assume that the Seshadri assertion fails for arbitrarily small \(t\). Jet counts show that the systems at all these levels are nonzero in sufficiently large divisible degree. A curve through the marked point with degree-to-multiplicity ratio below the lowest level is contained in that system’s base locus. Such a curve exists from the assumed Seshadri bound, whether or not the infimum defining the constant is attained. Lemma 97 supplies a moving proper component \(V\) and both estimates (207).

For all moving components under consideration, the effective boundary \(\Delta\) of Lemma 99 does not contain the component. Thus \(K_Z|_V\le c_tP|_V\) in effective order. Whenever \(V\) is of general type, the curve construction following Equation (208) and weight domination give a contradiction. We must also handle components that are not generically finite over a proper base image.

Proof. Put \(V=\Omega_W^1|_C\). For a vector bundle on the smooth curve \(C\), write \[L_{\min}(V)= \lim_{e\to\infty} \frac{\mu_{\min}((F_C^e)^*V)}{p^e}.\] Langer’s Frobenius-instability estimate (Langer 2004, Corollary 2.5) states that \[\mu_{\min}(V)-L_{\min}(V) \leq \frac{(n-1)\deg A_C}{p-1}\] if \(A_C\) is nef and \(T_C(A_C)\) is globally generated. The genus is fixed, so \(A_C\) can be chosen with degree bounded independently of \(p\). Equation (220) yields \[ L_{\min}(V)\geq-\frac{c}{p-1} \tag{226}\] with one fixed \(c\).

We need an estimate for tensor powers whose exponent grows with \(p\). For every \(i\geq0\), \[ \mu_{\min}(V^{\otimes i})\geq iL_{\min}(V). \tag{227}\] To prove this directly, twist \((F_C^e)^*V\) by a line bundle of degree at most \(-\mu_{\min}((F_C^e)^*V)+2g(C)+2\) so that it is globally generated. This follows from Serre duality and the slope criterion for the vanishing of \(H^1\) after subtracting any point. The \(i\)-th tensor power is then globally generated with the corresponding \(i\)-fold twist. Pull back any quotient of \(V^{\otimes i}\), take degrees, divide by \(p^e\), and let \(e\) tend to infinity. The bounded genus term disappears, proving Equation (227). Thus every quotient of a tensor power with \(i\leq n(p-1)\) has minimum slope at least \(-nc\).

Multiplication to the socle of the truncated polynomial algebra is a perfect pairing in complementary degrees. The top monomial transforms by \((\det V)^{p-1}\), so, equivariantly, \[A_j(V)^*\otimes(\det V)^{p-1}\simeq A_{u-j}(V).\] The right side is a quotient of \(V^{\otimes(u-j)}\). Therefore \[\mu_{\max}(A_j(V))\leq(p-1)\deg K_W|_C+nc.\] By Equations (222) and (219), \[\begin{align*} (2B_p+pqS_0)\cdot C &\leq4p\,l^{n-1}H^n+O(1),\\ (p-1)K_W\cdot C &\leq\frac{2p}{r}\,l^{n-1}H^n. \end{align*}\] Since \(r>1\), their sum is bounded by the claimed expression, with slack in its coefficient \(7\). ◻

Proof. Choose \(M_p=7p\,l^{n-1}H^n+c_0\), where \(c_0\) bounds the error in Lemma 102, and put \(d_C=h\cdot C=l^nH^n\). A bundle of rank \(e_j\) and maximal slope at most \(M_p\) has at most \(e_j(M_p+1)\) sections: evaluation at \(\lfloor M_p\rfloor+1\) distinct points is injective. Its twist by \(-kh\) has no sections if \(kd_C>M_p\).

The divisor sequences on the fixed flag give \[h^0(W,\mathcal G_j) \leq \sum_{k_1,\ldots,k_{n-1}\geq0} h^0\bigl(C,\mathcal G_j|_C (-(k_1+\cdots+k_{n-1})h)\bigr).\] One obtains this by successively summing the restriction inequalities; the remainders vanish for sufficiently negative ample twists at each stage. At most \((1+\lfloor M_p/d_C\rfloor)^{n-1}\) tuples contribute. Hence \[h^0(W,\mathcal G_j) \leq e_j(M_p+1)(1+M_p/d_C)^{n-1} =e_jp^n\bigl(7^nH^n+o(1)\bigr).\] The powers of \(l\) cancel in the leading coefficient. All error constants are fixed before \(p\), and the same bound applies to every \(j\). Summing over the filtration and using Equation (225) gives \[h^0(\Delta_F,\mathcal L_p) \leq s^2\bigl(7^nH^n+o(1)\bigr).\] Now \(7^nH^n\leq(14\epsilon)^n<1/4\) by Equations (217) and (219). This proves Equation (228). ◻