Proof. The pullback assertion is (C. Hacon and Xie 2026, Lemma 2.39); nef descent is (Das et al. 2024, Lemma 2.38). In particular, the first assertion applies to any proper bimeromorphic morphism between compact spaces with rational singularities whose source is in class \(\mathcal C\): its general fibre is a point. For a projective klt log Fano contraction, relative vanishing supplies the higher-direct-image hypothesis once rationality of the base has been established. ◻

Proof. Choose a smooth representative \(\theta\) of \(\alpha\) with local smooth potentials. On the smooth compact Kähler space \(U\), a positive representative of the given class has the form \[ S=p^*\theta+[E]+\mathrm{d}\mathrm{d}^{c}v \tag{1}\] for a global distribution \(v\). Remove from \(T\) its singular locus and the image of the exceptional locus, obtaining a smooth open set \(T^\circ\) whose complement has codimension at least two. The map \(p\) is an isomorphism there. Equation (1) gives a global \(\theta\)-plurisubharmonic function \(v^\circ\) on \(T^\circ\). We do not assert that \(v\) is quasi-plurisubharmonic along \(E\).

On a coordinate neighbourhood where \(\theta=\mathrm{d}\mathrm{d}^{c}\rho\), extend \(\rho+v^\circ\) first across the removed codimension-two subset of the regular locus, and then across the singular locus. The first extension is the plurisubharmonic Hartogs theorem; the second is its normal-space version of Grauert–Remmert, recalled in (Coman et al. 2017, sec. 2.1, pp. 927–928). These extensions are unique. On overlaps their differences are the original smooth pluriharmonic differences of the functions \(\rho\). Subtracting \(\rho\) therefore gives a global potential \(v_T\), and \[\theta+\mathrm{d}\mathrm{d}^{c}v_T\geq0\] is a current with local plurisubharmonic potentials representing exactly \(\alpha\). This proves descent, including for signed \(E\).

For pullback, compose local plurisubharmonic potentials with the resolution. They cannot become identically \(-\infty\) on a nonempty open subset, since a resolution is an isomorphism on a dense open set. Thus they define the pulled-back positive current; see (Coman et al. 2017, sec. 2.1, p. 928).

Finally, if \(\alpha\) is nef, then \(p^*\alpha\) is nef on \(U\). For a Kähler form \(\omega_U\), choose positive representatives of \(p^*\alpha+\varepsilon[\omega_U]\). Their masses are bounded as \(\varepsilon\downarrow0\), so a weak limit is a positive current in \(p^*\alpha\). On the smooth Kähler manifold \(U\) it has local plurisubharmonic potentials. The first part, with \(E=0\), descends this current to the required class on \(T\). ◻

Proof. Part (i) is the analytic negativity lemma (C. Hacon and Xie 2026, Lemma 2.5); apply it to both signs for the last assertion. For (ii), points of a connected projective fibre can be joined by a chain of irreducible curves. If the fibre met both \(\mathop{\mathrm{Supp}}E\) and its complement, some curve in such a chain would meet the support without being contained in it. Its intersection with the effective real Cartier divisor would be strictly positive, contrary to \(f\)-anti-nefness. The assertion for a real Cartier divisor follows locally from positive combinations of effective rational Cartier divisors, as explained below.

For (iii), let \(G\) be the normalization of the graph in \(T\times_ZT^+\), with projections \(p_G,q_G\) and map \(h:G\to Z\). Compatible canonical data give the actual real Cartier divisor \[F_G=p_G^*D-q_G^*D^+.\] This difference is defined locally by compatible meromorphic canonical generators and therefore glues, even when the canonical sheaves were specified as \(\mathbb Q\)-line bundles. Smallness makes it exceptional over both sides. It is effective by (i), applied over \(T\): on a \(p_G\)-vertical curve the degree of \(-F_G\) is the degree of \(q_G^*D^+\), which is nonnegative.

For every \(h\)-vertical curve \(C\) one has \[ F_G\cdot C<0. \tag{4}\] Indeed, normalization is finite onto the graph, so the two projections cannot both contract \(C\). The nonzero degrees contributed by \(p_G^*D\) and \(-q_G^*D^+\) are both negative. An effective real Cartier divisor has nonnegative degree on a curve outside its support. For completeness, this fact and its strict pullback analogue hold even if its individual prime components are not \(\mathbb Q\)-Cartier: locally express it in the real span of finitely many Cartier divisors. Vanishing of coefficients outside its support and nonnegativity of the remaining coefficients are rational linear conditions. The resulting rational polyhedral cone expresses it as a positive real combination of effective \(\mathbb Q\)-Cartier divisors. The usual effective Cartier assertions apply to each term; summing the nonnegative local intersection numbers gives the asserted global degree inequality. In particular, the pullback has positive coefficient at every prime whose centre is contained in the support.

The morphism \(p_G\) has connected fibres because \(T\) is normal; hence \(h=f p_G\) has connected fibres. Its fibres are projective because \(G\) is finite over the graph in the fibre product. Each point of a positive-dimensional fibre lies on a curve in that fibre. Equation (4) therefore places the entire fibre in \(\mathop{\mathrm{Supp}}F_G\). To identify the relevant fibres, suppose \(f\) is an isomorphism over an open subset of \(Z\). There the log divisor on \(T^+\) is pulled back from \(Z\), by smallness and agreement in codimension one. Relative ampleness excludes an \(f^+\)-vertical curve there; projectivity, connectedness and normality then make \(f^+\) an isomorphism there as well. Interchanging the sides proves that their exceptional images coincide. Thus \(h^{-1}(A)\subset\mathop{\mathrm{Supp}}F_G\).

Pull \(F_G\) back to a resolution on which \(P\) appears. Its coefficient at \(P\) is exactly \(a(P,T^+,B^+)-a(P,T,B)\), with our log-discrepancy convention. Effectivity gives monotonicity, and the support assertion gives the claimed strictness. This proves (iii).

For (iv), a small exceptional locus in a fourfold has dimension at most two and has positive-dimensional fibres, so its image has dimension at most one. Moreover \(F_G\ne0\) by (4). A component of its support has dimension three and maps finitely into \(\mathop{\mathrm{Exc}}(f)\times_Z\mathop{\mathrm{Exc}}(f^+)\). The dimension of this product is at most the sum of the dimensions of its two factors. This proves (iv).

For (v), choose a common smooth resolution \(p:V\to T\), \(q:V\to T'\) carrying the fixed b-datum and projective over the common contraction base. Such a resolution is obtained by taking projective modifications of the graph that resolve the maps to a carrier of the datum. Subtract the actual structure boundaries to obtain a real Cartier divisor \(F=B_{T,V}-B_{T',V}\). Its coefficients are the differences of generalized log discrepancies, and cancellation of the fixed b-part gives \[[F]=p^*L_T-q^*L_{T'}.\] Let \(h:V\to Z\) be the morphism to the common contraction base; in the divisorial case \(Z=T'\) and \(q=h\). It is projective by this choice and has connected fibres because it is bimeromorphic onto the normal space \(Z\). The boundary-pushforward condition makes \(F\) exceptional over \(Z\). The relative signs of the log classes give \(F\cdot C\leq0\) on every \(h\)-vertical curve, with strict inequality whenever \(p(C)\) or, in the small case, \(q(C)\) is a curve. Thus (i) gives \(F\geq0\).

If \(y\) lies in an exceptional image, a vertical curve downstairs can be lifted to a curve on \(V\) dominating it: use a component of its projective inverse image and cut by relatively ample hyperplanes. Such a curve has negative \(F\)-degree and hence lies in \(\mathop{\mathrm{Supp}}F\). Part (ii) now puts the entire fibre \(h^{-1}(y)\) in the support. Pullback to a higher resolution has positive coefficient at every prime centred in this support. This proves all the strict discrepancy inequalities, including for a centre on the flipped side. No trace current on the normalized graph is assumed to be Cartier; the actual divisor is constructed from the structure boundaries on the smooth resolution. ◻

Proof. On every \(p\)-vertical curve, \(-J_T\) has the same degree as the nef class \(q^*H\). Lemma 5(i) gives \(J_T\geq0\). Subtract two possible relations. The difference of the correction divisors is \(p\)-exceptional and numerically trivial over \(T\), hence zero. Pullback injectivity from Lemma 3 then gives uniqueness of the class as well.

Pulling back (6) to a higher resolution gives the claimed correction there by uniqueness. Conversely, if the relation is initially known on a higher resolution \(r:V'\to V\), push its correction down to \(V\). The difference between the upstairs correction and the pullback of this pushforward is \(r\)-exceptional and numerically trivial over \(V\), so is zero. Injectivity of \(r^*\) gives the relation on \(V\).

Equation (7) follows by adding \(K_V\) and the fixed nef class \(tq^*H\) to its two sides. Its coefficients give \[a(P,T,B_T+t\mathbf H) =a(P,T,B_T)-t\operatorname{coeff}_P J_T\] on every sufficiently high resolution. This proves the discrepancy assertions.

For the converse, let \(B_V\) be the given actual generalized structure boundary and put \(B_V^{\rm ord}=p^*(K_T+B_T)-K_V\). Both push forward to \(B_T\), so \[J_T=\frac{B_V-B_V^{\rm ord}}{t}\] is an actual \(p\)-exceptional divisor. Subtract the ordinary log class from (2) to obtain (6). This constructs the divisor before applying negativity; it does not infer a divisor from an arbitrary class difference. ◻

Proof. The positive current \(q^*h+[J_T]\) represents \(p^*H_T\). The proof of Lemma 4 constructs a positive current \(S_T\) in \(H_T\) with local potentials. On the big open where \(p\) is an isomorphism and the correction is absent, it agrees with \(p_*q^*h\). Their difference is a closed current of order zero supported in codimension at least two. Such a current vanishes: after a local embedding it has bidimension \((\dim T-1,\dim T-1)\), whereas its support has smaller dimension (Demailly 2012, III, Corollary 2.11). Thus \(S_T\) is precisely the actual trace in (5). This step is needed because a trace of a positive b-current need not have local potentials a priori (Das et al. 2026, Remark 2.6(ii)).

The class \(q^*H\) is nef and has positive top self-intersection: \(q^*h\) is semipositive and is positive on the dense open where \(q\) is an isomorphism. By (Demailly and Păun 2004, Theorem 0.5), it contains a Kähler current \(R\geq\delta\omega_V\) for some \(\delta>0\) and some Kähler form \(\omega_V\). Fix a Kähler form \(\omega_T\) and choose \(C>0\) with \(p^*\omega_T\leq C\omega_V\). For \(0<\varepsilon<\delta/C\), \[R+[J_T]-\varepsilon p^*\omega_T\geq0.\] Its class is \(p^*(H_T-\varepsilon[\omega_T])\). Lemma 4 shows that \(H_T-\varepsilon[\omega_T]\) is pseudo-effective. Hence \(H_T\) contains a Kähler current, as required. ◻

Proof. Subtract the two structure equations on \(V\). The trace of the fixed b-datum cancels, giving (8). At a prime dominating a divisor on \(T'\), the corresponding boundary coefficient on \(T\) is unchanged, so its coefficient in \(F\) is zero. This proves exceptionality over \(T'\). Smallness gives the same assertion over \(T\).

If \(L_{T'}\) is nef, then \(-F\) is \(p\)-nef, because its degree on a \(p\)-vertical curve is the degree of \(q^*L_{T'}\). Negativity gives \(F\geq0\). If \(L_T\) is also nef, then \(F\) is \(q\)-nef, so applying negativity with the opposite sign gives \(F\leq0\). The same proof uses only relative nefness when the models lie over a common base. For equal pullback classes, the already constructed exceptional divisor is relatively numerically trivial, and is zero by Lemma 5(i). ◻

Proof. Fix \(z\in Z\) and shrink to a connected Stein neighbourhood \(U\) on which the stated pair is defined. A line bundle \(M\) on \(f^{-1}U\) has a Cartier-divisor representative there: the coherent direct image \(f_*M\) has rank one over the nonempty open where \(f\) is an isomorphism, so Cartan’s Theorem A supplies a nonzero section. Its pullback is a section of \(M\) that is not identically zero, and its zero divisor is Cartier. This argument takes place over \(U\), not on the whole compact space.

Every rational line bundle \(N\) on \(f^{-1}U\) is also relatively big. Indeed, choose \(q>0\) with \(qN\) a line bundle and an \(f\)-ample line bundle \(A\). The same direct-image argument gives a nonzero section of \(qN-A\), hence \[N\sim_{\mathbb Q}q^{-1}A+q^{-1}E\qquad(E\geq0).\] This is relative bigness in the analytic sense (Fujino 2022, Definition 2.46). Apply it to \(N=L-(K_T+C_U)\). The klt base-point-free theorem gives, after shrinking around \(z\), relative generation of \(L^k\) for every sufficiently large integer \(k\) (Fujino 2022, Theorem 6.2 and Remark 6.3).

Choose finitely many such generating sections near the compact fibre \(F=f^{-1}(z)\). The morphism they define on the projective reduced fibre \(F_{\mathrm{red}}\) is constant on each irreducible component: a positive-dimensional image would yield a curve with positive degree against the pullback of the hyperplane bundle, contrary to the numerical triviality of \(L^k\). Such a curve is obtained by successive general hyperplane sections of a component. Connectedness of \(F\) makes the images of all its components the same point. A linear combination of the generating sections is consequently nonzero at every point of \(F_{\mathrm{red}}\).

This section is a generator of \(L^k\) in the local ring at every point of \(F\): its residue is nonzero, and hence its representing function is a unit. This also treats a nonreduced fibre. Properness allows us to shrink \(U\) away from the image of the section’s zero set. Thus \(L^k\) is trivial on \(f^{-1}U\). Apply the argument to consecutive integers \(k\) and \(k+1\), and use one common neighbourhood. Their quotient trivializes \(L\) itself. Since \(f_*\mathcal O_T=\mathcal O_Z\), on this neighbourhood \(f_*L\simeq \mathcal O_U\), and evaluation is an isomorphism. These local conclusions give (9) globally. No additional tensor power enters the integral conclusion. ◻

Proof. Fix a Kähler class \(\omega\) and write \(d=c_1(D)\). Cone duality applies because klt singularities are rational. The slice \[S=\{v\in\overline{\mathrm{NA}}(T):\omega\cdot v=1\}\] is compact. Indeed, fix smooth representatives of a basis of \(H^{1,1}_{\mathrm{BC}}(T)\). Each is bounded above and below by a multiple of a Kähler form, so all its pairings on \(S\) are bounded; the slice is closed. If \(D\) is not nef, the minimum of \(d\) on \(S\) is negative. An extreme point of its minimum face gives a negative extremal ray.

Fix any such ray and let \(r\in S\) be its normalized point. Choose \(\varepsilon>0\) with \((d+\varepsilon\omega)\cdot r<0\). The descending nef part \(\varepsilon\omega\) changes no discrepancy, and \(B+\varepsilon\omega\) is big since \(B\) is effective and \(\mathbb Q\)-Cartier. The finite form of the cone theorem gives \[ \overline{\mathrm{NA}}(T)=\overline{\mathrm{NA}}(T)_{d+\varepsilon\omega\geq0} +\sum_{j=1}^{N}\mathbb R_{\geq0}[C_j], \tag{11}\] where the \(C_j\) are rational curves. Normalize their classes in \(S\) and discard repetitions. By extremality, \(r\) is one of these points, say \(r_1\). Let \(K\) be the convex hull of the others and of \(S\cap\{d+\varepsilon\omega\geq0\}\). This is compact and excludes \(r\); otherwise extremality of \(r\) would fail.

If \(K\) is nonempty, strict separation gives a linear functional \(h\) with \(h(r)=0\) and \(h>0\) on \(K\), after subtracting a multiple of \(\omega\). Regard it as a Bott–Chern class by duality. For all sufficiently small \(\delta>0\), \(h-\delta d\) is strictly positive on \(K\) and at \(r\). Since \(S=\operatorname{conv}(\{r\}\cup K)\), it is strictly positive on \(S\), hence Kähler. Thus \(\alpha=h/\delta\) has the required properties. Its positive minimum on \(K\) persists under small perturbations in \(R^\perp\); openness of the Kähler cone also preserves \(\alpha-d\). These perturbations give \(\mathcal U_R\). If \(K\) is empty, then \(S=\{r\}\) and \(-d\) is Kähler; a small neighbourhood of \(0\) in \(R^\perp\) has the required properties. This includes the case \(R^\perp=\{0\}\). ◻

Proof. Clear the denominator of \(L\). Its Bott–Chern identity permits a smooth Hermitian metric whose curvature is the indicated difference of forms. On every fibre its curvature is positive. Positivity of a line bundle on a compact complex space implies ampleness, and for a proper analytic map ampleness on every fibre is equivalent to relative ampleness (Fujino 2026b, Corollary 1.12 and Definition 3.1–Remark 3.2). This criterion does not presuppose projectivity and applies to nonreduced fibres as well. Existence of the relatively ample line bundle makes \(f\) projective.

For the converse, local relative embeddings give metrics on a relatively ample bundle with positive curvature in fibre directions. Glue their weights by a partition of unity on the base. The resulting metric retains that positivity. Add a sufficiently large multiple of a base Kähler form to its curvature. The sum is strictly positive; compactness allows one multiple to work everywhere. ◻

Proof. Choose \(\alpha\in\mathcal U_R\) from Lemma 10 and put \(\kappa=\alpha-c_1(D)\). Proposition 45 gives \(f:T\to Z\) and \(\gamma\) Kähler with \(\alpha=f^*\gamma\). Thus \(c_1(-D)=\kappa-f^*\gamma\), and Lemma 11 proves projectivity and relative ampleness. This establishes projectivity without strong factoriality. For every curve \(C\subset T\), \[ f(C)\text{ is a point}\quad\Longleftrightarrow\quad \alpha\cdot C=0\quad\Longleftrightarrow\quad[C]\in R. \tag{16}\] The rational curve generating \(R\) is contracted, so \(f\) is nontrivial.

Relative vanishing gives \(R^j f_*\mathcal O_T=0\) for \(j>0\) (Fujino 2022, Theorem 5.2); the log Fano base has rational singularities (Das et al. 2024, Lemma 8.8). Lemma 3 now identifies \(f^*H^{1,1}_{\mathrm{BC}}(Z)\) with the classes of degree zero on all vertical curves, which is exactly \(R^\perp\). The global line-bundle rank is also one, since all those curves are proportional and the rational line bundle \(D\) detects their ray. This proves (12). The lower-dimensional case has all the stated Mori fibre-space properties. In equal dimension, connectedness gives generic degree one, so \(f\) is bimeromorphic.

Divisorial contractions. Suppose \(E\) is an exceptional prime. It is \(\mathbb Q\)-Cartier. If \(E\cdot R\geq0\), it would be \(f\)-nef, contradicting negativity for a nonzero effective exceptional divisor. Thus \(E\cdot R<0\). Every vertical curve lies in \(E\), since an effective \(\mathbb Q\)-Cartier divisor has nonnegative degree on a curve outside its support. Every positive-dimensional projective fibre is covered by curves. A zero-dimensional local component of a connected fibre would be isolated, and the birational form of Zariski’s main theorem would make \(f\) an isomorphism there. Consequently \(\mathop{\mathrm{Exc}}(f)=E\).

For a prime Weil divisor \(A_Z\) on \(Z\), let \(A_T\) be its strict transform. Choose \(t\in\mathbb Q\) so \((A_T+tE)\cdot R=0\). The coefficient is rational by taking the ratio of the two rational degrees on one integral curve. A Cartier multiple of \(A_T+tE\) descends by Lemma 9: its difference from \(D\) is relatively ample. The descended line bundle agrees with the corresponding reflexive power of \(\mathcal O_Z(A_Z)\) off \(f(E)\), a subset of codimension at least two. Reflexivity identifies them everywhere. Hence every global prime on \(Z\) is \(\mathbb Q\)-Cartier.

If a global canonical Weil divisor is given, push it forward. It agrees with the canonical sheaf on the big open where \(f\) is an isomorphism, so reflexivity shows that it remains a canonical Weil representative. Alternatively, start with an invertible reflexive power of \(\omega_T\), adjust it by a rational multiple of \(E\) to degree zero, and descend after clearing denominators. On the same big open the result is a reflexive power of \(\omega_Z\); hence that power is invertible everywhere. This proves the canonical assertion without assuming a global meromorphic canonical section.

Now \(D'\) is a rational line bundle and its canonical identification away from \(E\) gives (13). Intersection with \(R\) gives \(e>0\). If \(b_E\) is the coefficient of \(E\) in \(B\), then \[a(E;Z,B')=1-b_E+e>1-b_E=a(E;T,B)>0.\] Pullback of the effective divisor \(eE\) gives weak increase for all other log discrepancies. Thus the target pair is klt.

The small canonical model. Suppose \(f\) is small. Finite generation of the ordinary relative log canonical ring (Fujino 2022, Theorem 1.18) gives local finite generation of (14). When canonical data are written as sheaves, this application can be made over each Stein open in \(Z\): nonzero sections of rank-one proper direct images provide the meromorphic representatives there. Changing representatives induces compatible graded isomorphisms of the intrinsic algebra.

The ordinary analytic flip theorem (Fujino 2022, Theorem 1.14) constructs precisely its relative canonical model. It has no local factoriality hypothesis. Its proof glues the unique local log canonical models, giving a normal small projective \(f^+:T'\to Z\) globally. The fibres are connected because \(Z\) is normal. A finite cover of the compact base permits one common sufficiently divisible Veronese generated in degree one. Replace \(r\) by that multiple. The resulting tautological bundle is relatively ample and agrees with \(\mathcal O_T(rD)\) on the common big open. Its reflexive extension is \(\mathcal O_{T'}(rD')\), proving (15) as an actual bundle isomorphism. This also proves that \(D'\) is a rational line bundle. The relative Proj is invariant under this Veronese operation. By Lemma 11, \(T'\) is compact Kähler.

For a global prime \(A'\) on \(T'\), transform it to \(A\) on \(T\) and choose \(t\in\mathbb Q\) so \((A+tD)\cdot R=0\). Descend a Cartier multiple by Lemma 9. On \(T'\) its pullback agrees on the common big open with the same multiple of \(A'+tD'\). Reflexivity extends the identification; since \(D'\) is already a rational line bundle, \(A'\) is \(\mathbb Q\)-Cartier. This proves global Weil-divisor factoriality. A given canonical Weil divisor is transported on that common open and then reflexively. In the sheaf formulation, repeat the descent argument with an invertible power of \(\omega_T\) adjusted by \(tD\); undoing the twist by \(tD'\) proves the rational-line-bundle property of \(\omega_{T'}\).

On a common smooth resolution \(p:V\to T\), \(q:V\to T'\), Lemma 5 gives the actual discrepancy divisor \[ p^*D=q^*D'+F,\qquad F\geq0, \tag{17}\] exceptional over both sides, with the asserted strictness over the flipping and flipped loci. This also proves klt preservation.

Optional strong factoriality. In the divisorial case, take any coherent rank-one reflexive sheaf \(\mathcal F\) on \(Z\). Its reflexive pullback to \(T\) is coherent of rank one. Strong factoriality supplies an invertible reflexive power. Adjust it by a rational multiple of \(E\), descend as above, and compare off \(f(E)\). A reflexive power of \(\mathcal F\) is therefore a line bundle.

In the small case start with such a sheaf \(\mathcal F'\) on \(T'\). On a common smooth resolution put \[\mathcal M=(q^*\mathcal F'/\text{torsion})^{**},\qquad \mathcal F=(p_*\mathcal M)^{**}.\] Here \(\mathcal M\) is a line bundle and \(\mathcal F\) is coherent reflexive of rank one; both give the required transform on the common big open. An invertible power of \(\mathcal F\), adjusted by a rational multiple of \(D\), descends. Pull back the descended bundle and undo the twist by \(D'\). After clearing denominators, reflexivity identifies the resulting line bundle with a reflexive power of \(\mathcal F'\). Thus the strong condition is preserved. Only sheaves defined on the whole compact spaces have been used. ◻

Proof. Since \(D\cdot R\ne0\), subtracting a unique multiple of \(c_1(D)\) makes \(\theta\) vanish on \(R\). Equation (12) and pullback injectivity give the unique \(\eta\). On a common resolution over \(Z\) the two pullbacks of \(\eta\) agree. Therefore \[p^*\theta-q^*\theta' =s\,[p^*D-q^*D'].\] The expression in brackets is the actual exceptional discrepancy divisor already constructed in the proof of the proposition.

If \(\theta=c_1(A)\) for a global Weil divisor \(A\), the difference \(p^*A-q^*A'\) is likewise exceptional over \(T'\), where \(A'\) is its transform or pushforward. Subtracting the two identities represents \(q^*(\theta'-c_1(A'))\) by a \(q\)-exceptional divisor. That divisor is numerically trivial over \(q\), so negativity in both signs makes it zero. Pullback injectivity gives \(\theta'=c_1(A')\).

Finally suppose the old trace \(H_T\) of a fixed nef class on an earlier model has, on a common high resolution, the relation \[p^*H_T=p_0^*H+[J_T].\] If \(H_T=f^*\eta+s\,c_1(D)\), the new correction is the actual divisor \[J_{T'}=J_T-s(p^*D-q^*D'),\qquad q^*H_{T'}=p_0^*H+[J_{T'}].\] Every old exceptional divisor remains exceptional over \(T'\): neither kind of step extracts divisors. The resulting correction \(J_{T'}\) is therefore an actual \(q\)-exceptional divisor. Its negative is \(q\)-nef because \(p_0^*H\) is nef. Negativity gives \(J_{T'}\geq0\), and uniqueness and compatibility are those of Lemma 6.

For the stated real-boundary extension, Proposition 25 supplies the ordinary step and its negative-side Bott–Chern rank. The difference \(p^*D-q^*D'\) is an actual exceptional real discrepancy divisor, as shown in Section 6.2. The decomposition, transport formula, and correction argument above therefore apply unchanged. ◻

Proof. The Borel–Moore localization sequence for \(B\subset Y\) contains \[H_{2k}(B,\mathbb R)\longrightarrow H_{2k}(Y,\mathbb R) \xrightarrow{\ j_Y^*\ }H_{2k}^{\mathrm{BM}}(U,\mathbb R) \longrightarrow H_{2k-1}(B,\mathbb R).\] The outside groups vanish: the real dimension of \(B\) is at most \(2k-2\). Thus \(j_Y^*\) is an isomorphism. Compose restriction from \(X\) with its inverse to obtain a linear map \[H_{2k}(X,\mathbb R)\xrightarrow{\ j_X^*\ }H_{2k}^{\mathrm{BM}}(U,\mathbb R) \xrightarrow{\ (j_Y^*)^{-1}\ }H_{2k}(Y,\mathbb R).\] For an irreducible \(k\)-cycle \(S\subset X\) not contained in \(A\), its restriction is the fundamental class of \(S\cap U\). Closing its image in \(Y\) gives its strict transform: analyticity follows by taking the proper image of the corresponding strict transform in the given diagram. The displayed map therefore sends \([S]\) to this transformed cycle class. It sends \([S]\) to zero when \(S\subset A\).

Conversely, no \(k\)-dimensional subspace of \(Y\) is contained in \(B\). Its strict transform in \(X\) consequently maps to its fundamental class. This proves the asserted surjection of cycle spans. If \(S\subset A\) has dimension \(k\), its nonzero class, by (19), is in the kernel. This proves the strict inequality.

For a transformation extracting no prime divisor, choose the common isomorphic open so that the complement in the target has codimension at least two. Every contracted source prime lies in the source complement. Apply the result with \(k=n-1\) at every step. The nonnegative integer \(c_{n-1}\) cannot decrease infinitely often. For a small transformation both complements have codimension at least two, so applying the result in both directions gives equality. ◻

Proof. Let \(L\in H^{1,1}_{\mathrm{BC}}(X)\) be the generalized log class and let \(M_W=[\mathbf M_W]\) be the nef trace class on \(W\). Write the crepant formula as \[ [K_W+\Delta]+M_W=p^*L. \tag{20}\] This is the crepant boundary identity defining generalized discrepancies; it is an equality of Bott–Chern classes when the nef data are transcendental. The boundary \(\Delta\) is an actual real divisor with simple normal crossing support. Since the nef data descend to \(W\), every further generalized discrepancy is the ordinary discrepancy for \((W,\Delta)\). The coefficients of \(\Delta\) need not be nonnegative. We may enlarge its listed simple normal crossing divisor by components of coefficient zero. Here a closed stratum means an irreducible component of an intersection of listed components; it is smooth, and is the closure of its open stratum.

We give the local estimate and the realization argument explicitly; they are the log smooth calculations underlying (Kollár and Mori 1998, Proposition 2.36) and (Chen and Tsakanikas 2023, Proposition 2.8). Write \(\Delta=\sum_j d_jD_j\) and put \(w_j=1-d_j\). Suppose first that every \(w_j\) is positive, and set \[\delta=\min\bigl(\{1\}\cup\{w_j\}_j\bigr)>0.\] Let \(E\) be exceptional over a smooth model with this log smooth boundary. At a general point of its center \(C\), let \(c=\mathop{\mathrm{codim}}C\), and choose local parameters \(x_1,\ldots,x_c\) for \(C\) so that the boundary components containing \(C\) are \(x_1=0,\ldots,x_r=0\). For \(v=\operatorname{ord}_E\), the Jacobian calculation gives \[ a(E;W,\Delta)\geq \sum_{j=1}^{r}(1-d_j)v(x_j)+\sum_{j=r+1}^{c}v(x_j). \tag{21}\] Indeed, on a smooth model containing \(E\), with local parameter \(t\) for \(E\), write \(x_j=t^{v(x_j)}u_j\) at its general point. In a top exterior differential at most one factor can use the term involving \(dt\) that lowers the order by one. Thus the relative Jacobian has order at least \(\sum_{j=1}^c v(x_j)-1\). Adding one and subtracting the boundary orders proves (21). The remaining coordinates along \(C\) are holomorphic and contribute no negative orders.

In particular every discrepancy is at least \(\delta\): for a divisor already on the smooth model it is its listed weight, or one if it is not listed; for an exceptional divisor use (21). If \(C\) is not a stratum of the listed simple normal crossing divisor, then \(c>r\). When \(r>0\), the right hand side is at least \(\delta+1\); when \(r=0\), it is at least \(c\geq 2\geq\delta+1\). Hence \[ C\text{ not a stratum}\quad\Longrightarrow\quad a(E;W,\Delta)\geq 1+\delta. \tag{22}\]

Now fix a place \(E\) with discrepancy strictly below \(1+\delta\). If it is not already a divisor on \(W\), its center is a stratum by (22). Blow up that closed stratum. The center is smooth, the blowup is projective, and the new listed boundary is again simple normal crossing. If the center is the intersection of the components indexed by \(J\), then the new exceptional component has weight \[ w_{\mathrm{new}}=\sum_{j\in J}w_j. \tag{23}\] Thus all weights remain at least \(\delta\), so the same argument applies on the next model. It excludes a non-stratum center at every stage until \(E\) becomes a divisor.

This process is finite. While \(E\) is exceptional, a blowup of its smooth center of codimension \(c\geq2\) changes its ordinary empty-boundary discrepancy by \[a(E;W_{\mathrm{new}},0) =a(E;W_{\mathrm{old}},0)-(c-1)v(\mathcal I_C).\] The subtracted quantity is a positive integer, whereas an ordinary log discrepancy over a smooth space is a positive integer. Consequently only finitely many such blowups are possible. Thus \(E\) is obtained by successive stratum blowups, which is the meaning of a toroidal place here. This also realizes it on a projective model without first assuming that the original model on which \(E\) was given was projective over \(X\).

For completeness, there are only finitely many such toroidal places. A toroidal prime over a fixed irreducible stratum is determined by a primitive nonzero vector \((m_j)_{j\in J}\) of nonnegative integers in the normal boundary directions. Its discrepancy is \[ \sum_{j\in J}m_jw_j. \tag{24}\] These statements follow also directly from the successive stratum blowups: a blowup inserts the sum of the relevant coordinate rays, and the discrepancy transforms by (23). The vector determines the monomial order and is unchanged when a boundary equation is multiplied by a unit. There are finitely many irreducible strata on the compact model. Since each \(w_j\geq\delta\), an upper bound \(T\) on (24) bounds every \(m_j\) by \(T/\delta\). This leaves finitely many vectors and hence finitely many places. Add the finitely many prime divisors already on \(W\) which are exceptional over \(X\).

For a generalized klt pair, all weights in (20) are positive. The preceding argument with \(\epsilon=\delta\) proves (i) and its projective realization assertion. In particular \(a\leq1\) lies strictly below \(1+\delta\).

No rationality of the weights was used: their fixed positive minimum bounds the nonnegative integer vectors. Thus the conclusion applies to real boundaries and real nef data.

For a generalized terminal pair, the coefficient of every \(p\)-exceptional component in \(\Delta\) is negative, since its log discrepancy is greater than one. The other coefficients are the original boundary coefficients and are at most \(b<1\). All the weights on \(W\) are therefore at least \(1-b\). Repeat the argument with \(\delta=1-b\) and \(T=2-b\) to prove (ii). Subsequent stratum blowups preserve this lower bound by (23).

For (iii), the weights on \(W\) are nonnegative. Set \[\delta=\min\bigl(\{1\}\cup\{w_j:w_j>0\}\bigr)>0.\] The coefficient-one components have image in \(X\setminus U\). Thus, at the general center on \(W\) of a place whose center on \(X\) meets \(U\), no zero-weight component occurs. Work there over \(U\). The same estimates and stratum blowups use only positive weights, so give the uniform lower bound and the strict cutoff \(1+\delta\). The strata in question are restrictions of the finitely many global strata, and the blowups can be performed along their smooth closed closures on the compact model. Meeting a zero-weight component elsewhere does not change a weight computed at the general point of the center. This proves finiteness and projective realization also in this case.

Finally, to represent the entire finite set at once, take the dominating component of the fiber product of the finitely many projective realizations over \(W\), normalize, and resolve projectively. The strict transforms of the designated prime divisors remain divisors on this common model. Its composite map to \(X\) is projective. ◻

Proof. Dropping the effective boundary shows that \(T\) has terminal singularities. Suppose its singular locus has a codimension-two component. Near a general smooth point of this component choose a holomorphic projection to \(\mathbb C^2\) submersive along it. Take a general fibre transverse both to the regular locus and, on a fixed log resolution, to the exceptional divisor and its strata. These transversality conditions hold after shrinking and choosing a general value. The resulting surface is a complete intersection in the Cohen–Macaulay space \(T\), since klt singularities are rational, and is regular outside isolated points. It is therefore normal.

Adjunction of a local Cartier pluricanonical power, first on the regular locus and then by reflexivity, restricts the relative canonical equality to the surface resolution. All its exceptional discrepancy coefficients are positive. A normal surface germ with this property is smooth. Indeed, negative definiteness of the exceptional intersection matrix shows that a positive exceptional discrepancy divisor has negative intersection with some exceptional curve. That curve has negative canonical degree; adjunction makes it a smooth rational \((-1)\)-curve. Contract it smoothly and repeat. The remaining discrepancy coefficients stay positive by pushforward. Eventually no exceptional curve remains, and the proper bimeromorphic map to the normal surface is an isomorphism.

On the other hand, a slice of a singular fourfold point by two equations has embedding dimension at least the ambient embedding dimension minus two, and hence greater than two. The chosen surface is singular there, a contradiction. ◻

Proof. Choose a log resolution on which the positive strict boundary components are smooth and mutually disjoint. Starting with simple normal crossing support, blow up intersections of the positive components to separate them. Every exceptional crepant coefficient is negative by terminality. Thus the only positive components on this resolution are those disjoint strict transforms, with their original coefficients.

Consider a place not already on the resolution. At its general centre, apply (21), dropping negative boundary coefficients. A centre of codimension at least three has log discrepancy at least \(3-b>2\), if it lies in a positive component of coefficient \(b\), and at least three otherwise. A codimension-two centre outside the positive support has log discrepancy at least two. Thus a place with \(a<2\) has codimension-two centre in one positive component, and its log discrepancy is at least \(2-b\).

If this centre lies in the exceptional locus, it is an irreducible component of the intersection of that positive component and an exceptional divisor. There are finitely many such centres on the compact resolution. We check that each supports only finitely many places with \(a<2\). It suffices to retain only the smooth component of coefficient \(b\), since deleting the negative coefficients lowers discrepancies. Blow up the fixed centre at its general point. Unless the given place has appeared, the same estimate forces its next centre to be the intersection of the positive strict transform and the new exceptional divisor. This intersection is unique over the general point of the original centre. The successive exceptional log discrepancies are \[1+k(1-b),\qquad k=1,2,\ldots.\] At each further stage the negative coefficient of the newest exceptional divisor adds \(k(1-b)\) to the lower bound \(2-b\). Hence only \(k<1/(1-b)\) can occur before the cutoff two is reached. This proves finiteness over every fixed exceptional centre. These generic blowups are realized globally by blowing up the closures of the centres.

Add the finitely many exceptional primes on the chosen resolution. All other centres under consideration lie outside its exceptional locus at their general points, where the resolution is an isomorphism. Their images are precisely the surfaces described in the statement. ◻

Proof. By Lemma 5, discrepancies do not decrease, and they increase strictly for a place centred in either exceptional locus. Since the maps are small, the exceptional places are the same on all models. Thus every pair remains terminal in the stated sense, and its ambient fourfold is smooth in codimension two by Lemma 18.

Write the distinct positive coefficients of the boundary in decreasing order. We successively remove, by discarding finitely many initial steps, every flipping and flipped surface contained in a component of each coefficient. After the positive coefficients we treat the value zero, at which stage every flipped surface is considered.

Eliminating flipped surfaces. Fix the current coefficient \(b\), and suppose that no flipping or flipped surface is contained in a component of coefficient greater than \(b\). For \(b=0\) assume that all positive coefficients have already been treated. If \(S\) is a flipped surface contained in a coefficient-\(b\) component, or any flipped surface when \(b=0\), blow it up at its general point. The exceptional prime \(E\) has new log discrepancy \[ a(E;T_{i+1},\Phi_{i+1}) =2-\sum_{l=1}^{s}b_l\operatorname{mult}_{S}\Phi_{l,i+1}, \qquad 1<a(E;T_{i+1},\Phi_{i+1})\leq2-b, \tag{25}\] where \(\Phi_{l,i+1}\) are the positive prime boundary components and \(b_l\) their fixed coefficients. The blowup place is defined globally by blowing up the closed surface and resolving; its generic point suffices for this computation.

The values in (25) belong to a finite set, even for irrational coefficients. Indeed, terminality gives \[\sum_l b_l\operatorname{mult}_{S}\Phi_{l,i+1}<1, \qquad 0\leq\operatorname{mult}_{S}\Phi_{l,i+1}<1/b_l.\] Each multiplicity is an integer, and there are finitely many positive components. We use no discreteness assertion for all discrepancies.

Strictness in Lemma 5 gives \(a(E;T_i,\Phi_i)<2-b\). Its discrepancy on the first model of the current tail is therefore also strictly below \(2-b\). By Lemma 19, outside a finite set such a place would initially have a surface centre in one boundary component of coefficient \(c\), with discrepancy at least \(2-c\). Necessarily \(c>b\). That centre persists as a surface in this component at every later step: at each step the map is an isomorphism at its general point, since otherwise the centre itself would be a flipping surface in a higher-coefficient component. It cannot become a flipped surface either, by the same higher-coefficient exclusion. Thus it cannot be the place just constructed. All these witness places belong to a fixed finite set.

Each member of that finite set is used only finitely often. Between uses its discrepancy does not decrease, and at each use it strictly increases to one of the finitely many values (25). Consequently only finitely many steps have a flipped surface of the current kind. Pass beyond them.

Eliminating flipping surfaces inside the boundary. Suppose \(b>0\). Fix a coefficient-\(b\) component and normalize its transforms on the two sides of a flip. Denote the normalizations by \(D_i^\nu,D_{i+1}^\nu\), and normalize their common image in \(Z_i\) to obtain \(B_i\). The induced maps to \(B_i\) are proper and bimeromorphic. They are isomorphisms away from the inverse image of the common exceptional image in \(Z_i\), whose dimension is at most one by Lemma 5. Finite normalization preserves this bound. On \(D_{i+1}^\nu\) its inverse image also has dimension at most one, because no flipped surface is contained in the boundary component.

Every compact surface in \(B_i\) has a strict transform on \(D_i^\nu\), so pushforward gives a surjection \[\mathcal C_2(D_i^\nu)\longrightarrow\mathcal C_2(B_i).\] On the other side, Borel–Moore localization along the exceptional subset of dimension at most one makes restriction in degree four injective. Compatibility with proper pushforward and the common isomorphic open therefore gives an injection \[\mathcal C_2(D_{i+1}^\nu)\lhook\joinrel\longrightarrow \mathcal C_2(B_i).\] These are assertions about surface spans; no surjectivity on the full fourth homology of the negative side is needed.

If a flipping surface lies in this boundary component, a surface above it on \(D_i^\nu\) maps to a set of dimension at most one in \(B_i\). Its class is nonzero: its image in \(T_i\) is a surface, and the square of a Kähler class on \(T_i\) has strictly positive integral over it. One may compute this integral on a resolution of the surface. The finite normalization has positive generic degree on this surface, so the same nonvanishing holds upstairs. Its class is a nonzero kernel vector in the displayed surjection. Hence \[c_2(D_i^\nu)\geq c_2(D_{i+1}^\nu),\] with strict inequality whenever the component contains a flipping surface. Summing over the finitely many coefficient-\(b\) components shows that only finitely many such steps occur. This completes the induction at \(b\).

The remaining flips. After the positive coefficients have been treated, perform the flipped-surface argument with \(b=0\). No flipped surface remains. For every remaining flip, Lemma 5 gives \[\dim\mathop{\mathrm{Exc}}(f_i)+\dim\mathop{\mathrm{Exc}}(f_i^+)\geq3.\] Smallness bounds both dimensions by two, so \(\mathop{\mathrm{Exc}}(f_i)\) contains a surface and \(\dim\mathop{\mathrm{Exc}}(f_i^+)\leq1\). Apply Lemma 15 with \(k=2\) to the common isomorphic open. It gives \(c_2(T_i)>c_2(T_{i+1})\) at every remaining step, impossible for an infinite sequence of nonnegative integers. ◻

Proof. For flips, terminality persists by Lemma 5. For a divisorial contraction, (13) gives \(K_{X_i}=f^*K_{X_{i+1}}+eE\) with \(e>0\). The lost prime has target log discrepancy \(1+e>1\); every other exceptional place has target discrepancy at least its source discrepancy, which was greater than one. Thus terminality persists throughout the sequence. Lemma 15 with \(k=3\) bounds the number of divisorial steps. Any infinite sequence would therefore have a tail consisting only of flips, contrary to Proposition 20 with zero boundary. ◻

Proof. On \(X_0\), the nef datum descends, so its addition changes no discrepancies. Thus all the initial generalized pairs are gklt. Suppose the assertions have been established through \(X_i\). If \(D_i\) is nef, stop. Otherwise define \[\lambda_i=\min\{t\in[0,\lambda_{i-1}]:L_i(t)\text{ is nef}\}.\] The set is nonempty by induction and is closed and convex. Its minimum is positive because \(D_i\) is not nef.

We first find a ray at this threshold. Choose \(t_k\uparrow\lambda_i\) with \(\lambda_i/2<t_k<\lambda_i\). The generalized cone theorem gives an extremal ray negative for \(L_i(t_k)\). Since \(L_i(\lambda_i)\) is nef, that ray is also negative for \(L_i(\lambda_i/2)\). The trace \(H_i\) is big by Corollary 7. Hence \(\Delta_i+(\lambda_i/2)H_i\) is big, and the cone theorem gives only finitely many negative rays for this latter generalized pair (C. Hacon and Xie 2026, Theorem 1.3). One ray \(R_i\) therefore occurs for infinitely many \(k\). On this fixed ray, continuity gives \(L_i(\lambda_i)\cdot R_i=0\). Moreover \(H_i\cdot R_i>0\), so \(D_i\cdot R_i<0\).

Apply Proposition 12 to this ordinary negative ray, and denote its contraction by \(f_i:X_i\to Z_i\). A fibre contraction gives the stopping alternative. Otherwise make the ordinary divisorial step or flip. Lemma 13 transports the trace \(H_i\) to \(H_{i+1}\) while preserving its relation to the fixed b-datum. By Lemma 3, the class \(L_i(\lambda_i)\) descends to \(Z_i\). Its descended class is nef, and its pullback on the new model is \(L_{i+1}(\lambda_i)\). Thus the two threshold classes have equal pullbacks on a common resolution. Their difference is represented by the actual exceptional discrepancy divisor, which vanishes by negativity as in Lemma 8. The step is consequently generalized crepant at the threshold.

For completeness, this also proves the induction on singularities. On a common resolution, the log discrepancy at each prime \(E\) is affine in \(t\). Its difference across the step is zero at \(t=\lambda_i\) and is the ordinary discrepancy increase at zero. Consequently \[\begin{align*} &a(E,X_{i+1},\Delta_{i+1}+t\mathbf H) -a(E,X_i,\Delta_i+t\mathbf H)\\ &\qquad=\left(1-\frac{t}{\lambda_i}\right) \bigl(a(E,X_{i+1},\Delta_{i+1})-a(E,X_i,\Delta_i)\bigr) \geq0 \tag{26}\end{align*}\] for \(0\leq t\leq\lambda_i\). The generalized pairs on this interval remain gklt, and the new upper-threshold class is nef. The ordinary steps preserve all the stated properties of the original category by Proposition 12. ◻

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

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

Proof. Put \(D=K_T+B\) and fix a \(D\)-negative extremal ray \(R\). On the finite positive support of \(B\), effectiveness, the klt condition and negativity on \(R\) persist under small coefficient changes. Klt openness is checked on one log resolution of this support. Choose nearby effective rational klt boundaries \(B_0,\ldots,B_m\) and positive real numbers \(u_j\) with \[B=\sum_j u_jB_j,\qquad \sum_j u_j=1, \qquad D_j\cdot R<0,\quad D_j=K_T+B_j.\] For \(B=0\) take just \(B_0=0\). Every \(D_j\) is a global rational line bundle. The rational-step theorem for \((T,B_0)\) provides a projective contraction \(f:T\to Z\), a compact Kähler base, and \[f^*H^{1,1}_{\mathrm{BC}}(Z)=R^\perp,\qquad \rho(T/Z)=1, \qquad -D_0\text{ relatively ample}.\] Let \(C\) be a rational curve spanning \(R\), and set \(a_j=(D_j\cdot C)/(D_0\cdot C)\in\mathbb Q_{>0}\). On every projective fibre, \(-D_j\) is numerically equivalent to the positive multiple \(-a_jD_0\). Numerical invariance of ampleness on that fibre, followed by the relative-ampleness criterion, makes \(-D_j\) relatively ample. The positive combination \(-D\) is relatively ample as a real line bundle. This also handles a fibre-type contraction.

Suppose henceforth that \(f\) is birational. A Cartier multiple \(M_j=n_j(D_j-a_jD_0)\) is numerically trivial over \(Z\) and \(M_j-D_0\) is relatively nef. Integral descent, Lemma 9, applied to the rational klt pair \((T,B_0)\) gives a rational line bundle \(A_j\) on \(Z\) with the actual identity \[ D_j=a_jD_0+f^*A_j. \tag{29}\] Here and below an identity of rational line bundles means an isomorphism after a common integral multiple. The descent lemma is used only for birational \(f\).

For a small \(f\), let \(f^+:T^+\to Z\) be its rational \(D_0\)-flip. The rational theorem supplies a global relatively ample adjoint \(D_0^+\) and preserves the stated factoriality and canonical-sheaf conditions. The other transformed adjoints \(D_j^+\) are therefore rational line bundles. Transform (29) on the common big open and extend by reflexivity to obtain \[D_j^+=a_jD_0^++(f^+)^*A_j.\] All \(D_j^+\) are relatively ample, as is \(D^+=\sum_j u_jD_j^+\). Thus this same small map is the required ordinary real-boundary flip. The real ordinary discrepancy comparison proves that \((T^+,B^+)\) is klt and gives the strict increases at exceptional centres.

For a divisorial contraction, write \(D_0=f^*D_0'+e_0E\) with \(e_0>0\). Transforming (29) on the target and summing yields \[D-f^*D'=\left(\sum_j u_ja_j\right)e_0E.\] This coefficient is positive. The same discrepancy comparison proves the required assertions. The ambient category was already preserved by the rational step in both cases. ◻

Proof. Let \(V\) be an irreducible positive-dimensional subspace of a reduced fibre. It is Moishezon. Normalize \(V\) and take a smooth projective modification \(\pi:\widetilde V\to V^{\mathrm n}\), chosen so that an effective exceptional divisor \(E\) has \(-E\) relatively ample. The pullback of the Kähler class from \(V\) to its normalization is Kähler. Consequently \(\pi^*\omega-\delta[E]\) is Kähler for sufficiently small \(\delta>0\). On the projective manifold \(\widetilde V\) the actual real line bundle \(\pi^*L-\delta E\) has the same curve degrees as that Kähler class, and is therefore ample. For example, subtract a sufficiently small positive multiple of a fixed ample class from the Kähler class; the corresponding real line bundle is nef by the projective numerical criterion. Also \(\pi^*L\) is nef, and the decomposition \[\pi^*L=(\pi^*L-\delta E)+\delta E\] shows that it is big. Thus \[L^{\dim V}\cdot V=(\pi^*L)^{\dim V}>0.\] The real Nakai–Moishezon criterion for proper algebraic spaces (Fujino and Miyamoto 2020, Theorem 1.6), applied to the algebraizations of the Moishezon fibres, proves that \(L\) is ample on every reduced fibre. The criterion is insensitive to nilpotents and reducible components. Fibrewise ampleness is open for a proper analytic map; in the finite-dimensional span of the finitely many line bundles occurring in \(L\), compactness therefore gives a neighbourhood of \(L\) whose rational points are relatively ample. This proves relative ampleness of \(L\) and supplies a global relatively ample rational line bundle. Clearing its denominator proves projectivity. ◻

Proof. The inputs are the cone theorem, projective relative vanishing and minimal-model theory, projective small factorializations and dlt modifications, and the nef-and-big Kähler criterion from (C. Hacon and Xie 2026; Hacon et al. 2026). The higher-carrier canonical bundle formula (Hacon and Păun 2024, Theorem 2.3) is applied to globally gklt pairs; (Hacon and Păun 2024, Theorem 2.2) supplies the relative positivity used in Lemma 42. Each application below starts with a projective morphism.

The big non-klt contraction. Apply the construction of (C. Hacon and Xie 2026, sec. 3). On its smooth modification \(\nu:U\to X\) it writes \[\nu^*\alpha=[D_U]+\eta_U, \qquad D_U\geq0,\quad \eta_U\text{ K\"ahler}.\] The induction over the jumping coefficients of the multiplier ideal reduces the new reduced stratum to dimension at most \(d-1\). The given contraction on the old non-klt subspace and \(\mathsf C_{d-1}\) provide the Moishezon maps of those strata. Every map that is promoted to a projective map in that construction contracts exactly the curves on which \(\nu^*\alpha\) vanishes. On each such curve the actual real line bundle \(-D_U\) has degree \(\eta_U\cdot C\). On a further projective common resolution \(q:U''\to U\), choose an effective exceptional divisor \(E\) with \(-E\) relatively ample and choose \(\varepsilon>0\) sufficiently small. Use the decomposition \[q^*\nu^*\alpha=[D'']+\eta'',\qquad D''=q^*D_U+\varepsilon E,\qquad \eta''=q^*\eta_U-\varepsilon[E].\] Here \(\eta''\) is Kähler, and the actual real line bundle \(-D''\) has degree \(\eta''\cdot C\) on every contracted curve. Restrict this decomposition to the smooth reduced components in the gluing construction. Lemma 26 then supplies precisely the relative ampleness needed in place of (C. Hacon and Xie 2026, Lemma 2.42), including the common-resolution step in its Claim 3.5. The exceptional correction \(-\varepsilon[E]\) makes the positive part \(\eta^{\prime\prime}\) Kähler on the common resolution.

The exact sequences of multiplier ideals, relative vanishing, finite pushouts, and extension from sufficiently thickened \(\operatorname{Supp}D_U\) in that proof then apply unchanged. They give the birational Moishezon contraction in \(\mathsf C_{d,\mathrm{big}}\). In particular the gluing step uses neither generalized termination nor a projectivity criterion for an undetected ray.

Preparation for both cases of \(\mathsf B_d\). The modified-big perturbation (C. Hacon and Xie 2026, Lemma 2.23) supplies a nef datum that dominates a Kähler class on its carrier. Subtracting a sufficiently small multiple of the pullback of a Kähler class \(\omega_X\) still leaves a nef datum on that carrier. Thus \(\alpha\) is a gklt adjoint plus \(\varepsilon\omega_X\), and the cone argument of (C. Hacon and Xie 2026, Lemma 2.45) makes \(\alpha\) NQC. This preparation applies whether or not \(\alpha\) is big.

The nef-and-big case of \(\mathsf B_d\). For a gklt pair the multiplier ideal is the unit ideal, so the preceding big non-klt assertion now applies and gives a birational contraction \(h:X\to Y\). On a projective resolution \(p:W\to X\) chosen projective over \(Y\), relative Kawamata–Viehweg vanishing for the effective exceptional divisor \(-\lfloor B_W\rfloor\) gives \(R^ih_*\mathcal O_X=0\) for \(i>0\). Thus \(Y\) has rational singularities. Proper birational Bott–Chern descent (Das et al. 2024, Lemma 8.7) gives \(\alpha=h^*\gamma\). The descended adjoint is gklt, nef and big, and has no trivial curve. The criterion (Hacon et al. 2026, Theorem 4.3) makes \(\gamma\) Kähler.

For completeness, the projectivity input in that criterion is restricted to a map \(S\to Z'\) between smooth compact Kähler manifolds with rationally connected general fibre: \(S\) is the chosen smooth divisor on a resolution and \(Z'\) is the resolution of the null-locus component. The smooth-source, smooth-base argument in (Claudon and Höring 2024, Theorem 3.1, Step 1) applies. Its subsequent relative program starts with this projective map. Thus no general singular-source projectivity assertion is needed for this application of the criterion.

The non-big case of \(\mathsf B_d\). Use a projective small factorialization and the modified-big perturbation of the pair. The non-pseudo-effectivity of \(K_X\) gives an MRC fibration by (Ou 2025). Choose a smooth Kähler MRC base \(Z\) and a smooth modification \(\mu:U\to X\) such that \(U\to Z\) is projective. Only the smooth case of (Claudon and Höring 2024, Theorem 3.1, Step 1) is needed: pullback identifies the holomorphic two-forms since the general fibre is rationally connected. The same rational Hodge correction identifies any \((1,1)\)-class on \(U\), modulo a class pulled back from \(Z\), with an actual real line bundle.

Write, as in (C. Hacon and Xie 2026, Theorem 4.1, Claim 4.1), \[D_U(a)=K_U+\Delta_U+\mathbf M_U+a\alpha_U =(a+1)\alpha_U+F_U, \quad \alpha_U=\mu^*\alpha,\quad \Delta_U,F_U\geq0,\] where \(\Delta_U\) is klt and \(F_U\) is \(\mu\)-exceptional. The modified-big replacement is chosen, as in the cited Claim 4.1, so that the nef datum \(\mathbf M_U\) on this smooth carrier is Kähler. If a further projective resolution is needed, subtract a sufficiently small effective exceptional divisor from its nef part and add that divisor to the subboundary; this preserves the adjoint and the klt inequalities. Choose \(a_0\) using the uniform bound of Proposition 30. It makes the relative \(D_U(a_0)\)-program \(\alpha_U\)-trivial and preserves that choice through every step. This is a projective relative program from its first step: the nef data are represented by real line bundles modulo \(Z\). Lemma 28 applies on every intermediate model over this fixed smooth base. More precisely, put \(r:U\to Z\) and choose one actual global real line bundle \(L\) with \[\mathbf M_U+a_0\alpha_U=c_1(L)+r^*\gamma.\] The bundle \(L\) is relatively ample, because \(\mathbf M_U\) is Kähler and \(\alpha_U\) is nef. On each of a fixed finite collection of relatively compact Stein charts of \(Z\), choose an effective real divisor \(\Theta\) representing \(L\), with \((U,\Delta_U+\Theta)\) klt. The representatives can retain a positive relatively ample part in their boundaries. Here the equality \(K_U+\Delta_U+\Theta\sim_{\mathbb R}K_U+\Delta_U+L\) is an actual real linear equivalence of line bundles; the equality with \(D_U(a_0)\) modulo \(r^*\gamma\) is separately an equality of Bott–Chern classes.

For the ample-model comparison below, make the following additional choices before running the program. Take the fixed finite cover in the form \(W_\ell\Subset Y_\ell\subset Z\), where \(Y_\ell\) is a Stein coordinate chart and \(W_\ell\) is a closed coordinate polydisc, with the interiors of the \(W_\ell\) covering \(Z\). Each \(W_\ell\) is a semianalytic Stein compact, so it satisfies condition (P4) of (Fujino 2022); thus condition (P) holds for the restriction of every normal projective model over \(Z\) to \(Y_\ell\).

Write \[\alpha_i=c_1(N_i)+f_i^*\Gamma\] for the real line bundles \(N_i\) transported in Proposition 30; put \(N_i=0\) if \(\alpha_U=0\). These bundles are nef over \(Z\) and pull back from each contraction target, on both sides of a flip. Set \(a_j=a_0+j\) for \(j=1,2\). The real line bundles \(L+jN_U\) are relatively ample. On each initial chart choose, once and for all, \[\Theta_j\geq0,\qquad \Theta_j\sim_{\mathbb R}L+jN_U, \qquad (U,\Xi_{j,U}:=\Delta_U+\Theta_j)\ \text{klt}.\] Such representatives are obtained from general sufficiently divisible relative sections; their boundaries are relatively big because \(L+jN_U\) is relatively ample and \(\Delta_U\geq0\). With \(\Xi_{0,U}:=\Delta_U+\Theta\), these are actual real linear equivalences \[K_U+\Xi_{j,U}\sim_{\mathbb R}K_U+\Xi_{0,U}+jN_U\] on each chart, separately from the Bott–Chern comparison with the fixed base classes.

Let \(\Xi_{j,i}\) denote their birational transforms. Pushforward of these linear equivalences and the actual descent of \(N_i\) give \[K_{U_i}+\Xi_{j,i} \sim_{\mathbb R}K_{U_i}+\Xi_{0,i}+jN_i\] at every step. Thus the contracted adjoints have the same degrees on the negative side, and the flipped adjoints have the same degrees on the positive side. Every step of the chosen program is therefore also a \((K_{U_i}+\Xi_{j,i})\)-negative step. Ordinary discrepancy negativity preserves klt for these fixed pairs. Their boundaries remain effective, and relative bigness is preserved by these birational pushforwards, as in the proof of (Fujino 2022, Lemma 11.16).

Fix these ordinary pairs on the initial charts. Their actual linear equivalences push forward throughout the program. On a common resolution at each step, exceptional negativity preserves the Bott–Chern comparison with \(D_U(a_0)\) modulo the fixed base class. Thus every subsequent contraction or flip is a step for these fixed ordinary pairs. The projective ordinary theorems (Fujino 2022, 2026a) apply. In particular, the finiteness of weak log canonical models in (Fujino 2022, Theorem E) does not require local \(\mathbb Q\)-factoriality of each intermediate model. Applied on the finite chart cover, its chamber argument gives termination of the global projective program with scaling. Each small step is constructed by its global rational detector algebra as above; uniqueness of the relatively ample model identifies the local ordinary flips on overlaps. This supplies a good minimal model \(U\dasharrow V\) over \(Z\). Here one uses the projective case of (C. Hacon and Xie 2026, Proposition 2.30), not its general part (3).

Here is an explicit ample-model comparison that gives the required descent of \(\alpha_V\). Both \(D_V(a_0)\) and \(\alpha_V\) are nef over \(Z\). For \(j=1,2\), the ordinary adjoint \(K_V+\Xi_{j,V}\) is \(\mathbb R\)-Cartier and represents \[D_V(a_j)-f_V^*(\gamma+j\Gamma) =D_V(a_0)+j\alpha_V-f_V^*(\gamma+j\Gamma).\] It is consequently nef over every \(W_\ell\). The pair \((V,\Xi_{j,V})\) is klt and its effective boundary is relatively big. Apply (Fujino 2022, Lemma 11.15) with \(A=\Xi_{j,V}\) and \(B=0\). Its condition on non-klt centres is vacuous. After shrinking around \(W_\ell\), it gives \[\Xi_{j,V}\sim_{\mathbb R}A_j+B_j,\qquad A_j\geq0\ \text{relatively ample},\quad B_j\geq0, \qquad (V,A_j+B_j)\ \text{klt}.\] In particular this pair is log canonical, supplies the klt boundary required in (Fujino 2022, Theorem 8.3), and its adjoint is \(\mathbb R\)-Cartier and nef over \(W_\ell\). That theorem makes \(K_V+\Xi_{j,V}\) semiample on a neighbourhood of \(W_\ell\). The pushed-forward actual linear equivalences identify these local adjoints with restrictions of the same global real line-bundle data. Their ample models therefore agree on overlaps by uniqueness and glue as above. Hence \(D_V(a_1)\) and \(D_V(a_2)\) have projective relative ample-model contractions over \(Z\). Their zero curves are exactly the common zero curves of \(D_V(a_0)\) and \(\alpha_V\). Their projective ample-model contractions therefore have the same fibres: these fibres are connected by curves, and each contraction is constant on the fibres of the other. Identify the two normal targets, writing the common contraction as \(g:V\to S\). If \(D_V(a_j)=g^*\lambda_j\), with \(\lambda_j\) relatively Kähler over \(Z\), subtraction gives \[\alpha_V=g^*(\lambda_2-\lambda_1)=g^*\alpha_S.\] This is an equality of Bott–Chern classes, not merely of curve degrees. Since the initial program is \(\alpha_U\)-trivial, it is also \(D_U(a_1)\)-negative. From now on put \(a=a_1\) and suppress the argument in \(D_U(a)\) and \(D_V(a)\).

The maps \(V\to Z\) and \(S\to Z\) are projective. Moreover \(g\) is projective: a relatively ample bundle for \(V\to Z\) restricts to an ample bundle on every fibre of \(g\), and hence is \(g\)-ample. The argument of (C. Hacon and Xie 2026, Theorem 4.1, Claim 4.2), using projective relative semipositivity, gives \(\dim S<d\). The class \(D_V\) is the adjoint of a gklt generalized pair with effective trace boundary. Its nef datum on a common carrier is the pullback of \(\mathbf M_U+a\alpha_U\), hence is nef and big. The ordinary relative steps preserve the generalized klt condition by their negative discrepancy comparison, and preserve the nef b-datum. The space \(V\) is globally strongly \(\mathbb Q\)-factorial. Let \(F_V\) be the effective transform of \(F_U\) and put \(c=a+1\). On a common resolution \(p:W\to U\), \(q:W\to V\), \[p^*D_U=q^*D_V+[L],\qquad L\geq0, \qquad p^*\alpha_U=q^*\alpha_V,\] where \(L\) is \(q\)-exceptional. The divisor \(p^*F_U-L-q^*F_V\) is exceptional over \(V\) and numerically trivial over \(V\); applying negativity in both directions makes it zero. Consequently \(D_V=c\alpha_V+[F_V]\), with \(F_V\) real Cartier. Put \(D_S=\lambda_1\), so \(D_V=g^*D_S\) is the exact Bott–Chern pullback furnished by the relative ample model. The class \(\alpha_S\) is nef by descent through the surjective morphism \(g\).

Lemma 39 shows that \(D_V-c_1(K_V)\) is big. Apply its fixed-adjoint replacement to the pair on \(V\), and choose a Kähler class \(\omega_S\) and \(\varepsilon>0\) so small that the pullback of \(\varepsilon\omega_S\) can be subtracted from the Kähler nef datum. The resulting pair is globally gklt, its trace boundary is effective, and its adjoint is \(g^*(D_S-\varepsilon\omega_S)\). On a projective log resolution \(r:P\to V\) its structure equation is \[[K_P+B_P]+\mathbf M_P^{\rm new} =(gr)^*(D_S-\varepsilon\omega_S), \qquad \mathbf M_P^{\rm new}\ \text{nef}.\] All hypotheses of (Hacon and Păun 2024, Theorem 2.3) now hold: \(g\) is a projective contraction of normal compact Kähler spaces, the pair is globally generalized klt, and the adjoint equality is a Bott–Chern pullback. The theorem gives generalized klt data on \(S\) with effective trace boundary and adjoint exactly \(D_S-\varepsilon\omega_S\). On a sufficiently high smooth carrier \(u:S_1\to S\) these satisfy \[u^*(D_S-\varepsilon\omega_S) =[K_{S_1}+B_{S_1}]+\mathbf N_{S_1}, \qquad \mathbf N_{S_1}\ \text{nef}.\] The subboundary \(B_{S_1}\) can have negative coefficients. Add \(\varepsilon\overline{\omega_S}\) to the nef b-datum. This restores the exact adjoint \(D_S\) and preserves every boundary and discrepancy. Its carrier trace \(\mathbf N_{S_1}+\varepsilon u^*\omega_S\) is nef and big. Consequently the effective trace boundary plus nef trace on \(S\) is modified big: lift the effective boundary by strict transform to a common carrier and push down the resulting big class. The nef datum throughout is a datum on the carrier.

Take a projective small strong \(\mathbb Q\)-factorialization \(s:S^{\mathrm q}\to S\) and resolve the main component of \(V\times_S S^{\mathrm q}\). The resulting maps \(\pi:\widetilde V\to V\) and \(\widetilde g:\widetilde V\to S^{\mathrm q}\) are projective, and \(\widetilde V\) is compact Kähler. Pull back \(D_V\), \(\alpha_V\) and \(F_V\) to \(\widetilde V\), and \(D_S\) and \(\alpha_S\) to \(S^{\mathrm q}\). The pair on \(S^{\mathrm q}\) is gklt and its boundary-plus-nef class remains modified big: smallness preserves the effective boundary trace, and on a common carrier the nef datum is still nef and big. Lemma 34 first gives \(N_\sigma(D_V)=F_V\). Lemma 37 then gives \(N_\sigma(\pi^*D_V)=\pi^*F_V\). The new total space is used only as a carrier for this equality and for pullbacks of positive currents; no effective boundary on it is needed. Rename these spaces \(V\) and \(S\). No extra relative program from Claim 4.4 of the cited proof is needed.

The vertical-divisor descent and negative-part lemmas above now give actual effective divisors \(F_S,F_V\) with \[F_V=g^*F_S,\qquad D_S=(a+1)\alpha_S+[F_S],\qquad N_\sigma(D_S)=F_S.\] Apply \(\mathsf G_{d-1}\) to the generalized pair with adjoint \(D_S\). On a common resolution \(p:W\to S\), \(q:W\to S_m\) of the resulting good model, write \[p^*D_S=q^*D_m+[E],\qquad E\geq0 \text{ exceptional over }S_m.\] The negative-part comparison proved above gives \(E=p^*F_S\), and hence \[q^*D_m=(a+1)p^*\alpha_S.\] Since \(D_m\) is semiample, \(\alpha_S\) is semiample after descent through \(p\). Pulling back by \(g\) and using the \(\alpha\)-trivial initial program gives semiampleness of \(\alpha\) on \(X\). The target is compact Kähler, and the resulting map is Moishezon, as follows by taking common projective modifications of the maps just constructed. The holomorphic descent and the Moishezon witness are those of Proposition 38.

Selected good models in dimensions at most three. Lemma 33 gives \(\mathsf G_d\) from \(\mathsf B_d\) and ordinary rational termination. Its smooth input has the form \[\mu^*A+[E_\mu]=K_U+B_q+H, \qquad B_q\text{ rational klt},\quad H\text{ K\"ahler},\] where \(E_\mu\) is effective with all \(\mu\)-exceptional divisors in its support. The ordinary \((K_U+B_q)\)-program with scaling of \(H\) is stopped at the first threshold at most one. Every performed step is negative for the displayed adjoint. Termination is the ordinary specialization of Theorem 67, with zero nef b-divisor. Its proof uses the projective relative and discrepancy arguments of Section 10, not \(\mathsf B_d\) or \(\mathsf G_d\). The final nef adjoint is semiample by \(\mathsf B_d\). Negativity removes the added exceptional divisors and gives the claimed model of \(X\). All forward transforms used here are those of the detected ordinary steps; no inverse identification of Bott–Chern groups is needed.

The non-big non-klt contraction in dimensions at most three. Assume \(\mathsf B_d\), \(\mathsf C_{d-1}\) and termination of existing ordinary rational klt programs in dimensions at most \(d\), where \(2\leq d\leq3\). The big case having been proved, let \(\alpha\) be non-big. Take a projective dlt modification and a projective small strong factorialization, as in (C. Hacon and Xie 2026, Theorems 2.26 and 2.27). Denote their composite by \(p:T\to X\) and use the full crepant generalized boundary on \(T\). The underlying space \(T\) is klt, compact Kähler and globally strongly \(\mathbb Q\)-factorial. The crepant boundary is effective: its nonexceptional coefficients are the original ones, and each extracted prime has log discrepancy at most zero. The pulled-back class \(\alpha_T\) remains nef, weakly NQC and non-big. Lemma 41 transports the EMC hypothesis on the multiplier-ideal closed subspace to \(T\).

We verify the bigness needed for the ordinary program. On \(T\) put \[Q_T=\alpha_T-c_1(K_T)=B_T+\mathbf M_T.\] The modified-bigness resolution lemma (C. Hacon and Xie 2026, Lemma 2.10) gives bigness after adding an effective exceptional divisor \(E\) whose support contains all modification exceptional primes. Each of these primes already occurs in \(B_T\) with positive coefficient. Since \(\mathbf M_T\) is pseudo-effective by Lemma 39, for sufficiently small \(s>0\) the identity \[Q_T=s(Q_T+[E])+(1-s)\mathbf M_T+ [(1-s)B_T-sE]\] expresses \(Q_T\) as a big class plus a pseudo-effective class. For the small factorialization alone the exceptional divisor is empty. Hence \(Q_T\) is big.

Lemma 32 gives a finite ordinary \(K\)-negative, \(\alpha_T\)-trivial program ending in a projective Mori fibre contraction \[f:Y\to Z.\] Its class descends in Bott–Chern cohomology: \(\alpha_Y=f^*\gamma\), with \(\gamma\) nef and weakly NQC on the normal compact Kähler base. Transport the generalized nef b-datum and the full crepant boundary along the program. On common resolutions the adjoints and structure subboundaries agree; since no step extracts a divisor, the boundary remains effective. The ordinary discrepancy comparison gives \[q^*(\alpha_{i+1}-c_1(K_{Y_{i+1}})) =p^*(\alpha_i-c_1(K_{Y_i}))+[F_i],\qquad F_i\geq0,\] so \(Q_Y=\alpha_Y-c_1(K_Y)\) remains big. Lemmas 40 and 41 transport both the EMC hypothesis on the multiplier-ideal scheme and, once obtained, the EMC conclusion across these projective comparisons. It therefore suffices to prove EMC for \(\gamma\) on \(Z\).

If \(\dim Z\leq1\), this follows from degree on a normal compact curve, using the identity for positive degree and the map to a point for degree zero. The only remaining case is \(\dim Y=3\), \(\dim Z=2\). The auxiliary application of (Hacon and Păun 2024, Theorem 2.3) in Lemma 42 shows that \(Z\) is klt and globally strongly \(\mathbb Q\)-factorial.

Apply Lemma 39 on \(Y\) with adjoint \(\alpha_Y\). The multiplier ideal can only increase, so its new closed subspace \(N\) is a closed subspace of the previous one. It has EMC for \(\alpha_Y|_N\) by Lemma 40. On a projective log resolution \(r:W_0\to Y\) the new data satisfy \[[K_{W_0}+B_{W_0}]+\mathbf M_{W_0}=(fr)^*\gamma, \qquad \mathbf M_{W_0}\ \text{K\"ahler}.\]

Suppose first that \(N\) dominates \(Z\). Write \(\mathcal J=r_*\mathcal O_{W_0}(-\lfloor B_{W_0}\rfloor)\). Relative Kawamata–Viehweg vanishing for \(fr\) gives \[R^1(fr)_*\mathcal O_{W_0}(-\lfloor B_{W_0}\rfloor)=0:\] the difference with \(K_{W_0}+\{B_{W_0}\}\) is the relatively nef and big class \(\mathbf M_{W_0}-(fr)^*\gamma\). The low-degree Leray sequence implies \(R^1f_*\mathcal J=0\). The exact sequence of the actual multiplier-ideal scheme therefore gives a surjection \[\mathcal O_Z\longrightarrow f_*\mathcal O_N.\] It is injective because \(Z\) is reduced and \(N\) dominates it. Thus \(N\to Z\) is a projective contraction, including on structure sheaves. EMC on \(N\) descends to \(Z\) by Lemma 40.

Suppose next that \(N\) does not dominate \(Z\). The pair is gklt over a nonempty open subset of \(Z\). Lemma 42, applied to these data, gives \[\gamma-c_1(K_Z)\ \text{big}.\] If \(\gamma\) is not big, apply Lemma 32 in dimension two to \((Z,\gamma)\). Its klt, factoriality, nefness, NQC and bigness hypotheses have all been verified; \(\mathsf B_2\) and termination of existing ordinary surface programs are available. Its class-trivial Mori contraction has base of dimension at most one. EMC there, followed by the projective comparisons of Lemma 40, gives EMC on \(Z\).

Finally suppose \(\gamma\) is big. Take a projective resolution \(\tau:S\to Z\) with \(S\) smooth and compact Kähler. The class \(\tau^*\gamma\) is big and nef. Its null curves are finite: a Kähler current with analytic singularities in this class has positive degree on every curve not contained in its singular locus. Their intersection matrix is negative definite. Indeed Hodge index is negative definite on the orthogonal complement of \(\tau^*\gamma\), whose square is positive. The null-curve classes are independent: a relation can be written as equality of classes of two effective real divisors with disjoint component supports; their mutual intersection is nonnegative, contradicting negative definiteness unless the common class is zero, which its positive Kähler degree excludes. Grauert’s contraction criterion (Grauert 1962) gives a contraction of precisely these curves onto a normal compact complex surface. The contraction is projective: solve using their negative-definite matrix for an integral combination having positive degree on every contracted curve, and apply fibrewise relative ampleness. This is EMC on \(S\), and it descends to \(Z\) by Lemma 40. Thus \(\mathsf C_d\) holds in all required dimensions.

Order of the induction. The assertions in dimensions zero and one are immediate from the degree of a class on a compact curve (and the ordinary curve minimal model program). For \(d=2,3\), prove successively \[\mathsf C_{d,\mathrm{big}},\quad \mathsf B_d,\quad \mathsf G_d,\quad\mathsf C_d.\] At \(d=4\) prove only \(\mathsf C_{4,\mathrm{big}}\) and \(\mathsf B_4\). The ordinary rational termination invoked at the intermediate stages has dimension at most three. This closes the induction without using (C. Hacon and Xie 2026, Theorems 1.5 or 5.15). ◻

Proof. Take the fixed nef b-divisor in (OpenAI 2026) to be zero, represented on any projective log resolution of the initial space. Its class is nef. Klt implies log canonical; the boundary is rational and is transported by strict transform. All the remaining conditions in that theorem are exactly the hypotheses just stated. No pseudo-effectivity or scaling parameter is needed. ◻

Proof. Proper direct image preserves coherence. On the isomorphism locus its rank is one, and double dual gives a rank-one reflexive sheaf on the normal target. Locally on \(T\), choose meromorphic generators of this coherent sheaf and of a cleared invertible power. Pulling back their comparison gives a meromorphic comparison with the bundle upstairs. One can see that the comparison is meromorphic, rather than merely defined on a punctured open set, by using a local finite presentation of \(p_*L\): its generators evaluate to holomorphic sections of \(L\), and their ratios supply local denominators. The comparisons agree on overlaps because they agree on the isomorphism locus. Their divisor is exceptional. This also proves compatibility with powers and pullback. A global meromorphic frame was not used.

For a common resolution \(W\) with maps to the two spaces, first pull back the sheaf, remove torsion, and take its double dual; after an additional projective principalization it is a line bundle. Proper direct image to the other space followed by double dual is coherent. On a common codimension-one open set it is the required transform. Normality gives uniqueness of a reflexive extension across codimension at least two, proving the assertion for small maps. The additive trace identity is checked at every codimension-one point of \(T\), where the modification is an isomorphism, and then extended reflexively. The same observation applies to canonical sheaves and their pullbacks. It requires compatible local canonical forms only.

Global strong \(\mathbb Q\)-factoriality makes \(M_T\) rationally invertible, so the first assertion defines \(F_M\). Its negative is \(p\)-nef and has zero pushforward. Lemma 5 gives \(F_M\geq0\). The ordinary crepant boundary is \(B_W-F_M\): (43) gives explicitly \[p^*(K_T+B)-K_W=B_W+M_W-p^*M_T=B_W-F_M.\] Thus ordinary log discrepancies are at least the generalized ones. Every ordinary zero center is a generalized zero center and is therefore generically in the specified simple normal crossing open set. This proves the klt and dlt assertions.

The effective divisor \(\lfloor B\rfloor\) is rational Cartier on \(T\). Decreasing it subtracts the effective divisor \(\epsilon p^*\lfloor B\rfloor\) from the crepant boundary. A former positive discrepancy stays positive. Every former zero-discrepancy place has center in the floor, since its center is generically a coefficient-one stratum, and hence has strictly positive order on the pullback of the floor. Its new discrepancy is positive as well. The boundary on \(T\) remains effective for \(0<\epsilon<1\). ◻

Proof. Write \(p\colon W\to T\) for a smooth carrier, with \(p\) projective, and put \(\rho=\pi p\). Work on a Stein neighborhood \(U\) of the prescribed compact. Choose a \(\rho\)-ample rational line bundle \(A\). For an integer \(m\) clearing denominators, the sheaf \[\rho_*\mathcal O_W\bigl(m(M_W-A)\bigr)\] is coherent and has rank one over the bimeromorphic isomorphism locus. It is nonzero. Cartan’s Theorem A gives a nonzero section over \(U\), whose pullback defines an effective divisor. Hence, after division by \(m\), \[ M_W\sim_{\mathbb Q}A+E,\qquad E\geq0. \tag{46}\] This proves relative bigness directly, including when \(M_W=0\). The divisor \(E\) need not be exceptional. The equivalence comes from a section of the specified bundle, and introduces no unspecified numerical or base twist.

Keep \(W\) and the \(\rho\)-ample bundle \(A\) fixed. Choose an auxiliary log resolution of the finite fixed supports only to test the singularities. On that auxiliary resolution there are finitely many crepant coefficients over the compact preimage, all strictly below one. Hence a sufficiently small rational \(\epsilon>0\) makes \((W,B_W+\epsilon E)\) sub-klt. The rational bundle \[A_\epsilon=(1-\epsilon)M_W+\epsilon A\] is \(\rho\)-ample. Choose a rational \(\eta>0\) such that \(A_\epsilon-\eta p^*H\) is still \(\rho\)-ample. On a smaller neighborhood, sufficiently large powers of these relatively ample bundles are relatively generated. Cartan’s Theorem A supplies finitely many sections generating over the compact preimage; shrink once more so that they generate throughout. Divided general members give \[H_0\geq0,\quad H_0\sim_{\mathbb Q}H, \qquad G\geq0,\quad G\sim_{\mathbb Q}A_\epsilon-\eta p^*H.\] Choose the members on \(T\) and \(W\), respectively, with large denominators, and test generality on the auxiliary log resolution. Their pulled-back systems remain free; we do not claim that a pulled-back ample bundle stays ample. Analytic Bertini gives \[C_W=B_W+\epsilon E+G+\eta p^*H_0\] sub-klt. Here generation is used for the ample systems only; the nonzero section producing \(E\) was not claimed to generate its bundle. The analytic generation and Bertini statements used in this step are (Das et al. 2024, Theorems 2.12 and 2.20).

Push down to obtain \[\Delta=B+\epsilon p_*E+p_*G+\eta H_0\geq\eta H_0.\] Only exceptional coefficients of \(B_W\) could be negative, so \(\Delta\) is effective. The actual upstairs identity is \(K_W+C_W\sim_{\mathbb Q}p^*D_T\). Choose an integer clearing this identity, push forward the resulting line-bundle isomorphism, and take double duals. Agreement in codimension one gives \[ \mathcal O_T\bigl(m(K_T+\Delta)\bigr)\simeq\mathcal O_T(mD_T). \tag{47}\] In particular, the new ordinary adjoint is rational Cartier. We have not inferred Cartierness of newly chosen local divisors from global strong \(\mathbb Q\)-factoriality. The divisor \[K_W+C_W-p^*(K_T+\Delta)\] is exceptional and \(p\)-numerically trivial by the same line-bundle identity. Lemma 5 in both signs makes it zero. Thus \(C_W\) is the ordinary crepant boundary, and \((T,\Delta)\) is klt. Subtracting \(\delta H_0\) gives the asserted representation of \(D_T-\delta H\) and leaves the common ample part \((\eta/2)H_0\).

For finitely many vertices, use one carrier and one \(H_0\), choose their effective divisors in (46), and resolve all fixed supports together. Take a common sufficiently small \(\epsilon\) and then a common \(\eta\). Choose the finitely many free members generally. On the resulting common resolution all the crepant coefficients are uniformly below one. Convex combinations are therefore klt and keep the same ample part. The parameter space can be the simplex of vertex weights; no affine independence of the original vertices is required.

For real data, first express the bundle in a finite real span of rational bundles on the carrier. Every signed rational generator has an effective representative over the Stein neighborhood by the same Cartan argument. Thus a real nef bundle also has an expression \(M_W\sim_{\mathbb R}A+E\) with \(A\) relatively ample and \(E\geq0\) in a fixed finite divisor span. The preceding small-\(\epsilon\) construction applies. A real relatively ample bundle can be written as a positive combination of finitely many nearby rational relatively ample bundles; divided general members of those bundles give the required real boundary. The ordinary real klt cone theorem is consequently applicable. Only the rational construction is used for line-bundle descent and for the finite adjoint rings. ◻

Proof. For an exceptional prime \(E\), global strong \(\mathbb Q\)-factoriality makes \(E\) rational Cartier. If it were \(f\)-nef, Lemma 5(i) applied to \(-E\) would give \(-E\geq0\). Thus \(E\cdot R<0\). Every contracted curve must lie in \(E\), since an effective rational Cartier divisor has nonnegative intersection with a curve not contained in its support. Every point of a nontrivial projective fiber lies on a curve in that fiber. Hence the entire exceptional locus is \(E\), and no second exceptional prime exists.

For a global rank-one reflexive sheaf \(\mathcal F\) on \(Z\), form its reflexive pullback to \(T\). A positive power is a line bundle. Subtract a rational multiple of \(E\) so that its degree is zero on \(R\) and apply Lemma 9. The descended rational bundle agrees with \(\mathcal F\) to that power on the complement of a codimension-two set in \(Z\). Reflexivity extends the identity. This proves the global strong condition and, in particular, applies to the pushed-forward canonical, boundary, and nef-trace sheaves.

In the small case, \(L-cJ\) has zero degree and descends by Lemma 9. The right-hand side of (48) is a rational line bundle on \(T^+\) and agrees with the transform of \(L\) on the common open set. It is therefore the reflexive transform. The sign conclusions follow from relative ampleness of \(J^+\) and the sign of \(c\).

Conversely, transport an arbitrary global rank-one reflexive sheaf \(\mathcal G\) on \(T^+\) to \(T\) as in Lemma 53. A reflexive power of its transform is invertible on \(T\). The preceding argument makes its transform a rational line bundle on \(T^+\). It agrees with the corresponding reflexive power of \(\mathcal G\) on the common codimension-one open set, and hence everywhere. This is global strong \(\mathbb Q\)-factoriality on \(T^+\); no assertion about arbitrary open subsets is involved. ◻

Proof. First assume gklt. The global cone construction above gives a negative ray and a nef rational supporting line bundle \(L\) which vanishes precisely on that ray and for which a multiple minus the adjoint is relatively ample. Apply the ordinary base point free theorem on the finite Stein cover, using Lemma 54. It generates a common positive power of \(L\) relative to \(V\). The relative section morphism and its Stein factorization give \(f\colon T\to Z\) with connected fibers and normal target, contracting exactly the curves on the selected ray. The target is projective over \(V\): it is finite over the image in the projectivization of the coherent sheaf of relative sections. Since \(T\to V\) is bimeromorphic, \(Z\) has the same dimension and \(f\) is bimeromorphic. The negative adjoint is relatively ample, because on every \(f\)-fiber its class is a positive multiple of a relatively ample class in the one-dimensional global degree space.

If \(f\) is divisorial, apply Lemma 55. If it is small, apply Lemma 54 locally over \(Z\), followed by the ordinary analytic klt flip theorem (Fujino 2022, Theorem 1.14). Equivalently, the actual relative adjoint algebra is locally finitely generated by (Das et al. 2024, Corollary 3.7); the flip theorem also supplies its normality and smallness. The local models glue by the intrinsic algebra. Compactness permits a common Veronese generated in degree one. On the common codimension-one open set its tautological bundle is the stated adjoint power. Both sheaves are reflexive on the normal positive side, so this identity extends globally. Thus the positive adjoint is rational Cartier and relatively ample as an actual bundle. Lemma 55 gives global strong \(\mathbb Q\)-factoriality.

All spaces constructed are projective over a compact Kähler space. A relatively ample line bundle has a metric with positive curvature in the fiber directions; adding a sufficiently large pullback of a Kähler form on the base gives a Kähler form. Thus the spaces are compact Kähler. The common carrier remains projective over the models: take the main component of the graph over \(Z\) and resolve it. The pulled-back nef data are still nef over \(V\).

For gdlt data, replace \(B\) by \(B-\epsilon\lfloor B\rfloor\) with small rational \(\epsilon>0\). By Lemma 53 this is gklt; denote its adjoint by \(D_\epsilon\). Normalize by a relatively ample bundle \(H\), and choose a relative curve class \(z\) with \(H\cdot z=1\) and \(D\cdot z=-a<0\). Choose \(\epsilon\) so that \(|(D_\epsilon-D)\cdot u|<a/4\) on the whole compact normalized slice. Choose an extreme point \(u_0\) of the face where \(D\) attains its minimum on this slice. It is an extreme point of the entire slice, so it spans a global extremal ray \(R\). Since \(D\cdot u_0\leq D\cdot z=-a\), the uniform perturbation bound gives \(D_\epsilon\cdot u_0<-3a/4\). Thus \(R\) is negative for both adjoints. The gklt construction contracts this global extremal ray for the perturbed pair. The original adjoint has the same negative sign on the ray. In the small case Lemma 55 identifies its positive transform with a positive multiple of the perturbed positive adjoint modulo a base pullback. In the divisorial case that lemma makes its pushdown rational Cartier. These are therefore steps for the original data.

Finally Lemma 5 proves the discrepancy comparisons. For gklt, positivity is preserved. For gdlt, a new zero-discrepancy center cannot have its general point in an exceptional locus, by strictness. Its corresponding old zero center is generically in the unchanged good open set. The transformed boundary is SNC and the b-data descend there. This gives a gdlt good open set on the output. The same hypotheses hold for the next relative morphism to \(V\), so the construction continues whenever relative nefness fails. ◻

Proof. Lemma 54 supplies ordinary klt representatives with a common positive relatively ample part. Choose a common projective log resolution \(a\colon S\to T\) by successive blowups and principalizations with centers exceptional over \(T\), resolving all fixed supports. This choice can be made with an effective exceptional divisor \(F\) such that \(-F\) is \(a\)-ample: for each blowup take its effective tautological exceptional divisor, and inductively add it to a sufficiently large positive multiple of the pullback of the divisor for the preceding composite. Relative ampleness under composition proves the assertion; further such blowups preserve it. For the identity resolution take \(F=0\). This is the blowup-resolution construction used in (Das et al. 2024, Theorem 2.13, Remark 2.14, and the proof of Corollary 3.7). For each vertex choose every exceptional coefficient strictly between the maximum of zero and its crepant coefficient and one. We obtain \[K_S+\Gamma_j=a^*D_j+E_j, \qquad E_j\geq0\text{ exceptional},\quad 0\leq\Gamma_j<1.\] All supports, including that of \(F\), are SNC. The ample part downstairs permits a common relatively ample part upstairs. More explicitly, scale the chosen \(F\) by a small positive rational number so that \(a^*H-F\) is ample over \(V\), where \(H\) is the fixed ample part downstairs. For a sufficiently small \(\tau>0\) put \(A=\tau(a^*H-F)\). The finitely many exceptional coefficients of the \(\Gamma_j\) have positive margins from both zero and one. Then \(\Gamma_j-A\) is effective with coefficients below one; their supports, including that of \(F\), are SNC. One can either use this relatively ample rational divisor \(A\) directly or replace it by a divided general member and make the corresponding simultaneous linear-equivalence changes in the boundaries.

The multigraded ring of the \(K_S+\Gamma_j\) is locally finitely generated by (Das et al. 2024, Theorem 3.1). For nonnegative integers \(n_j\), the correction \(\sum n_jE_j\) is effective and exceptional over \(T\). Normality gives \(a_*\mathcal O_S(\sum n_jE_j)=\mathcal O_T\). The projection formula therefore identifies every multigraded piece with the corresponding piece for the \(D_j\), compatibly with multiplication. Passing to one common Veronese removes all rounding. Thus their actual multigraded ring is locally finitely generated as well.

For completeness, finite generation gives the required finiteness as follows. On a sufficiently small neighborhood choose homogeneous generators with multidegrees \(d_1,\ldots,d_N\in\mathbb N^{s+1}\). For a rational weight \(w\) in the simplex, a point of the relative spectrum is semistable for the diagonal ring of weight \(w\) precisely when \(w\) is in the cone generated by those \(d_i\) whose generators are nonzero at the point. To verify this equivalence, clear the rational coefficients of such a cone expression and form the corresponding monomial; conversely a nonvanishing homogeneous section contains a nonvanishing monomial of that weight. There are only finitely many subsets of the generators and hence finitely many resulting open sets. The diagonal Proj is the quotient of its open set, obtained by the degree-zero localizations of these monomials. Uniqueness of this quotient shows that weights with the same open set give the same ample model. All identifications commute with the map from \(T\), which is determined on the bimeromorphic isomorphism locus. A finite cover of the Stein compact makes the number of possibilities finite there. This is also the marked finite-model conclusion of (Fujino 2022, Theorem E), now with the ordinary representatives and the smooth reduction justified. ◻

Proof. Let \(D=K_T+B+M_T\). We may suppose \(D\) is not already relatively nef.

A scaling class valid on every later model. Choose global rational line bundles \(L_1,\ldots,L_r\) whose degrees form a basis of \(N^1_{\mathrm{gl}}(T/V)_{\mathbb Q}\), and write the class of \(D\) as \(d\in\mathbb Q^r\).

These bundles continue to span the degree space after every finite sequence of elementary steps. Indeed every line bundle on a later model can be transported to a rational line bundle on the preceding one by Lemma 53 and global strong \(\mathbb Q\)-factoriality. For a divisorial contraction its pullback is also such a bundle; exceptional-divisor adjustments are available on the source. Moreover a bundle numerically trivial over \(V\) remains so after a step. It first descends across that contraction by Lemma 9. Any curve on the new side over \(V\) has a projective lift to a common model and then a curve upstairs mapping onto it with positive degree. Intersecting the pullbacks of the descended bundle proves its zero degree. Thus every later relative curve determines a rational functional \(v\in\mathbb Q^r\), nonzero because the later space has a relatively ample line bundle.

Choose a real vector \(h\in\mathbb R^r\) defining a real linear combination \(H\) of the chosen rational line bundles, with \(H\) and \(D+H\) relatively ample. We choose it outside the following countable family of proper rational hyperplanes: \[\begin{align*} v(h)&=0 &&(0\ne v\in\mathbb Q^r),\tag{49}\\ v(d)w(h)-w(d)v(h)&=0 &&\left(\begin{array}{l}v,w\in\mathbb Q^r\text{ independent},\\ (v(d),w(d))\ne(0,0)\end{array}\right). \tag{50}\end{align*}\] The ample conditions define a nonempty open set, and a countable union of proper hyperplanes cannot exhaust it. The rational functionals here range over all possibilities, so this single choice works at every later model; no enumeration of the models is required.

The scaling steps and their strictly decreasing thresholds. On a current model, let \(D_i\) be the reflexive transform of \(D\), and let \(H_i\) be the same real linear combination of the reflexive transforms of the rational line bundles defining \(H\); across a divisorial step these transforms are the corresponding pushforwards. Regard \(H\) also as nef b-data over \(V\), carried on the initial space. Its pullback to a common carrier is relatively nef, so for every \(t\geq0\) the initial generalized pair with adjoint \(D+tH\) is gklt. At each preceding step of threshold \(t_j\), its adjoint has negative degree for \(0\leq t<t_j\) and zero degree for \(t=t_j\). Lemma 5 consequently shows that the transformed pair is gklt throughout the parameter interval below the preceding threshold.

Start with the relatively ample class \(D+H\). On a current model take the lower endpoint \(t_i\) of the nef segment extending downward from the preceding parameter. If \(t_i=0\), then \(D_i\) is nef and the process stops. Otherwise \(J_i=D_i+t_iH_i\) is nef and not ample. The local cone argument preceding Lemma 55, applied to this real gklt adjoint, shows that its zero face is generated by finitely many curves. A generating curve cannot have \(D_i\)-degree zero, because then its \(H_i\)-degree would be zero, contrary to (49). Two independent such rays would violate (50). Hence the face is one ray in the global degree space. Its \(D_i\)-degree is negative: if it were positive, lowering \(t\) would be positive on this face, and the uniform cone margin would give nefness below \(t_i\), a contradiction.

Apply Proposition 57 to this \(D_i\)-negative ray. The threshold class descends. Here descent for a real class requires no real-bundle base point free theorem: the perpendicular hyperplane of the rational ray is rational, so express the threshold class as a real combination of rational line-bundle classes of degree zero and descend each by Lemma 9. The threshold comparison is crepant and its transform is nef.

The thresholds strictly decrease. In a small step, the zero face on the positive side is again generated by curves by the real gklt local cone argument. Equations (49)– (50) allow at most one ray. Flipped curves supply that ray and have positive \(D\)-degree. Lowering the parameter is therefore positive on it, and the uniform margin makes the class ample for a nonempty interval below the old threshold. In a divisorial step there are no zero curves at the old threshold: lifting one to the old space would make it lie on the contracted zero ray while still mapping onto a curve. The same margin again gives an ample interval. At every stage, the open interval between successive thresholds consists of relatively ample classes: an interior zero curve would have zero degree for two parameters, hence zero \(H_i\)-degree, contrary to (49).

Finitely many marked ample models. It remains to exclude infinitely many such intervals. Choose finitely many rational relatively ample bundles \(H^{(1)},\ldots,H^{(s)}\) near \(H\), with \(H\) in the interior of their convex hull and with \(D+H^{(j)}\) relatively ample. Consider the fixed rational simplex of weights with vertices \[ D,\ D+H^{(1)},\ldots,D+H^{(s)}. \tag{51}\] At an interior parameter of any one of the alleged infinitely many ample intervals, all preceding intersections for that parameter are strictly negative and final ampleness is open. Choose a nearby rational point of (51) retaining these finitely many strict conditions. The current model is its marked ample model: on a common resolution its pullback difference from the initial adjoint is effective and exceptional over the current model by composition of the preceding comparisons. The corresponding relative section algebras agree, since pushing an effective exceptional correction to a normal target gives the structure sheaf.

Each vertex in (51) has nef b-data on the initial model and is gklt. Apply Lemma 58 on each member of a finite Stein compact cover of \(V\). It gives finitely many local marked ample models, hence finitely many possible tuples of such local models. Two global models giving the same tuple are uniquely isomorphic over the interiors of these compacts: the identification is fixed on the common bimeromorphic open set and extends by the marked local isomorphism. The identifications agree on overlaps by normality, so they glue. Thus only finitely many global marked ample models occur.

A marked model cannot recur across a nontrivial intervening step. For a fixed marking \(T\dashrightarrow Y\), choose a common resolution. The pullback difference for \(D+tH\) is affine in \(t\). Its coefficient inequalities of effectivity are convex, and the relatively ample cone of \(Y\) is convex. The parameters for which \(Y\) is its marked ample model therefore form an interval. If it occurred in two separated ample intervals of the run, it would also be the ample model at every intermediate ample parameter. Uniqueness of the relative Proj would identify each intermediate model with the same marking. But adjacent models across a nontrivial elementary step have different markings, as their adjoint has opposite relative signs. This is impossible. Finitely many marked models therefore imply finitely many steps. ◻

Proof. The projective realization statement in Lemma 16 places the finitely many designated divisors on a projective log resolution carrying the nef data. Its proof uses only log-smooth discrepancy calculations and stratum blowups, independently of extraction. Write this resolution as \(p\colon W\to T\). Keep every designated coefficient at its crepant value \(1-a(E)\), which lies in \([0,1)\). Keep the nonexceptional coefficients as well. For every other exceptional prime choose a coefficient in \([0,1)\) strictly greater than its crepant coefficient. Since the pair is gklt, there are only finitely many such choices and they are possible. With this effective SNC boundary \(\Gamma\) we have \[K_W+\Gamma+M_W=p^*D_T+G, \qquad G\geq0,\] where the support of \(G\) is exactly the set of undesired exceptional primes. The new pair is gklt.

Apply Proposition 59 over \(T\). At every stage the adjoint is the pullback of \(D_T\) plus the effective transform of \(G\), and the latter remains exceptional over \(T\). A divisorial negative step cannot contract a prime outside its support: a general covering contracted curve in that prime avoids containment in the support and has nonnegative intersection with the effective rational Cartier divisor \(G\), whereas its adjoint degree is negative. Small steps contract no prime divisor. Hence every designated prime survives, and no nonexceptional prime of the resolution is lost.

At the relative nef endpoint \(h\colon U\to T\), the remaining effective exceptional correction \(G_U\) is \(h\)-nef. Negativity gives \(G_U\leq0\), hence \(G_U=0\). Every undesired prime has therefore been contracted. The equality is now crepant, and the retained boundary is effective. The global strong condition, Kählerness, and gklt property follow at each step from Proposition 57. The exceptional primes are exactly the specified ones, as claimed. ◻

Proof. Write the sequence as \[(X_i,B_i+\mathbf M)\dashrightarrow (X_{i+1},B_{i+1}+\mathbf M), \qquad X_i\xrightarrow{f_i}Z_i\xleftarrow{f_i^+}X_{i+1}.\] The boundary is transformed strictly, so its coefficients belong to one finite set. Smallness and Lemma 5 preserve generalized terminality. Forgetting the effective boundary and the effective nef defect increases discrepancies, so each \(X_i\) is an ordinary terminal threefold. In particular its singularities are isolated, and the general point of every flipped curve is smooth.

Choose an integer \(p>0\) for which \(p\mathbf M\) is represented by an integral line bundle on a carrier, and an integer \(N\) divisible by \(p\) and by the denominators of the boundary coefficients. These integers continue to work on higher carriers throughout any finite part of the sequence. If \(C\) is a flipped curve on \(X_{i+1}\), the exceptional valuation \(E_C\) of its generic blowup has log discrepancy \[ a(E_C;X_{i+1},B_{i+1}+\mathbf M) =2-\sum_j m_jb_j-\frac{q}{p}, \qquad m_j,q\in\mathbb Z_{\geq0}. \tag{52}\] Indeed, the ordinary blowup contribution is two, the \(m_j\) are the generic multiplicities of the boundary components, and the remaining term is the coefficient of the nef defect. At this smooth generic point the integral trace of \(p\mathbf M\) is Cartier. The defect therefore has coefficients in \(p^{-1}\mathbb Z\) and is effective by Lemma 53. This proves (52), including its signs.

Let \(b\) be the largest boundary coefficient, with \(b=0\) when the boundary is empty. For \(b>0\) test the flipped curves contained in a coefficient-\(b\) component; for \(b=0\) test every flipped curve. The value in (52) belongs to the fixed finite set \[\mathcal A_b=N^{-1}\mathbb Z\cap(0,2-b].\] For each \(t\in\mathcal A_b\), set \[d_t(i)=\#\{E\text{ exceptional over }X_i: a(E;X_i,B_i+\mathbf M)<t\}.\] Lemma 16 makes these numbers finite, since the terminal cutoff is the strict inequality \(a<2-b\). Small transformations identify the sets of exceptional valuations, and discrepancy monotonicity makes every \(d_t(i)\) nonincreasing. For the valuation of a tested curve, Lemma 5 gives \[a(E_C;X_i,B_i+\mathbf M) <a(E_C;X_{i+1},B_{i+1}+\mathbf M)=t.\] Thus \(d_t\) drops at that step. The nonnegative integer \(\sum_{t\in\mathcal A_b}d_t(i)\) can drop only finitely often. If \(b=0\), every nontrivial small step has a flipped curve, so the proof is complete.

Suppose \(b>0\). On a tail, no flipped curve is contained in any of the coefficient-\(b\) components. Fix such a component \(D_i\), and let \(D_i^\nu\) denote its normalization. The two images of \(D_i\) and \(D_{i+1}\) in \(Z_i\) agree, because the ambient transformation is an isomorphism in codimension one. Denote the normalization of this common image by \(V_i\). There are projective birational morphisms \[D_i^\nu\longrightarrow V_i\longleftarrow D_{i+1}^\nu.\] The morphism on the right contracts no curve and is therefore finite: every positive-dimensional projective fiber contains a curve. A finite birational morphism to a normal space is an isomorphism. Hence there is a projective birational morphism \(D_i^\nu\to D_{i+1}^\nu\). It contracts exactly the curves whose images in \(X_i\) are flipping curves contained in \(D_i\). By Lemma 15, with \(k=1\), each such contraction strictly decreases the dimension of the span of curve classes of the compact normalized surface. There are finitely many coefficient-\(b\) components, so only finitely many further steps have a flipping curve contained in one of them.

On the resulting tail put \(D_i^{\max}=\sum_{b_j=b}D_{i,j}\) and replace \(B_i\) by \(B_i-bD_i^{\max}\). Every contracted curve \(C\) has \(D_i^{\max}\cdot C\geq0\), because it is not contained in this effective \(\mathbb Q\)-Cartier divisor. Thus the new adjoint \[A_i'=K_{X_i}+B_i-bD_i^{\max}+M_{X_i}\] is negative on the old ray. To check its positive transform, write \(A_i=K_{X_i}+B_i+M_{X_i}\) and choose the rational number \(c=(A_i'\cdot C)/(A_i\cdot C)>0\). The global one-ray condition and Lemma 9 give an actual identity of rational line bundles \[A_i'=cA_i+f_i^*L_i,\qquad L_i\in\mathop{\mathrm{Pic}}(Z_i)\otimes\mathbb Q.\] Its strict transform is \(A_{i+1}'=cA_{i+1}+(f_i^+)^*L_i\), which is \(f_i^+\)-ample. Consequently the same tail is a sequence of elementary steps for the new generalized pair. Decreasing the effective boundary preserves generalized terminality. Induction on the number of distinct positive boundary coefficients proves the proposition. ◻

Proof. For the first assertion, use the analytic index-one cover \(\pi:(T^\sharp,x^\sharp)\to(T,x)\), a cyclic cover of degree \(r\) that is étale away from the distinguished point. The terminal threefold classification makes \(T^\sharp\) smooth or an isolated cDV hypersurface germ; see (Mori 1985). A sufficiently small punctured representative \(U^\sharp\) is simply connected, by the connectivity theorem for links of isolated hypersurfaces (Milnor 1968, Theorems 2.10 and 5.2). For a convergent analytic defining equation, finite determinacy gives an analytically equivalent sufficiently high Taylor polynomial (Jong et al. 1998, Theorem 6.2, p. 24), so the same connectivity statement applies; the smooth case has the same property. For every \(m>0\), the analytic Kummer sequence implies that the \(m\)-torsion of \(\mathop{\mathrm{Pic}}(U^\sharp)\) is an image of \(H^1(U^\sharp,\boldsymbol\mu_m)=0\). The local divisor class group injects into this Picard group: a rank-one reflexive sheaf trivial on the punctured normal germ is trivial on the germ by reflexive extension. It follows that the local class group of \(T^\sharp\) is torsion-free.

Since \(D\) is \(\mathbb Q\)-Cartier, the integral divisor \(\pi^*D\) has torsion class and is therefore principal, say \(\pi^*D=\operatorname{div}(u)\) for a meromorphic germ \(u\). The meromorphic norm gives \[\operatorname{div}\operatorname{Norm}_{\pi}(u) =\pi_*\operatorname{div}(u) =\pi_*\pi^*D=rD.\] Thus \(rD\) is Cartier; compare the index divisibility statement in (Kawamata 1988, Lemma 5.1). The argument above proves the analytic assertion for divisor germs already known to be \(\mathbb Q\)-Cartier.

The second assertion is the terminal small-discrepancy theorem (Kawamata 1993, Appendix, pp. 193–195). Its proof uses the analytic quotient classification and gives ordinary discrepancy \(1/r\), or log discrepancy \(1+1/r\) in our convention. ◻

Proof. Let \(\nu:P^\nu\to P\) be the normalization and let \(\mu:S\to P^\nu\) be its minimal resolution. Divisorial adjunction gives \[ \nu^*((K_U+P)|_P)=K_{P^\nu}+\Delta_{P^\nu}, \qquad \Delta_{P^\nu}\geq0, \tag{53}\] as an identity of rational adjoint bundles. Neither normality of \(P\) nor log canonicity of \((U,P)\) is needed for effectivity of the different in this formula. This is the ordinary different of a reduced prime, computed at codimension-two points of \(U\); the construction and effectivity are given in (Fujino 2010, secs. 14.1(i)–(iii), pp. 40–41), and analytic adjunction is included in (Fujino 2023, Theorem 1.1 and the following setup, p. 1). Here is the local analytic effectivity argument. On a transverse klt surface \(T\), take a small smooth quotient \(q:V\to T\) and write \(P_T\) for the reduced boundary curve and \(C=(q^{-1}P_T)_{\mathrm{red}}\). The quotient is unramified at the general points of these curves, so \(q^*(K_T+P_T)=K_V+C\). Let \(\rho:C^\nu\to P_T^\nu\) be the induced map of normalizations. Residue adjunction for the reduced plane curve \(C\) has an effective conductor divisor \(A\), and the canonical ramification formula for \(\rho\) has an effective divisor \(R\). Compatibility of residues with pullback gives \[\rho^*\operatorname{Diff}_{P_T^\nu}(0)=A+R.\] At a branch of ramification degree \(e\), its coefficient downstairs is \((c+e-1)/e\geq0\), where \(c\) is the conductor coefficient. This proves the analytic effectivity in (53) without an lc hypothesis on \((U,P)\) or normality of \(P\).

The sum on the right of (53) is rational Cartier; its individual canonical term need not be. If that term is used separately, take its numerical pullback in the sense of Mumford, determined by intersection zero with all exceptional curves. The negative definite exceptional intersection matrix makes this pullback linear and makes the pullback of an effective divisor effective. In particular the calculation does not presume \(\mathbb Q\)-factoriality of \(P^\nu\).

For completeness, the effective correction can also be checked directly on \(S\). For every irreducible \(\mu\)-exceptional curve \(E\), adjunction on the smooth surface gives \[K_S\cdot E=2p_a(E)-2-E^2\geq0.\] Indeed, an exceptional smooth rational \((-1)\)-curve is excluded by minimality over \(P^\nu\); every other exceptional curve satisfies the displayed inequality. The pullback of the adjoint in (53) therefore has the form \[ (\nu\mu)^*((K_U+P)|_P)=K_S+\Delta_S, \qquad \Delta_S\geq0. \tag{54}\] To obtain the sign, the residue comparison defines \(\Delta_S\) as a rational divisor with pushforward \(\Delta_{P^\nu}\geq0\). Since \(K_S\) is \(\mu\)-nef, the divisor \(-\Delta_S\) is \(\mu\)-nef. Apply Lemma 5(i) to \(E=\Delta_S\) to obtain \(\Delta_S\geq0\). This is equivalently the numerical-pullback calculation above, and proves (54) as an actual bundle comparison.

The image \(h(P)\) has dimension zero or one. In the latter case, general smooth fiber components of \(S\to h(P)\) are moving curves \(F\) with \(K_S\cdot F=2g(F)-2\geq-2\). One may use the Stein factorization to describe these fibers over a smooth compact curve. Choose a general fiber component not contained in \(\mathop{\mathrm{Supp}}\Delta_S\), the conductor, or the pullbacks of the \(D_j\).

If \(h(P)\) is a point, \(P\) is a projective fiber subspace, and \(S\) is a smooth projective surface. Take a surface minimal model \(\rho:S\to S_0\). If \(K_{S_0}\) is nef, use general ample curves on \(S_0\). Otherwise the classification of smooth projective surfaces gives ruling fibers with canonical degree \(-2\), or lines on \(\mathbb P^2\) with canonical degree \(-3\). Their general strict transforms \(F\) satisfy \(K_S\cdot F\geq-3\), because \(K_S-\rho^*K_{S_0}\) is effective. In all cases the curves cover \(S\), so \(F\) may be chosen not contained in the finitely many excluded divisors. This is the surface argument also used in (Chen and Tsakanikas 2023, Lemma 3.3).

The curve \(F\) maps birationally to a curve \(C\subset P\): it is not contained in the exceptional or conductor locus. By the projection formula and (54), \[(K_U+P)\cdot C=(K_S+\Delta_S)\cdot F \geq K_S\cdot F\geq-3.\] Its choice also ensures \(C\not\subset\mathop{\mathrm{Supp}}D_j\) for every \(j\). ◻

Proof. We induct on the number of exceptional places with log discrepancy at most one. The terminal case is controlled by local canonical indices. For the induction step, extract one place, bound its intersection degree using the preceding surface estimate, and descend an integral line bundle.

Replace \(\varepsilon\) by \(\min\{\varepsilon,1\}\), so assume \(0<\varepsilon\leq1\). We construct integers divisible by those from the preceding induction stage. When \(n=0\), the pair is generalized terminal. The underlying space is ordinary terminal and \[a(E;T,0)\geq a(E;T,B+\mathbf M)\geq1+\varepsilon\] for every exceptional valuation. By Lemma 63 and the observation following that lemma, every canonical point index \(r>1\) satisfies \(r\leq1/\varepsilon\). Each local representative of \(\mathcal F\) is \(\mathbb Q\)-Cartier by the global strong condition, and its Cartier index divides \(r\). Thus one may take \[I(0,\varepsilon)=\operatorname{lcm} \{1,\ldots,\lfloor1/\varepsilon\rfloor\}.\] Local invertibility of this reflexive power at every point is precisely global invertibility; no trivialization on all of \(T\) is asserted.

Suppose \(n>0\) and put \(I'=I(n-1,\varepsilon)\). If there is no exceptional valuation of discrepancy at most one, the preceding case applies. Otherwise choose one such place, of discrepancy \(a\in[\varepsilon,1]\), and apply Proposition 61 to obtain a projective extraction \(h:U\to T\) with that single exceptional prime \(P\). Write its effective crepant boundary as \[B_U=B_U^{\mathrm{str}}+(1-a)P.\] Crepancy preserves all discrepancies and lowers the number of exceptional low places by one. The induction hypotheses therefore hold on \(U\), so \(I'\) clears the index of every global rank-one reflexive sheaf on \(U\).

First, \(-P\) is \(h\)-ample. If \(H\) is an \(h\)-ample line bundle, its reflexive pushforward is rational Cartier on \(T\). Lemma 53 gives \[H=h^*H_T+cP\] as a rational bundle comparison. The divisor \(cP\) is \(h\)-ample and \(h\)-exceptional, hence is nonpositive by negativity. Since \(h\) has a positive-dimensional fiber, \(c\neq0\), and therefore \(c<0\).

The same trace comparison for the nef data gives \[M_U=h^*M_T-sP,\qquad s\geq0.\] Apply Lemma 64 with the components of \(B_U^{\mathrm{str}}\) as the excluded divisors. For its contracted curve \(C\subset P\), the other boundary has nonnegative degree and \(M_U\cdot C=-sP\cdot C\geq0\). Crepancy yields \[-3\leq(K_U+P+B_U^{\mathrm{str}}+M_U)\cdot C =aP\cdot C<0.\] Consequently \[ 0<-P\cdot C\leq\frac{3}{\varepsilon}. \tag{55}\]

Now let \(\mathcal F\) be any global rank-one reflexive sheaf on \(T\), and let \(\mathcal F_U\) be its reflexive strict transform. In additive rational-bundle notation, Lemma 53 gives \[ h^*\mathcal F=\mathcal F_U+qP,\qquad q\in\mathbb Q. \tag{56}\] This comparison is defined by local meromorphic frames agreeing off the exceptional locus, and requires no global meromorphic section of \(\mathcal F\). The sheaves \(\mathcal F_U^{[I']}\) and \(\mathcal O_U(I'P)\) are line bundles. Intersecting (56) with \(C\) gives \[q=\frac{I'\mathcal F_U\cdot C}{-I'P\cdot C}, \qquad k:=-I'P\cdot C\in\mathbb Z, \quad 1\leq k\leq\frac{3I'}{\varepsilon}.\] The numerator is an integer. Define \[ I(n,\varepsilon) =I'\operatorname{lcm} \{1,\ldots,\lceil3I'/\varepsilon\rceil\}. \tag{57}\] Then \(I:=I(n,\varepsilon)\) is divisible by \(I'\) and \(Iq\in I'\mathbb Z\). Thus \(I\mathcal F_U+(Iq)P\) represents an integral line bundle \(L\) on \(U\), with its prescribed comparison to \(h^*\mathcal F^{[I]}\) off the exceptional locus. Equation (56) shows that \(L\) has degree zero on every \(h\)-contracted curve.

Increase the coefficient \(1-a\) of \(P\) by a sufficiently small positive rational number \(\delta\). On a fixed log resolution carrying the data, the inequalities defining generalized klt remain strict for all sufficiently small \(\delta\), so the new pair is effective and generalized klt. Its adjoint is \[h^*(K_T+B+M_T)+\delta P,\] which is \(h\)-antiample. Lemma 9 now descends \(L\) integrally. More explicitly, local ordinary klt replacement and the analytic base-point-free theorem generate \(L^m\) for every sufficiently large integer \(m\); descend two consecutive powers and take their quotient. No further index is introduced. The descended line bundle agrees with \(\mathcal F^{[I]}\) away from \(h(P)\), a set of codimension at least two in \(T\). Reflexive extension identifies them on \(T\). This proves the induction, including the case \(a=1\). ◻

Proof. Set \(E=P-P_+=F-G\). Over \(Y\), the divisor \(-E=P_+-P\) is nef, while \[p_*(-E)=-p_*F\leq0,\] since \(G\) is \(p\)-exceptional. Lemma 5 gives \(-E\leq0\), proving the first assertion. If \(A\) is nef over \(S\), then \(E\) is nef over \(T_+\) and \[q_*E=-q_*G\leq0,\] since \(F\) is \(q\)-exceptional. Negativity gives \(E\leq0\), and hence \(E=0\). The two exceptional supports need not coincide.

Assume \(A_+\) is relatively ample. A fiber of \(p\) maps by \(q\) into a projective fiber of \(T_+\to S\). Equality \(p^*A=q^*A_+\) gives degree zero on every curve in that fiber. A nonconstant image would be detected by a curve of positive degree against \(A_+\), so \(q\) is constant on each connected fiber of \(p\). Proper factorization through the normal space \(Y\) gives \(q=g\circ p\). The map \(g\) is projective because \(Y\to S\) is projective and \(T_+\to S\) is separated. Finally, pullback by \(p\) is injective on rational line bundles, by \(p_*\mathcal O_W=\mathcal O_Y\) and the projection formula; therefore \(A=g^*A_+\). This proves the actual bundle identity, not merely numerical equality. Compare (Lazić et al. 2023, Lemma 2.12) for the algebraic form of this comparison. ◻

Proof. In dimension at most two there are no nontrivial small birational contractions, and Lemma 15 gives the result. In dimension three that lemma first removes a finite divisorial prefix. Relabel the remaining small sequence as \[(X_i,B_i+\mathbf M)\dashrightarrow(X_{i+1},B_{i+1}+\mathbf M),\qquad i\geq1.\] All boundaries are now strict transforms of \(B_1\).

By Lemma 16, there is \(\varepsilon_0>0\) such that the initial set of exceptional valuations with discrepancy below \(1+\varepsilon_0\) is finite. Since discrepancies increase and small maps preserve exceptionality, the sets \[\mathcal L_i=\{E\text{ exceptional over }X_i: a(E;X_i,B_i+\mathbf M)\leq1\}\] form a decreasing sequence of finite sets. After discarding a finite prefix, they equal one set \(\mathcal L=\{E_1,\ldots,E_n\}\). At the first model of this tail, the finitely many exceptional values in \((1,1+\varepsilon_0)\) have a positive minimum distance from one; use \(\varepsilon_0\) itself if there are no such values. Choose \(\varepsilon>0\) no larger than this distance, than \(\varepsilon_0\), than one, and than the positive lower bound for all discrepancies of the initial gklt pair furnished by Lemma 16. Monotonicity and the stabilization of \(\mathcal L\) show that, on every model of this tail, all discrepancies are at least \(\varepsilon\), and every exceptional discrepancy greater than one is at least \(1+\varepsilon\).

Apply Lemma 65 with these fixed \(n\) and \(\varepsilon\). Let \(r\) clear the coefficients of \(B_1\) and let \(p\) clear the higher nef datum. For example \[J=I(n,\varepsilon)\operatorname{lcm}(r,p)\] makes each \(J(K_{X_i}+B_i+M_{X_i})\) an integral line bundle, and makes \(J\mathbf M\) integral on every higher carrier. Comparing the actual adjoint bundles on a smooth carrier shows that every discrepancy belongs to \(J^{-1}\mathbb Z\): the canonical bundle there is integral and the coefficients of the corresponding integral divisor difference are integers. The same argument applies after any further resolution. Thus each nondecreasing sequence \[a(E_\ell;X_i,B_i+\mathbf M)\in J^{-1}\mathbb Z\cap[\varepsilon,1]\] is eventually constant. After another finite prefix, all these values are constant simultaneously.

Use Proposition 61 to extract exactly \(E_1,\ldots,E_n\) over the first model of this final tail: \[h_1:(Y_1,\Delta_1+\mathbf M)\longrightarrow(X_1,B_1+\mathbf M).\] The boundary is effective and crepant. Every valuation exceptional over \(Y_1\) is exceptional over \(X_1\) and is not in \(\mathcal L\), so its discrepancy is greater than one. Hence this is a globally strongly \(\mathbb Q\)-factorial generalized terminal model. If \(n=0\), take \(h_1=\mathrm{id}\).

Consider the first remaining flip over \(Z_1\). Starting from \(Y_1\), use Proposition 57 to take relative steps over \(Z_1\) while the adjoint is not nef. We show that no divisorial contraction can occur in this run. At any finite stage \(Y_j\), a common smooth model gives \[P_0=P_j+G_j=P_++F,\] where \(P_0\) is pulled back from \(Y_1\), \(P_j\) from the current lifted adjoint, and \(P_+\) from \(K_{X_2}+B_2+M_{X_2}\). These are the prescribed adjoint comparisons. The accumulated error \(G_j\) is effective and exceptional over \(Y_j\); the error \(F\) of the original flip is effective and exceptional over \(X_2\). The positive adjoint is ample over \(Z_1\). The first assertion of Lemma 66, which does not require nefness on \(Y_j\), gives \(G_j\leq F\).

If a divisor were lost, its strict transform from \(Y_1\) would have a positive coefficient in \(G_j\), by the strict discrepancy comparison at its contraction. But every divisor on \(Y_1\) is either the transform of a divisor on \(X_1\), or one of the \(E_\ell\). In the first case its coefficient in \(F\) is zero because the original map is small. In the second case that coefficient is \[a(E_\ell;X_2,B_2+\mathbf M)-a(E_\ell;X_1,B_1+\mathbf M)=0\] by stabilization. Both cases contradict \(G_j\leq F\). Thus the entire lifted run is small. It stays generalized terminal, and Proposition 62 forces it to reach a nef endpoint after finitely many steps.

At that endpoint, Lemma 66 gives equality of the two adjoint pullbacks and a projective crepant morphism \[h_2:(Y_2,\Delta_2+\mathbf M)\longrightarrow(X_2,B_2+\mathbf M).\] Here \(Y_2\) denotes the endpoint, rather than the first intermediate model of the finite run. Since the lifted transformations were small, the exceptional primes of \(h_2\) are exactly the same \(E_\ell\), with the same boundary coefficients. We may therefore repeat this construction over each successive base \(Z_i\).

Each nontrivial flip below forces at least one step above. Indeed, take a negative curve \(C\subset X_i\) over \(Z_i\). Projectivity of \(h_i\) supplies a curve \(\Gamma\subset Y_i\) mapping onto \(C\) with positive degree. The crepant identity gives \[(K_{Y_i}+\Delta_i+M_{Y_i})\cdot\Gamma =\deg(\Gamma/C)(K_{X_i}+B_i+M_{X_i})\cdot C<0.\] Thus the starting lifted adjoint is not nef over \(Z_i\). An infinite original sequence would consequently concatenate to an infinite sequence of small generalized terminal elementary steps, with fixed rational boundary coefficients and the same rational nef b-line data. All spaces remain compact Kähler and globally strongly \(\mathbb Q\)-factorial. The bases may change, exactly as allowed in Proposition 62, which gives the contradiction. ◻

Proof. Choose a projective log resolution \(h:W\to V\) carrying the nef data which is an isomorphism at the general points of the good strata under consideration. We may replace \(W\) further whenever necessary, preserving these general points. Since \(V\) is globally strongly \(\mathbb Q\)-factorial, the trace \(M_V\) is a rational line bundle. Lemma 53 and Lemma 5 give \[ H=h^*M_V-M_W\geq0, \qquad H\text{ is }h\text{-exceptional}. \tag{61}\] Use compatible local canonical representatives to write \[K_W+A_W+M_W=h^*(K_V+A+M_V),\qquad K_W+A_W^{\rm o}=h^*(K_V+A).\] These are actual rational sheaf comparisons, and \(A_W=A_W^{\rm o}+H\).

We first justify the ordinary companion in this construction. The inequality above says that ordinary log discrepancies of \((V,A)\) are at least the generalized log discrepancies. In particular \((V,A)\) is lc. An ordinary zero-discrepancy place is also a generalized zero-discrepancy place. Its center therefore meets the good open set. Conversely, every generalized center meets that set, where \(H=0\), the data descend, and the log formula is the ordinary snc formula. Such a center is an ordinary coefficient-one stratum. Thus \((V,A)\) is ordinary dlt and its lc centers are exactly the generalized ones. This argument also explains why forgetting the nef part does not introduce an additional zero center outside the good open set; compare the algebraic formulation in (Chen and Tsakanikas 2023, Remark 2.6(1)).

Let \(S\subset V\) be a prime component of \(\lfloor A\rfloor\) and let \(S_W\) be its strict transform. It is smooth after the chosen resolution. The coefficient of \(S_W\) is one in both formulas. The residue isomorphism restricts them to \(S_W\) as \[\begin{align*} (K_W+A_W+M_W)|_{S_W} &=K_{S_W}+(A_W-S_W)|_{S_W}+M_W|_{S_W},\\ (K_W+A_W^{\rm o})|_{S_W} &=K_{S_W}+(A_W^{\rm o}-S_W)|_{S_W}. \end{align*}\] The normal space \(S_W\) maps to the normalization \(S^\nu\). Push forward the indicated boundaries and the nef trace to \(S^\nu\). Over its smooth large open set, the residue identification is the adjunction identification of the canonical sheaf. It therefore defines, by reflexive extension, the rational adjoint bundle on \(S^\nu\) as the pullback of the ambient adjoint. This construction gives an actual line-bundle isomorphism after a common positive multiple. The prescribed residue identification, and not equality of first Chern classes, specifies the comparison.

The restricted log boundary on \(S_W\) is snc with coefficients at most one. Thus the resulting generalized pair is sub-lc before effectivity is checked. Every zero stratum of that restricted log formula is an intersection of \(S_W\) with coefficient-one components of the ambient log formula. Its image is an ambient generalized center and hence meets the good open set. Over that set the restriction is snc and its nef data descend. The same reasoning applies to the ordinary companion. In particular, after effectivity is established, both restricted pairs have the appropriate dlt good-open property. Their zero centers agree: on a common carrier their log boundaries differ by the effective restriction of \(H\), while every generalized zero center meets the open set where that restriction vanishes.

Here is the local effectivity and coefficient calculation, with the issue of nonfactorial intermediate strata made explicit. Suppose at some stage of the chain that \(U\) is the normal space already obtained, \(C\) is its effective generalized boundary, and \(S\subset\lfloor C\rfloor\) is the next prime. Carry along the ordinary dlt companion just constructed. Its floor has the same strata. Both adjoints are rational Cartier, although the individual trace \(M_U\) need not be rational Cartier on \(U\). Fix a prime divisor on \(S^\nu\) whose coefficient is to be computed, and work at an analytically general point of its image, a codimension-two locus of \(U\).

Take a general transverse surface slice there. This operation can be performed by fixing general values of local parameters along that locus on a simultaneous embedded resolution. We choose the parameters transverse to the finitely many resolution strata dominating the locus. At a general point in question the normal ambient space is Cohen–Macaulay: a normal space is \(S_2\), and its non-Cohen–Macaulay locus has codimension at least three. The sliced surface is \(S_2\) and regular in codimension one, hence normal. The transverse restrictions of the resolution formulas are crepant surface formulas, by complete-intersection adjunction. One can check this first away from the distinguished point and then extend the isomorphism of the rational Cartier adjoints. In particular, the ordinary surface pair is dlt along the coefficient-one branch. The boundary multiplicities computed on the slice are the multiplicities at the original general point.

We use the following local surface fact. An ordinary dlt surface germ along a coefficient-one branch is either snc for the reduced boundary, or is a plt germ with a smooth pair chart modulo a small cyclic group of some order \(r\). The branch in the chart is a coordinate curve. The different of that branch alone is \(1-1/r\), and the local Cartier index of any integral divisorial sheaf divides \(r\). This is the log surface classification of (Kollár and Mori 1998, Theorem 4.15(1),(3), pp. 119–120, and Proposition 4.18). Its use here is in the analytic-germ category: the numerical plt resolution criterion gives a Hirzebruch–Jung string with the strict boundary meeting an end; the corresponding normal analytic surface germ is the cyclic quotient. It is a classification of the surface germ, and does not assert local factoriality of the higher-dimensional space \(U\). In the snc case set \(r=1\).

Write the additional effective boundary on the slice as a sum with coefficients \(d_j\). Pulling to the cyclic chart and restricting to its smooth coordinate branch gives nonnegative integral intersection multiplicities \(k_j\). Its contribution to the different is \(\sum_j k_jd_j/r\). For the generalized part, restrict a common higher model and the line bundle representing \(p\mathbf M\) to the higher surface. Define its trace on the sliced surface by this proper pushforward; the restricted higher log formula fixes the identification on the punctured surface. This avoids interchanging an arbitrary reflexive hull with specialization. The cyclic quotient fact makes \(rp\) times this trace Cartier. Consequently its pullback minus the higher nef bundle is an exceptional divisor with denominator dividing \(rp\). It is effective by Lemma 5, because the higher bundle is nef on every curve of the projective surface resolution. It does not contain the strict coefficient-one branch, and its restriction to that branch has coefficient \(k/(rp)\) for some \(k\in\mathbb Z_{\geq0}\). The resulting coefficient is therefore \[ 1-\frac1r+\frac{\sum_jk_jd_j+k/p}{r}. \tag{62}\] This proves effectivity of the generalized different. Omitting the nef correction proves effectivity of the ordinary different carried along in the construction. The coefficient is at most one by the sub-lc log formula already established. The local analytic adjunction and nef-defect calculation also appears explicitly in the proof of (C. D. Hacon and Xie 2026, Lemma 3.1, pp. 5–6); the local gdlt modification in that proof is not used here as a global modification theorem.

We can now repeat the construction. On a resolution the chosen chain is a chain of snc strata, and its successive restrictions have the two log formulas just considered. A zero stratum of any restricted formula is a zero stratum of the preceding one; its image is therefore an ambient center meeting the original good open set. This proves the center and gdlt assertions at every stage. Closures and normalizations introduce no finite cover of the chosen center: each selected component is generically isomorphic over it, so its normalization is its unique normalization. The restricted higher model is projective over that normalization and is compact Kähler. The higher line bundle remains nef, and its denominator still divides \(p\), under restriction and subsequent pullback. This proves (59) and the asserted description of the b-data. A fixed chain on a common higher model also shows that these b-data are compatible under any birational comparison which preserves the general points of the chain.

It remains to prove the DCC assertion rather than infer a bound on all Cartier indices. If \(I\) is DCC, its positive elements, when present, have a positive minimum. The same is true after adjoining \(1/p\). A sum of such elements bounded by one consequently has a bounded number of nonzero summands. Finite sums of a nonnegative DCC set satisfy DCC: from any sequence of tuples of a fixed length one can successively choose a subsequence on which each coordinate is nondecreasing. Thus the numerator sums in (60) satisfy DCC. For bounded \(r\), their images in that formula satisfy DCC. In a strictly decreasing sequence of displayed coefficients with first term \(c_0<1\), the inequality \[1-\frac1r\leq c\leq c_0 \quad\Longrightarrow\quad r\leq\frac1{1-c_0}\] bounds \(r\). A sequence starting at one reduces to this case after its first strict decrease. Therefore \(\mathcal D_p(I)\) is DCC, and induction proves the same for each finite iterate. ◻

Proof. The restricted morphisms factor through \(S\) by normality. They are projective, since restriction and this finite factorization preserve relative ampleness; they are bimeromorphic because the ambient step is an isomorphism at the general point of the stratum. Restriction of the relatively ample adjoint, and then pullback by the normalization, gives the asserted signs.

Take a common projective log resolution of the ambient step, carrying its common nef data and preserving the general points of the chosen snc chain. If \(a,b\) are its projections, Lemma 5 gives the actual divisor comparison \[F=a^*(K_V+A+M_V)-b^*(K_{V^+}+A^++M_{V^+})\geq0.\] This difference is anti-nef over \(Z\). Its support contains the full fibers over the non-isomorphism locus in \(Z\): opposite relative ampleness makes the difference nontrivial over each such point, and the fiber-support assertion of Lemma 5 applies. The strict transform of the chosen stratum is not contained in \(\mathop{\mathrm{Supp}}F\), since its general point lies in the common isomorphism open set.

Restrict the two log formulas along the common snc chain. At each restriction the residue canonical bundles are the same, and the common higher nef bundles are the same. They cancel. Each restricted difference is effective, because the next strict stratum is not contained in its support. At the last stage this gives (63) as an equality with its prescribed meromorphic identification. The coefficient of \(E\) in this difference is the increase of its generalized log discrepancy.

For strictness, first make \(E\) divisorial on a smooth higher stratum. This can be achieved by projective blowups of its successive centers; perform the same blowups in the ambient resolution and resolve further, preserving the generic chain. If the center of \(E\) is in the ambient exceptional locus, its entire inverse image in the common ambient model lies in \(\mathop{\mathrm{Supp}}F\), by the full-fiber assertion. Therefore the center on the higher stratum lies in the support of the restricted effective Cartier divisor. Its defining local equation has positive order at \(E\). This proves strictness on either side. In particular, the argument detects an intersection in arbitrary codimension on the original stratum; it does not require that intersection already to be a divisor there. The algebraic counterparts are (Lazić et al. 2023, Lemma 2.8) and (Moraga 2025, Proposition 3.2); the argument above uses the analytic projective negativity statement directly. ◻

Joint proof of Theorem 70, Proposition 71, and Theorem 72. For each \(d\) in increasing order we prove special termination, then the modification assertion, and, if \(d\leq3\), gdlt termination. In dimension zero these statements are immediate. Fix \(d>0\), assume all the appropriate assertions in smaller dimensions, and first prove special termination in dimension \(d\).

Stabilizing centers. There are only finitely many divisorial steps, by Lemma 15; hence we may discard a finite prefix and consider only small steps. Generalized discrepancies do not decrease. A zero-discrepancy place on the positive side was consequently a zero-discrepancy place before the step, and strictness in Lemma 5 excludes its center from either exceptional locus. Thus every new center is the transform of an old center with unchanged general point. If the exceptional locus contains the general point of an old center, that center disappears. There are finitely many centers on the initial model, so after another finite prefix their list is unchanged and all steps are isomorphisms at their general points.

For each center choose a chain of coefficient-one strata at its general point. Transform this choice along the sequence. Lemma 68 gives the corresponding normalized strata \(T_i\) and their adjoints \(D_{T_i}\). The b-line data on these strata are the same under their birational identifications, with one fixed higher denominator. The ambient boundary coefficients belong to a fixed finite set. The coefficients after adjunction therefore belong to one fixed DCC set, depending only on that set, the denominator, and the length of the chosen chain.

We now induct on the dimension \(e\) of a surviving center to show eventual disjointness. A point center is already disjoint after the stabilization just made. For a center of positive dimension \(e\), its proper zero substrata have smaller dimension and there are finitely many of them. Discarding a further finite prefix, we may assume that both exceptional loci avoid every such substratum at every remaining step. In particular, the induced maps on \(T_i\) are isomorphisms in neighborhoods of their non-gklt loci.

Finitely many extractions on the stratum. Identify divisorial places over the \(T_i\) through common proper bimeromorphic analytic models. If a prime divisor \(E\) is extracted by \(T_i\dashrightarrow T_{i+1}\), then \(E\) is a divisor on \(T_{i+1}\) and its center there is contained in the ambient exceptional locus. Otherwise the ambient isomorphism near its general point would provide its strict transform as a divisor on \(T_i\). Lemma 69 gives \[a(E;T_i,A_{T_i}+\mathbf M^T) <a(E;T_{i+1},A_{T_{i+1}}+\mathbf M^T)\leq1,\] where the last inequality uses effectivity of the boundary. By monotonicity its discrepancy on \(T_0\) is less than one. Its center on \(T_0\) cannot be contained in \(\operatorname{Nklt}(T_0,A_{T_0}+\mathbf M^T)\): the maps are isomorphisms near that locus and its transforms, so a valuation centered there cannot acquire a divisorial center through the transformations under consideration.

There are only finitely many such valuations by the off-non-gklt assertion of Lemma 16. In addition to its exceptional valuations, include the finitely many divisors already on \(T_0\) having discrepancy less than one; these are components of its positive boundary. For any one of this finite set, the coefficients on the models where it is extracted form a strictly decreasing sequence: discrepancy never decreases, and each new extraction gives a strict increase. Those coefficients are in the fixed DCC set. Thus each valuation is extracted only finitely often. Altogether there are finitely many extractions.

Removing divisorial changes and stabilizing coefficients. After these extractions, the induced bimeromorphic maps extract no divisors. Lemma 15, applied to the span of prime \((e-1)\)-cycles of the compact Kähler strata, shows that only finitely many further maps contract a divisor. In dimension one a proper bimeromorphic map between normal curves is already an isomorphism. We may therefore assume that every induced map is an isomorphism in codimension one.

With primes matched on this tail, their boundary coefficients do not increase, by Lemma 69. No component with coefficient zero can acquire a positive coefficient. Only the finite boundary support at the start of the tail needs to be considered, and DCC stabilizes each coefficient. Consequently \[(T_i\dashrightarrow T_{i+1})_*A_{T_i}=A_{T_{i+1}}.\] Let \(S_i\) be the normalization of their common image in the contraction base. Neither \(T_i\to S_i\) nor \(T_{i+1}\to S_i\) contracts a prime divisor. Indeed such a prime is matched in codimension one on the other stratum, but its center is in the corresponding ambient exceptional locus. Strict discrepancy increase would contradict coefficient stabilization.

We have thus obtained small diagrams over the varying \(S_i\), with antiample and ample adjoints, respectively, and strict-transform boundary and b-data. The effective difference of Lemma 69 is now exceptional over both stratum models: its coefficient at a divisor on either model is zero by the matched data. If one of the induced birational maps is an isomorphism, the matched boundary and b-line identify its adjoints as actual rational bundles with their canonical meromorphic comparison. The restricted difference is then zero on a common model. The strictness assertion shows that this step has no ambient exceptional intersection with the stratum. Thus only nontrivial small stratum surgeries remain to be excluded.

Lifting the small surgeries in dimension \(e<d\). The induction hypothesis in dimension \(e\) gives a projective crepant gdlt modification \(h_0:Y_0\to T_0\) with globally strongly \(\mathbb Q\)-factorial source. Over \(S_0\) run the relative elementary steps of Proposition 57 for its crepant adjoint. By gdlt termination in dimension \(e\), this run is finite and ends at a model \(Y_1\) whose adjoint is nef over \(S_0\).

On a common smooth model of the initial lift, its endpoint, and \(T_1\), write the actual comparisons as \[P_0=P_1+G=P_++F.\] Here \(P_0\) pulls back the initial crepant adjoint, \(P_1\) pulls back the adjoint of \(Y_1\), and \(P_+\) pulls back \(D_{T_1}\). The relative steps give \(G\geq0\) exceptional over \(Y_1\). The restricted comparison just established, pulled through the initial crepant modification, gives \(F\geq0\) exceptional over \(T_1\). Lemma 66 applies: \(P_1\) is nef over \(S_0\) and \(D_{T_1}\) is ample there. It gives \(P_1=P_+\) and a projective crepant morphism \(h_1:Y_1\to T_1\).

This endpoint is again a gdlt modification of the required type. The elementary steps preserve gdlt and effectivity. They extract no prime divisor, and the stratum map below is small. Any prime on \(Y_1\) exceptional over \(T_1\) was therefore exceptional on \(Y_0\) over \(T_0\). If it survives, its coefficient is still one, since boundaries are pushed forward. Crepancy identifies that coefficient with its zero-discrepancy extraction over \(T_1\).

Repeat over \(S_1,S_2,\ldots\). Each nontrivial stratum surgery forces at least one lifted elementary step. Indeed its negative morphism has a contracted curve \(C\) with \(D_{T_i}\cdot C<0\). Projectivity gives a curve on the lift dominating \(C\) with positive degree, on which the pulled-back adjoint is negative. The lifted model is therefore not already nef over \(S_i\). Infinitely many nontrivial stratum surgeries would give an infinite concatenation of elementary gdlt steps in dimension \(e\). This contradicts the lower-dimensional gdlt termination hypothesis, which allows varying compact contraction bases. There are only finitely many such surgeries. The preceding isomorphism case now proves disjointness for the chosen center.

This completes the induction on \(e\). Taking the maximum of the finitely many resulting indices gives special termination in dimension \(d\), on both sides of every remaining step. This proves Theorem 70 in that dimension.

Constructing the modification in dimension \(d\). Now let \((V,A+\mathbf M)\) be an arbitrary rational glc pair as in Proposition 71 in dimension \(d\). Take a projective log resolution \(h:W\to V\) with \(W\) smooth, carrying the higher nef bundle. Let \[\Gamma=h_*^{-1}A+\sum_{E\subset\mathop{\mathrm{Exc}}(h)}E.\] The pair \((W,\Gamma+\mathbf M)\) is gdlt: its boundary is snc and its data descend to \(W\). Its adjoint has the actual comparison \[ K_W+\Gamma+M_W=h^*(K_V+A+M_V)+Q, \qquad Q=\sum_{E\subset\mathop{\mathrm{Exc}}(h)} a(E;V,A+\mathbf M)E\geq0. \tag{64}\] The support of \(Q\) is contained in \(\lfloor\Gamma\rfloor\). Run relative elementary steps over \(V\) using Proposition 57. On every model the pushforward of (64) remains the identity of its adjoint with the pulled-back original adjoint plus the effective exceptional transform of \(Q\). Every surviving component of that transform is a floor component.

If this run were infinite, special termination in dimension \(d\), just proved, would make its tail disjoint from the floor. A contracted curve on such a tail lies outside the support of \(Q\) and has nonnegative degree against this effective rational Cartier divisor. Its degree against the pulled-back original adjoint is zero. This contradicts negativity of the elementary step. Thus the run is finite; continuation ensures it ends at relative nefness. On the endpoint \(Y\), the remaining transform \(Q_Y\) is nef over \(V\) and exceptional, so Lemma 5 gives \(Q_Y\leq0\). It is also effective, hence zero. The comparison is crepant. The resulting model is projective over \(V\), compact Kähler, globally strongly \(\mathbb Q\)-factorial, and gdlt, by the relative construction. Any prime exceptional over \(V\) is the transform of an exceptional prime on \(W\), and has coefficient one in the pushed boundary. This proves Proposition 71 in dimension \(d\).

Gdlt termination in dimension \(d\leq3\). Finally suppose \(d\leq3\) and consider any infinite elementary gdlt sequence in this dimension. Discard its finite divisorial prefix and apply special termination. On the resulting tail all exceptional loci avoid \(\lfloor A_i\rfloor\). Replace its boundary by \(A_i-\lfloor A_i\rfloor\) on each model, retaining the same b-line. The resulting pairs are effective and gklt. To check the latter, discrepancies cannot decrease on removing an effective rational Cartier floor. Each old zero-discrepancy place has center in the floor, and the order of its pullback is positive, so its discrepancy becomes positive. Equivalently this is the snc calculation on a fixed carrier and the gdlt good-open condition.

The floor has degree zero on every contracted curve on both sides, because it is disjoint from the exceptional loci. Thus all steps of this tail are still negative and positive, respectively, for the new adjoints. Their boundaries are strict transforms, their nef data are unchanged, and their one-ray condition for global line-bundle degrees is unchanged. They are consequently elementary gklt steps in the category of Theorem 67. That theorem excludes an infinite tail. This proves Theorem 72 in dimension \(d\) and completes the triangular induction through dimension four. ◻

Proof. Choose a projective log resolution \(p:W\to V\) carrying \(\mathbf M\) and resolving all the divisor comparisons below. Write \[K_W+C_W=p^*(K_V+B),\qquad H=p^*(N+M_V)-M_W.\] These are the comparisons of actual rational sheaves from Lemma 53. Strong \(\mathbb Q\)-factoriality makes \(N\) and \(M_V\) rational Cartier, and \[ H=p^*N+(p^*M_V-M_W)\geq0 \tag{66}\] by Lemma 5. After increasing \(W\), the support of \(C_W+H\) is simple normal crossing. The crepant higher boundary at parameter \(t\) is \(C_W+tH\).

The divisor \(H\) is nonzero. Otherwise \(p^*T=M_W\) is nef. Since \(p\) is projective, there is a compact curve \(\widetilde C\subset W\) mapping onto \(C\) with positive degree: take a component dominating \(C\) in \(p^{-1}(C)\) and intersect it with sufficiently many general relative hyperplanes. The projection formula would give \[0\leq M_W\cdot\widetilde C =\deg(\widetilde C/C)\,T\cdot C<0.\] Here only degrees on curves are used; no absolute curve criterion for analytic nefness is needed.

For the finitely many prime components \(E\) with \(h_E=\operatorname{coeff}_E H>0\), put \(c_E=\operatorname{coeff}_E C_W\). The log-smooth discrepancy criterion gives \[ s=\min_{h_E>0}\frac{1-c_E}{h_E}. \tag{67}\] Indeed all coefficients of \(C_W+aH\) are strictly below one, and components with \(h_E=0\) impose no new bound. Formula (67) proves finiteness, rationality, attainment, and \(s>a\), as well as the claimed singularities at and below \(s\). ◻

Proof. Take \(h_1\) by Proposition 71. Suppose the current \(h_i\) has been constructed. We choose a nearby klt parameter, run a finite relative program, and identify its endpoint with a crepant model of \(X_{i+1}\).

Choosing a nearby klt parameter. For \(t\in[a,s]\), define its crepant boundary by \[K_{Y_i}+\Delta_i(t)+tM_{Y_i}=h_i^*A_i(t).\] This boundary depends affinely on \(t\). Its nonexceptional coefficients are \(b+tn\) with \(b,n\geq0\). Its exceptional coefficients equal one at \(s\). There are only finitely many exceptional primes, so for a rational \(t_i<s\) sufficiently close to \(s\), with \(t_i\geq a\), the boundary \(\Delta_i(t_i)\) is effective. In particular, a nonexceptional coefficient zero at the positive parameter \(s\) has \(b=n=0\) and stays zero. The pair downstairs is generalized klt for \(a\leq t<s\): its discrepancies are affine combinations of the positive discrepancies at \(a\) and the nonnegative ones at \(s\). Thus the crepant pair at \(t_i\) upstairs is effective and generalized klt.

Choose a curve in the ray of \(f_i\). By the one-ray condition on global rational line-bundle degrees, there is a positive rational number \(q_i\) such that \(A_i(t_i)-q_iA_i(s)\) has zero degree on all \(f_i\)-contracted curves. Apply Lemma 9, using the generalized klt antiample adjoint at \(t_i\), to obtain a rational line bundle \(L_i\) on \(Z_i\) with \[ A_i(t_i)\sim_{\mathbb Q}q_i A_i(s)+f_i^*L_i. \tag{68}\] After pullback by \(h_i\), this is an actual identity modulo a rational bundle from \(Z_i\), not merely an equality of numerical classes.

Running the finite relative program. The composite \(Y_i\to Z_i\) is projective and bimeromorphic. Apply Proposition 59 to the effective generalized klt pair at \(t_i\) to obtain some finite relative elementary run ending nef over \(Z_i\). Transforming (68) through each step shows that the same run is negative, and positive after each flip, for the full adjoint at \(s\). It is therefore a generalized dlt run at \(s\), by Proposition 57, and its endpoint is nef for this adjoint as well.

Identifying the endpoint. To check the endpoint, on a common resolution write \[ P_0=P_j+G_j=P_++F. \tag{69}\] Here \(P_0\) is the pullback of \(h_i^*A_i(s)\), \(P_j\) is the pullback of the lifted endpoint adjoint, and \(P_+\) is the pullback of \(A_{i+1}(s)\). Negativity for the lifted run gives \(G_j\geq0\), exceptional over its endpoint, and negativity for the original small step gives \(F\geq0\), exceptional over \(X_{i+1}\). Both endpoint adjoints are nef over \(Z_i\), and \(A_{i+1}(s)\) is ample there. Lemma 66 applies. Explicitly, if \(p\) and \(q\) denote the maps to these two targets, then \(P_j-P_+=F-G_j\) is \(q\)-nef with pushforward \(-q_*G_j\leq0\); its negative is \(p\)-nef with pushforward \(-p_*F\leq0\). The two negativity inequalities give \(P_j=P_+\). Relative ampleness and projective connected fibers then give the crepant projective morphism from the lifted endpoint to \(X_{i+1}\).

Every prime exceptional for this morphism is the transform of an \(h_i\)-exceptional prime: the original step is small and the lifted run extracts none. If it survives, its coefficient one is preserved by the boundary pushforward rule. This proves the required assertion for \(h_{i+1}\). The string is nonempty, since \(h_i^*A_i(s)\) has negative degree on a curve lifting an \(f_i\)-contracted curve and so is not nef over \(Z_i\). Induction proves the lemma. The choice of \(t_i\) is made separately at each finite stage. ◻

Proof of Theorem 74. From an infinite sequence we will construct successively larger thresholds with fixed coefficient data. At each repetition, special termination will let us remove the floor while retaining an infinite sequence with the required signs.

Step 1: the infinite tail and its first threshold. Suppose there is an infinite sequence. By Lemma 15, it has only finitely many divisorial steps. Truncate after the last of them and relabel the resulting sequence by \(X_1\dashrightarrow X_2 \dashrightarrow\cdots\). Taking actual reflexive traces transports (65) through this finite prefix. The effective divisor remains effective and rational, and a projective common higher model carries the pullback of the original nef line bundle. Thus the relabelled input still has the required expression.

We will repeat the following construction. Its input is an infinite small elementary sequence with rational families \[ (X_i,B_i+tN_i+t\mathbf M),\qquad A_i(t)=K_{X_i}+B_i+tN_i+tM_{X_i}, \quad T_i=N_i+M_{X_i}, \tag{70}\] and a rational \(a\geq0\) such that the pairs at \(a\) are generalized klt, and both \(T_i\) and \(A_i(t)\) have the negative-to-positive signs of each step for all rational \(t\geq a\). The divisors \(B_i,N_i\) are effective strict transforms. Initially \(a=0\), the nef part at zero is zero, and \[T_i\sim_{\mathbb Q}K_{X_i}+B_i, \qquad A_i(t)\sim_{\mathbb Q}(1+t)(K_{X_i}+B_i),\] so these conditions hold.

Let \(s\) be the threshold of the first pair in (70). Lemma 75 applies to a contracted curve and gives \(s\in\mathbb Q\), \(a<s<\infty\). Since \(A_i(s)\) is negative on the input side and ample on the positive side at each step, discrepancy monotonicity shows that every pair in the sequence at \(s\) is generalized lc. Lemma 76 gives an infinite concatenation of generalized dlt elementary steps at \(s\), with compatible crepant endpoint morphisms to the \(X_i\).

Step 2: deleting the floor after special termination. Apply Lemma 15 and Theorem 70 to this concatenation. It has a tail consisting of small steps whose exceptional loci, on both sides, avoid the support of the coefficient-one boundary. Every lifted string is finite and nonempty, so an endpoint occurs beyond any prescribed finite prefix. Start the tail at such an endpoint \(h:Y\to X_i\) and relabel. Write the union of the supports of \(B_i,N_i\) as \(\bigcup P\), with coefficients \(b_P,n_P\geq0\). If \(\widehat P\) denotes the strict transform on \(Y\), the crepant boundary at \(s\) is \[\Delta(s)=\sum_P(b_P+sn_P)\widehat P+\sum_{E\text{ exceptional for }h}E.\] Set \[ \widetilde B=\sum_{b_P+sn_P<1}b_P\widehat P, \qquad \widetilde N=\sum_{b_P+sn_P<1}n_P\widehat P. \tag{71}\] Then \[\widetilde B+s\widetilde N =\Delta(s)-\lfloor\Delta(s)\rfloor.\] Deleting the floor of a generalized dlt pair gives a generalized klt pair with the same nef data. Indeed every place of discrepancy zero has center in that floor, whose rational Cartier pullback has positive order at the place; all other discrepancies were already positive. Consequently \((Y,\widetilde B+s\widetilde N+s\mathbf M)\) is generalized klt. Both divisors in (71) are effective.

Step 3: retaining the signs and increasing the threshold. Let their strict transforms define the new family on every model of the lifted tail. Its signs require checking for the testing direction, as well as for its adjoint. Do this over each original flip base \(Z_i\). At the initial endpoint of its lifted string, the trace comparison gives \[ h_i^*T_i=\widetilde N_i+M_{Y_i}+F_i, \qquad \mathop{\mathrm{Supp}}F_i\subseteq\mathop{\mathrm{Supp}}\lfloor\Delta_i(s)\rfloor. \tag{72}\] To see the support assertion, the difference between pullback and strict transform of \(N_i\), and the difference between pullback and b-line trace of \(M_{X_i}\), are \(h_i\)-exceptional. Every such prime has coefficient one in \(\Delta_i(s)\). The remaining difference consists of the discarded \(n_P\widehat P\), also supported in the floor. This is a comparison of actual rational sheaves; a sign for \(F_i\) is unnecessary.

On \(X_i\), both \(A_i(s)\) and \(T_i\) have negative degree on the contracted ray, and their ratio is a positive rational number \(c_i\). Lemma 9, applied using any nearby generalized klt parameter below \(s\), gives a rational line bundle \(Q_i\) on \(Z_i\) such that \[ h_i^*A_i(s)\sim_{\mathbb Q} c_i(\widetilde N_i+M_{Y_i}+F_i)+(f_i h_i)^*Q_i. \tag{73}\] Transform this identity through the finite lifted string. The tail is small, so all its boundary divisors, traces, and \(F_i\) transform strictly. The support of the transform of \(F_i\) remains in the transformed floor. All contracted and flipped curves are disjoint from that floor. Equation (73) therefore shows that \(\widetilde T=\widetilde N+M_{\mathrm{trace}}\) has negative degree on every contracted ray and positive relatively ample transform on the other side. For the ampleness assertion, the same identity, with the floor bundles trivial near the exceptional fibers, identifies its restriction there with a positive rational multiple of the full ample adjoint; fiberwise ampleness is the relative ampleness criterion.

Likewise subtracting the floor changes neither sign of the adjoint at \(s\). On each side, for every rational \(u\geq s\), the new adjoint is \[\widetilde A(u)=\widetilde A(s)+(u-s)\widetilde T,\] so it has the same negative-to-positive signs. At later endpoints the retained coefficients still obey (71): the downstairs maps are small, and an exceptional divisor of an endpoint morphism cannot become the transform of a downstairs prime. All boundaries upstairs are pushforwards, hence strict transforms on this tail. Thus the families constructed over consecutive bases agree.

We have obtained another infinite small elementary sequence satisfying the same conditions, now with \(a=s\). The generalized klt condition at \(s\) holds throughout the tail by discrepancy monotonicity. It also implies the generalized klt condition at parameter zero, by (66). Lemma 75 shows that its next threshold is finite, rational, and strictly larger than \(s\). The construction can therefore be repeated indefinitely if the original sequence was infinite. It produces \[ 0<s_1<s_2<s_3<\cdots. \tag{74}\]

Step 4: the fixed data for analytic ACC. Let \(I\) be the finite set consisting of zero and the coefficients of the first \(B\), and let \(J\) consist of zero, the coefficients of the first \(N\), and \(1/p\), where \(p>0\) makes \(pM_{W_0}\) integral on the original nef carrier \(W_0\). At every repetition, the new boundary and testing coefficients are obtained solely by omissions from the previous ones, as in (71). Newly extracted coefficient-one primes are removed before the next threshold input. The boundary coefficients therefore stay in \(I\), the testing-boundary coefficients stay in \(J\), and the nef part at parameter zero remains zero. On every required higher model the testing nef data are \(\frac1p\) times the pullback of the same integral nef line bundle. The denominator \(p\) is fixed. Common projective models are needed only for finitely many maps at a time and do not change \(p\).

We verify explicitly that each \(s_j\) belongs to the analytic generalized threshold set with these data. Choose a general point of a computing center on its compact input, and a relatively compact Stein neighborhood \(U\) of that point. Restrict a projective log resolution carrying the data to \(U\), and denote it by \(g:W\to U\). The computing place still meets this restriction. Hence the local threshold equals \(s_j\): the pair is lc at \(s_j\), whereas the same place has negative discrepancy for every larger parameter.

Let \(\mathcal L\) represent the integral line bundle \(pM_W\). The coherent sheaf \(g_*\mathcal L\) has rank one on the isomorphism locus of \(g\). Cartan’s Theorem A supplies a section nonzero at a point of that locus, and thus a nonzero section of \(\mathcal L\) on \(W\). Its divisor \(P'\) is Cartier and represents \(\mathcal L\). In particular \(P'\) is \(g\)-nef and \(M_W\) is represented by \(\frac1pP'\). The induced meromorphic identification gives precisely the original trace and pullback defect. Compatible local canonical representatives may be chosen in the same way from the coherent canonical sheaf. Neither choice changes the boundary or testing-boundary coefficients, the discrepancies, or the factor \(1/p\). The testing boundary is rational Cartier because its components are restrictions of global rational Cartier divisors on the strongly \(\mathbb Q\)-factorial input.

The analytic generalized threshold theorem (C. D. Hacon and Xie 2026, Theorem 1.1 and Theorem 3.3; see also Definition 2.3 and Remark 2.5) consequently applies in dimension four with the fixed DCC sets \(I,J\). Its nef summands are actual relatively nef Cartier divisors with the fixed weight \(1/p\); it requires no common bound for Cartier indices of their downstairs traces. It prohibits the strictly increasing sequence (74). This contradiction proves the theorem. ◻