Proof. The pullback assertion is (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). In the latter statement, every plurisubharmonic function on the regular locus of a normal space extends uniquely to the whole space; local upper boundedness at the singular points is not a further hypothesis to be deduced from \(v\). We apply it only after obtaining a plurisubharmonic function on the entire regular locus. 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 (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 \(L\) and choose any smooth Hermitian metric on that line bundle. The Bott–Chern identity says that the difference between its curvature and the indicated difference of forms is \(\mathrm{d}\mathrm{d}^{c}\) of a global smooth function. Adjusting the metric by the exponential of that function gives the required curvature, with the same denominator restored at the end. Smooth functions and metrics here have their normal-space meaning in local embeddings. 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 2026, 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 \(\kappa=\alpha-c_1(D)\). Insert the descending nef part \(\kappa\) into the ordinary pair. It changes no discrepancy, while \(B+\kappa\) is big. Transcendental base-point-freeness 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, the cited contraction theorem supplies the negative-side Bott–Chern rank and the ordinary real flip. The difference \(p^*D-q^*D'\) is still an actual exceptional real discrepancy divisor, by Lemma 5. The decomposition, transport formula, and correction argument above therefore apply unchanged. Rationality of \(D\) was not used in these arguments. ◻

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. ◻

Topological justification. We give the abstract-space reduction for the finite compatible triangulation used above. Teissier’s compatible Whitney stratification (Teissier 1982, III, Propositions 1.5 and 2.2.2) has finitely many strata on a compact space. Choose finitely many analytic charts \(z_j:U_j\hookrightarrow\mathbb C^{N_j}\) and nonnegative smooth cutoffs \(\rho_j\) compactly supported in these charts whose positivity sets cover the space. Extend \(\rho_j\) and \(\rho_j z_j\) by zero. The map \[x\longmapsto\bigl(\rho_j(x),\rho_j(x)z_j(x)\bigr)_j\] is a topological embedding into a Euclidean space. Near a point where \(\rho_j>0\), its components extend smoothly to the ambient chart and the \(j\)-th block has the smooth left inverse \((t,w)\mapsto w/t\). It is therefore locally an ambient smooth embedding, which preserves the Whitney conditions. Mather’s control data (Mather 1970, sec. 7, Proposition 7.1, and Section 8) and Goresky’s triangulation theorem (Goresky 1978, sec. 5) now give a compatible triangulation, smooth on each stratum. Compactness makes it finite, and smoothness on strata gives the dimension bound. Borel–Moore localization is then the relative simplicial-chain sequence for a closed analytic subset and its complement. ◻

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).

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. Make this reduction once on the fixed resolution; on subsequent blowups retain all exceptional coefficients in the crepant boundary. 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. Indeed, the strict transform of the positive divisor maps isomorphically to that divisor near the generic point of the original centre, because the centre is Cartier within it. A codimension-two centre over that generic point must therefore be this unique intersection. A proper subset has codimension at least three and discrepancy at least \(3-b\); a centre outside the positive divisor has discrepancy at least two. Thus no other branch is possible. The process reaches the given place after finitely many blowups, by the decreasing positive integer empty-boundary discrepancy used in Lemma 16. For this reduced pair, the \(k\)th exceptional divisor on the branch has crepant coefficient \(-k(1-b)\) and log discrepancy \[1+k(1-b),\qquad k=1,2,\ldots.\] Indeed, the first coefficient is \(b-1\), and blowing up the intersection of the newest exceptional divisor with the coefficient-\(b\) strict transform changes \(-k(1-b)\) to \(b-k(1-b)-1=-(k+1)(1-b)\). Thus 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. Only compact analytic topology is used on the normalized components and their common image; the positivity needed next comes from the ambient Kähler fourfold.

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 (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 \(0<\epsilon<1\) and a resolution \(\nu:U\to X\). Consider the compact segment of generalized data \[\mathcal P_\epsilon =\{(\Delta,t\nu^*H):\epsilon\leq t\leq1\} \subset \operatorname{Div}_{\mathbb R}(X)\times H^{1,1}_{\mathrm{BC}}(U).\] The upstairs nef data here are the pullbacks of the fixed smooth form. Every pair in this segment is gklt, its upstairs class is nef, and its boundary-plus-trace is modified big, since \(tH\) is Kähler on \(X\). This real segment is allowed in the finiteness theorem for weak log canonical models (Hacon and Xie 2026, Theorem 5.15); neither rationality of \(H\) nor strong \(\mathbb Q\)-factoriality of \(X\) is required there.

Let \(\phi_i:X\dashrightarrow X_i\) be the accumulated map, and suppose \(\lambda_i\geq\epsilon\). It is a weak log canonical model of the original generalized pair at parameter \(\lambda_i\). Indeed, it extracts no divisors, its target adjoint is nef, and discrepancies do not decrease: apply (26) to every preceding step at the fixed parameter \(\lambda_i\), which is no larger than any preceding threshold. On a common resolution the original adjoint is the pullback of the target nef class plus an effective exceptional divisor. Pseudo-effectivity descends by Lemma 4, so the original data belong to the pseudo-effective portion of \(\mathcal P_\epsilon\).

The finiteness conclusion concerns the maps \(\phi_i\), up to isomorphisms of their targets commuting with the maps from \(X\). These marked maps are pairwise distinct in the flip tail. In fact, given two stages, an exceptional prime has strictly increasing ordinary discrepancy at the first intervening flip by Lemma 5, and later steps cannot decrease it. A target isomorphism compatible with the marking would identify the transformed boundary and all its discrepancies, a contradiction. This is also the distinction used in the proof of (Hacon and Xie 2026, Theorem 5.16).

There are therefore only finitely many \(i\) with \(\lambda_i\geq\epsilon\). Since \(\epsilon>0\) was arbitrary, the thresholds tend to zero. ◻

Proof. Discard the finitely many divisorial steps. All remaining maps \(X_i\dashrightarrow X_{i+1}\) are small. For each such \(i\), put \[\mathcal E_i=\{E\text{ exceptional over }X_i: a(E,X_i,\Delta_i+\lambda_i\mathbf H)\leq1\}.\] The fixed nef datum has a carrier projective over \(X_i\): take a common resolution of the finitely many preceding projective ordinary step diagrams. For a flip its graph is projective over either side, so this construction and a further projective log resolution retain projectivity over \(X_i\). On this resolution the datum is the pullback of \(h\). Lemma 16 therefore makes \(\mathcal E_i\) finite. It allows real coefficients and counts places of log discrepancy exactly one as well as those below one.

Smallness identifies the primes exceptional over successive \(X_i\). Their log discrepancies at the current threshold agree by crepancy. On the new model, decreasing the coefficient of the nef part increases discrepancies: in the trace formula of Lemma 6, the correction divisor is effective. Thus, for every such prime, \[ \begin{split} a(E,X_{i+1},\Delta_{i+1}+\lambda_{i+1}\mathbf H) &\geq a(E,X_{i+1},\Delta_{i+1}+\lambda_i\mathbf H)\\ &=a(E,X_i,\Delta_i+\lambda_i\mathbf H). \end{split} \tag{28}\] It follows that \(\mathcal E_{i+1}\subseteq\mathcal E_i\). After another truncation they equal one finite set \(\mathcal E\). Only this set stabilizes; no discreteness of the real discrepancy values is asserted.

Apply the exact extraction theorem (Hacon and Xie 2026, Corollary 2.29) to the generalized pair at \(\lambda_i\) and the prescribed set \(\mathcal E\). It gives a strongly \(\mathbb Q\)-factorial compact Kähler model \(\mu_i:Y_i\to X_i\) extracting precisely those primes. Its compact Kähler output is supplied by the relative MMP in its proof. The input of this theorem need not be strongly \(\mathbb Q\)-factorial, and its nef datum and crepant boundary may be real.

Let \(\Theta_i\) be the crepant generalized boundary on \(Y_i\). On the transform of a divisor of \(X_i\), its coefficient is the coefficient in \(\Delta_i\); on an extracted prime \(E\), it is \[1-a(E,X_i,\Delta_i+\lambda_i\mathbf H)\in[0,1).\] Thus \(\Theta_i\) is effective. The generalized crepant relation gives (27). More explicitly, subtract the ordinary class \(K_{Y_i}+\Theta_i\) from the pulled-back generalized adjoint and divide by \(\lambda_i>0\). Lemma 6 identifies the result as the trace \(H_{Y_i}\) of our fixed b-datum, with an effective exceptional correction on a common resolution. Equation (27) is nef because its right-hand side is a pullback of a nef class.

Every prime exceptional over \(Y_i\) is exceptional over \(X_i\) and is absent from \(\mathcal E\), so its generalized log discrepancy is greater than one. Dropping the nef part weakly increases this discrepancy. Hence \((Y_i,\Theta_i)\) is ordinary terminal. Together with the boundary coefficients below one, this also proves that it is klt.

The flip tail identifies all prime divisors on the \(X_i\), and each \(Y_i\) extracts exactly the same additional primes. Consequently the \(Y_i\) are naturally isomorphic in codimension one. Fix the first one, \(Y=Y_{i_0}\). On its primes inherited from \(X_{i_0}\), the coefficients of \(\Theta_{i,Y}\) are constant. On the extracted primes, they decrease by (28). They are nonnegative and have a fixed finite containing support, so their coefficientwise limit \(\Theta_Y\) exists and \[0\leq\Theta_Y\leq\Theta_{i,Y}\leq\Theta_{i_0,Y}.\]

Terminality of the limit follows by comparison with the first boundary on this fixed model. Since \(Y\) is \(\mathbb Q\)-factorial, the effective real divisor \(\Theta_{i_0,Y}-\Theta_Y\) has nonnegative order at every divisorial valuation. Therefore \[a(E,Y,\Theta_Y)\geq a(E,Y,\Theta_{i_0,Y})>1\] for every exceptional prime \(E\). All boundary coefficients remain below one, so the limit is klt as well. This argument uses decrease of the boundary, rather than any assertion that terminality is closed under limits. Finally, only coefficients of \(\mu_i\)-exceptional primes vary. The transformed limiting boundary thus pushes forward to \(\Delta_i\), proving (iii). ◻

Proof. Run the ordinary real-boundary program from \((Y,\Theta_Y)\) using (Hacon and Xie 2026, Proposition 4.2), with zero nef part. We show inductively that every step is a flip. Let \(T\) be a current model, so that \(T\) is small to every \(Y_i\), and write \[P_T=K_T+\Theta_T,\qquad Q_{i,T}=K_T+\Theta_{i,T}+\lambda_i H_T .\] The trace \(H_T\) exists by Lemma 6: along these strongly \(\mathbb Q\)-factorial real-boundary flips we use the canonical Bott–Chern transport of (Hacon and Xie 2026, Proposition 4.2). Its pullback comparison has an actual exceptional-divisor correction, so the same trace relation propagates with the fixed b-data. Together with the effective transformed boundary, the actual trace relation supplies the generalized structure boundary for \(Q_{i,T}\); no generalized klt condition is required for the small-model comparison below. The finite boundary support and \(\lambda_i\to0\) give \(Q_{i,T}\to P_T\) in \(H^{1,1}_{\mathrm{BC}}(T)\).

Fix a \(P_T\)-negative extremal ray and its contraction. For all sufficiently large \(i\) this same ray is \(Q_{i,T}\)-negative. Take a common smooth Kähler resolution \(p:U\to T\), \(q:U\to Y_i\). The boundaries are transforms of one another and the b-part is unchanged. Thus Lemma 8, together with (29), gives an actual divisor \(E_i\) such that \[ p^*Q_{i,T}=q^*Q_{i,Y_i}+[E_i], \qquad E_i\ge0, \qquad E_i\text{ is exceptional over both models}. \tag{30}\]

Suppose that the contraction were divisorial. Choose a general point of an exceptional prime outside the codimension-at-least-two image of \(\mathop{\mathrm{Exc}}(p)\). A positive-dimensional projective fibre through this point contains an irreducible curve \(C\). Its strict transform \(\widetilde C\) on \(U\) is not contained in \(\mathop{\mathrm{Supp}}(E_i)\). Consequently \[Q_{i,T}\cdot C =q^*Q_{i,Y_i}\cdot\widetilde C+E_i\cdot\widetilde C\ge0,\] contrary to the choice of the ray. For a fibre-type contraction the same argument uses a general point of \(T\) instead. It also rules out ending the program with a Mori fibre space.

Therefore every step is an ordinary flip. Terminality persists by the ordinary discrepancy comparison. These flips have the exact relative properties needed for the auxiliary termination theorem. The negative contraction is projective by (Hacon and Xie 2026, Proposition 4.2); the positive one is proper birational between rational-singularity spaces, hence strongly Moishezon by (Hacon and Xie 2026, Lemma 2.39), and projective by (Hacon and Xie 2026, Lemma 2.40). For a step \(T\xrightarrow{f}Z\xleftarrow{f^+}T^+\), the signed ordinary adjoints \(-P_T\) and \(P_{T^+}\) are relatively Kähler real line bundles. To obtain relative ampleness in this intrinsic line-bundle setting, write either signed adjoint \(L\) in a finite real span of integral line bundles, and choose a base class \(\gamma\) such that \(c_1(L)+h^*\gamma\) is Kähler, where \(h=f\) or \(f^+\). Openness of the Kähler cone gives a rational simplex around the coefficient vector of \(L\) whose rational vertices \(L_\nu\) still have \(c_1(L_\nu)+h^*\gamma\) Kähler. Lemma 11 makes every \(L_\nu\) relatively ample, and their positive convex combination \(L\) is therefore relatively ample as a real line bundle. Thus Proposition 20 makes this sequence finite. Its endpoint has nef ordinary adjoint, as asserted. ◻

Proof. If \(B=0\), take \(r=1\) and \(B^1=0\). Otherwise work in the finite-dimensional coefficient space on \(\mathop{\mathrm{Supp}}(B)\). A sufficiently small simplex with rational vertices \(C^0,\ldots,C^d\) contains \(B\) in its interior and consists of effective klt boundaries. Indeed, all coefficients of \(B\) on this support are positive, and the klt inequalities are open on one log resolution. Put \(D_k=K_W+C^k\).

Choose a rational number \(\delta>0\) smaller than all the barycentric coordinates of \(B\). Let \(\mathcal P\) be the smaller rational polytope in the simplex defined by requiring every barycentric coordinate to be at least \(\delta\). We claim that its nef locus \[\mathcal N=\{C\in\mathcal P:K_W+C\text{ is nef}\}\] is a rational polytope.

Consider an extremal ray \(R\) that is negative for \(K_W+C\) for some \(C\in\mathcal P\). At least one vertex adjoint, say \(D_a\), is negative on \(R\). Among all irreducible rational curves whose classes span \(R\), choose \(\Gamma\) minimizing \(-D_a\cdot\Gamma\). Such curves exist by the cone theorem (Hacon and Xie 2026, Theorem 1.3), and a minimum exists because a Cartier multiple of \(D_a\) puts their positive degrees in a discrete subset of \(\mathbb R_{>0}\).

This choice minimizes the numerical scale of the curve class on \(R\). For any other vertex \(D_k\) negative on \(R\), the cone theorem supplies a rational curve \(\Gamma_k\) on \(R\) with \(0<-D_k\cdot\Gamma_k\le8\). Since \([\Gamma]\) is a positive multiple of \([\Gamma_k]\) with multiplier at most one, it follows that \(-D_k\cdot\Gamma\le8\) as well. Vertices nonnegative on \(R\) need no estimate. Thus all the numbers \(d_k=D_k\cdot\Gamma\) satisfy \(d_k\ge-8\).

Write a member of \(\mathcal P\) that is negative on \(R\) as \(\sum t_kC^k\), where \(t_k\ge\delta\) and \(\sum t_k=1\). Then \(\sum t_kd_k<0\), so for each \(k\), \[t_kd_k<8(1-t_k).\] In particular, \[ -8\le d_k<\frac8\delta\qquad(0\le k\le d). \tag{31}\] Each \(D_k\) has a fixed global Cartier index on \(W\). Hence its degrees belong to a fixed discrete subgroup of \(\mathbb R\), and (31) permits only finitely many vectors \((d_0,\ldots,d_d)\). Their nonnegativity inequalities are rational.

These finitely many inequalities define exactly \(\mathcal N\) inside \(\mathcal P\). Indeed, a ray never negative on \(\mathcal P\) imposes no further condition. For any other ray the chosen vector tests its nonnegativity. Finally, positivity on all extremal rays is positivity on \(\overline{\mathrm{NA}}(W)\): intersect this cone with the hyperplane where a fixed Kähler class has value one and use the resulting compact convex slice. Nef-cone duality now proves the claim. Express \(B\) as a convex combination of rational vertices of \(\mathcal N\), omitting vertices with zero weight. These are the required \(B^j\). ◻

Proof. On \(W\), set \(G_i=\Theta_{i,W}-\Theta_W\ge0\). Keep the integer \(m\) in (33) fixed and consider the generalized adjoint \[ A_W=K_W+\Theta_W+mG_i+m\lambda_iH_W =mQ_{i,W}-(m-1)P . \tag{35}\] For large \(i\) its pair is gklt. To see this, choose one log resolution carrying the finite boundary support and the fixed b-data. The crepant boundary coefficients vary continuously with \(G_i\) and \(\lambda_i\); at the limit they are those of the ordinary klt pair \((W,\Theta_W)\). Thus they remain strictly less than one. The boundary \(\Theta_W+mG_i\) is effective. The pair for \(Q_{i,W}\) is the convex interpolation with weight \(1/m\) between this pair and \((W,\Theta_W)\), so it is gklt too.

The sum \(\Theta_{i,W}+\lambda_iH_W\) is big: \(\lambda_i>0\), the trace \(H_W\) is big by Corollary 7, and \(\Theta_{i,W}\) is effective. The model \(W\) is strongly \(\mathbb Q\)-factorial. Choose a sufficiently large Kähler scaling class and apply (Hacon and Xie 2026, Theorem 1.5) to obtain a finite \(Q_{i,W}\)-program.

All its steps are flips and its endpoint is nef. Indeed, as long as the current model \(T\) is small to \(Y_i\), the comparison \[p^*Q_{i,T}=q^*Q_{i,Y_i}+[E],\qquad E\ge0\] holds on a common resolution, with \(E\) exceptional over both models. The curve-through-a-general-point argument in Proposition 25 excludes a negative divisorial contraction or a fibre-type contraction. It applies with this fixed \(i\), without any limiting argument.

We prove the remaining assertions by induction along these flips. Suppose that on the current model \(T\) the ordinary pairs \((T,B^j_T)\) are klt, the \(P^j_T=K_T+B^j_T\) are nef with \(\ell_jP^j_T\) line bundles, and the amplified pair with class \[A_T=mQ_{i,T}-(m-1)P_T\] is gklt. These conditions hold initially. A \(Q_{i,T}\)-negative extremal ray is \(A_T\)-negative because \(P_T\) is nef. The cone theorem for the amplified pair gives a rational curve \(\Gamma\) spanning that ray with \(A_T\cdot\Gamma\ge-8\). If \(P_T\cdot\Gamma>0\), then (34) implies \[mQ_{i,T}\cdot\Gamma =A_T\cdot\Gamma+(m-1)P_T\cdot\Gamma \ge-8+(m-1)c>0,\] a contradiction. Hence the ray is \(P_T\)-trivial. Since every \(u_j\) is positive and every \(P^j_T\) is nef, it is also \(P^j_T\)-trivial for each \(j\).

Let \(f:T\to Z\leftarrow T^+:f^+\) be this flip. For the ordinary rational klt pair \((T,B^j_T)\), the line bundle \(M_j=\ell_jP^j_T\) is numerically trivial over \(Z\), and \[M_j-(K_T+B^j_T)=(\ell_j-1)P^j_T\] is relatively nef. Lemma 9 gives an actual line bundle \(M_{j,Z}\) with \(M_j=f^*M_{j,Z}\). The pullback \((f^+)^*M_{j,Z}\) agrees on the common big open set with the reflexive sheaf of \(\ell_j(K_{T^+}+B^j_{T^+})\). Normality and reflexivity extend the agreement. Thus the same integer \(\ell_j\) is a Cartier index on \(T^+\); no further multiple is taken.

It follows that both adjoints \(P^j_T\) and \(P^j_{T^+}\) pull back from the same class on \(Z\). On a common smooth compact Kähler resolution \(p:V\to T\), \(q:V\to T^+\), their pullbacks are therefore equal. The class \(p^*P^j_T\) is nef, and nef descent along \(q\) in Lemma 3 shows that \(P^j_{T^+}\) is nef: the target \(T^+\) is generalized klt and hence has rational singularities. Their actual discrepancy-change divisor is zero by negativity, so the pairs remain klt and the comparison is crepant. In particular the same is true of their convex combination \(P_T\).

Finally, on a common resolution the discrepancy-change divisor for the amplified pair is now \(m\) times that for the \(Q_i\)-pair, because the contribution from \(P_T\) is zero. The latter divisor is effective along the chosen generalized flip. The amplified pair therefore remains gklt. This completes the induction, including the fixed-index assertion and the validity of (34) at the endpoint.

It remains to prove assertion (4). Write \(Q=Q_{i,W_i}\) and \(P'=P_{W_i}\). Both are nef. Suppose a \(Q\)-trivial curve has positive \(P'\)-degree. The face \[\mathcal F=\{\gamma\in\overline{\mathrm{NA}}(W_i):Q\cdot\gamma=0\}\] then has a compact Kähler slice on which \(P'\) takes a positive value. Maximize \(P'\) on that slice and choose an extreme point of its maximizing face. This point is extreme in the slice of \(\mathcal F\). Since \(\mathcal F\) is a face of \(\overline{\mathrm{NA}}(W_i)\), it spans an extremal ray of the full cone. On this ray \(Q=0\) and \(P'>0\), so the amplified class \(A=mQ-(m-1)P'\) is negative. Its cone theorem gives a rational generator \(\Gamma\) with \[-8\le A\cdot\Gamma =-(m-1)P'\cdot\Gamma \le-(m-1)c<-8,\] which is impossible. ◻

Proof. Suppose it were infinite. Use Proposition 24 and choose \(W\), then a sufficiently large \(i\) and \(W_i\), as in Propositions 25 and 27. Let \(f_i:X_i\to Z_i\) be the contraction of the chosen \(D_i\)-negative ray in the original flip tail. The threshold class \(L_i(\lambda_i)\) descends to \(Z_i\).

Take a common smooth compact Kähler resolution \(p:V\to W_i\) and \(q:V\to Y_i\). Put \(s=\mu_iq\) and \(r=f_is\). The relevant maps are shown in Figure 2.

The models \(W_i\) and \(Y_i\) are small modifications with the same transformed boundary and b-data. Their \(Q_i\)-classes are both nef. Applying Lemma 8 in both directions gives \[ p^*Q_{i,W_i}=q^*Q_{i,Y_i} =s^*L_i(\lambda_i). \tag{36}\] Consequently this class has zero degree on every \(r\)-vertical curve. For such a curve \(C\), if \(p(C)\) is a point then \(p^*P_{W_i}\cdot C=0\) immediately. Otherwise the projection formula and (36) show that \(p(C)\) is \(Q_{i,W_i}\)-trivial. Proposition 27(4) then gives \(p^*P_{W_i}\cdot C=0\) as well.

The map \(r\) is proper and birational. Its target \(Z_i\) is normal compact Kähler with rational singularities by the ordinary-step construction, and its source \(V\) is smooth compact Kähler. Its fibres are connected, and its general fibre is a point. Thus all the hypotheses of Lemma 3 apply to \(r\). There is a class \(\xi\in H^{1,1}_{\mathrm{BC}}(Z_i)\) with \[ p^*P_{W_i}=r^*\xi. \tag{37}\]

We next form the ordinary canonical-and-boundary comparison. Using compatible canonical data, the difference \[ F=p^*(K_{W_i}+\Theta_{W_i}) -s^*(K_{X_i}+\Delta_i) \tag{38}\] is an actual real divisor on \(V\), with the relative canonical terms interpreted as in Section 2. It is exceptional over \(X_i\): on every prime that is not \(s\)-exceptional the two boundary coefficients agree, since the limiting boundary on \(Y_i\) pushes to \(\Delta_i\) and \(W_i\) is small to \(Y_i\). This constructs the divisor before any appeal to its cohomology class.

By (37), \[[F]=p^*P_{W_i}-s^*D_i =s^*(f_i^*\xi-D_i).\] Hence \(F\) is numerically trivial over \(X_i\). Lemma 5, applied to \(F\) and to \(-F\), gives \(F=0\). Pullback injectivity for \(s\) now yields \[D_i=f_i^*\xi\quad\text{in }H^{1,1}_{\mathrm{BC}}(X_i).\] Its intersection with the ray contracted by \(f_i\) is therefore zero, contradicting the strict \(D_i\)-negativity with which that ray was chosen. The scaling program is finite. ◻