Finite ordinary minimal model programs on compact Kähler fourfolds

Proof. The pullback assertion is [13, Lemma 2.39]; nef descent is [4, 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

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 (2.1) 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=\ddc\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 [3, Section 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

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 [3, Section 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 [13, 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 \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

This difference is defined locally by compatible meromorphic canonical generators and therefore glues, even when the canonical sheaves were specified as \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

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 \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 \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 (2.4) therefore places the entire fibre in \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\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 (2.4). A component of its support has dimension three and maps finitely into \Exc(f)\times_Z\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

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 \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 2.3(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 2.1 then gives uniqueness of the class as well.

Pulling back (2.6) 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 (2.7) follows by adding K_V and the fixed nef class tq^*H to its two sides. Its coefficients give

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

is an actual p-exceptional divisor. Subtract the ordinary log class from (2.2) to obtain (2.6). 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 2.2 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 [6, Chapter III, Corollary 2.11]. Thus S_T is precisely the actual trace in (2.5). This step is needed because a trace of a positive b-current need not have local potentials a priori [5, 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 [7, 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,

Its class is p^*(H_T-\varepsilon[\omega_T]). Lemma 2.2 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 (2.8). 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 2.3(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

This is relative bigness in the analytic sense [10, 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 [10, 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_*\cO_T=\cO_Z, on this neighbourhood f_*L\simeq \cO_U, and evaluation is an isomorphism. These local conclusions give (3.1) 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

is compact. Indeed, fix smooth representatives of a basis of \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 \Q-Cartier. The finite form of the cone theorem gives

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 \ddc 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 [11, 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 3.2 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 3.3 proves projectivity and relative ampleness. This establishes projectivity without strong factoriality. For every curve C\subset T,

The rational curve generating R is contracted, so f is nontrivial.

Relative vanishing gives R^j f_*\cO_T=0 for j>0 [10, Theorem 5.2]; the log Fano base has rational singularities [4, Lemma 8.8]. Lemma 2.1 now identifies f^*\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 (3.4). 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 \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 \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 \Exc(f)=E.

For a prime Weil divisor A_Z on Z, let A_T be its strict transform. Choose t\in\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 3.1: its difference from D is relatively ample. The descended line bundle agrees with the corresponding reflexive power of \cO_Z(A_Z) off f(E), a subset of codimension at least two. Reflexivity identifies them everywhere. Hence every global prime on Z is \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 (3.5). Intersection with R gives e>0. If b_E is the coefficient of E in B, then

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 [10, Theorem 1.18] gives local finite generation of (3.6). 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 [10, 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 \cO_T(rD) on the common big open. Its reflexive extension is \cO_{T'}(rD'), proving (3.7) 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 3.3, T' is compact Kähler.

For a global prime A' on T', transform it to A on T and choose t\in\Q so (A+tD)\cdot R=0. Descend a Cartier multiple by Lemma 3.1. 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 \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 2.3 gives the actual discrepancy divisor

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 \cF 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 \cF is therefore a line bundle.

In the small case start with such a sheaf \cF' on T'. On a common smooth resolution put

Here \cM is a line bundle and \cF is coherent reflexive of rank one; both give the required transform on the common big open. An invertible power of \cF, 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 \cF'. 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 (3.4) and pullback injectivity give the unique \eta. On a common resolution over Z the two pullbacks of \eta agree. Therefore

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

If H_T=f^*\eta+s\,c_1(D), the new correction is the actual divisor

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 2.4.

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 2.3. 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

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

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 (4.1), 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 [22, Chapter 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\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

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 [17, Section 7, Proposition 7.1, and Section 8] and Goresky’s triangulation theorem [12, Section 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\BC(X) be the generalized log class and let M_W=[\bM_W] be the nef trace class on W. Write the crepant formula as

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 [16, Proposition 2.36] and [2, 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

Let E be exceptional over a smooth model with this log smooth boundary. At a general point of its center C, let c=\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

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 (4.3). 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 (4.3). 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

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 (4.4). 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

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

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

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 (4.5). 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 (4.6) 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 (4.2) 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 (4.5).

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 \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 (4.3), 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 4.2. For this reduced pair, the kth exceptional divisor on the branch has crepant coefficient -k(1-b) and log discrepancy

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 2.3, 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 4.4.

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

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 (4.7) belong to a finite set, even for irrational coefficients. Indeed, terminality gives

Each multiplicity is an integer, and there are finitely many positive components. We use no discreteness assertion for all discrepancies.

Strictness in Lemma 2.3 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 4.5, 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 (4.7). 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 2.3. 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

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

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

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 2.3 gives

Smallness bounds both dimensions by two, so \Exc(f_i) contains a surface and \dim\Exc(f_i^+)\leq1. Apply Lemma 4.1 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 2.3. For a divisorial contraction, (3.5) 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 4.1 with k=3 bounds the number of divisorial steps. Any infinite sequence would therefore have a tail consisting only of flips, contrary to Proposition 4.6 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

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 2.5. 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 [13, 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 3.4 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 3.5 transports the trace H_i to H_{i+1} while preserving its relation to the fixed b-datum. By Lemma 2.1, 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 2.6. 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

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 3.4.□

Proof. Fix 0<\epsilon<1 and a resolution \nu:U\to X. Consider the compact segment of generalized data

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 [13, Theorem 5.15]; neither rationality of H nor strong \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 (5.1) 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 2.2, 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 2.3, 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 [13, 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

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 4.2 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 2.4, the correction divisor is effective. Thus, for every such prime,

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 [13, Corollary 2.29] to the generalized pair at \lambda_i and the prescribed set \mathcal E. It gives a strongly \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 \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

Thus \Theta_i is effective. The generalized crepant relation gives (5.2). More explicitly, subtract the ordinary class K_{Y_i}+\Theta_i from the pulled-back generalized adjoint and divide by \lambda_i>0. Lemma 2.4 identifies the result as the trace H_{Y_i} of our fixed b-datum, with an effective exceptional correction on a common resolution. Equation (5.2) 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 (5.3). They are nonnegative and have a fixed finite containing support, so their coefficientwise limit \Theta_Y exists and

Terminality of the limit follows by comparison with the first boundary on this fixed model. Since Y is \Q-factorial, the effective real divisor \Theta_{i_0,Y}-\Theta_Y has nonnegative order at every divisorial valuation. Therefore

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 [13, 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

The trace H_T exists by Lemma 2.4: along these strongly \Q-factorial real-boundary flips we use the canonical Bott–Chern transport of [13, 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 \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 2.6, together with (6.1), gives an actual divisor E_i such that

Suppose that the contraction were divisorial. Choose a general point of an exceptional prime outside the codimension-at-least-two image of \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 \Supp(E_i). Consequently

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 [13, Proposition 4.2]; the positive one is proper birational between rational-singularity spaces, hence strongly Moishezon by [13, Lemma 2.39], and projective by [13, 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 3.3 makes every L_\nu relatively ample, and their positive convex combination L is therefore relatively ample as a real line bundle. Thus Proposition 4.6 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 \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

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 [13, Theorem 1.3], and a minimum exists because a Cartier multiple of D_a puts their positive degrees in a discrete subset of \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,

In particular,

Each D_k has a fixed global Cartier index on W. Hence its degrees belong to a fixed discrete subgroup of \R, and (6.3) 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 \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 (6.5) fixed and consider the generalized adjoint

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 2.5, and \Theta_{i,W} is effective. The model W is strongly \Q-factorial. Choose a sufficiently large Kähler scaling class and apply [13, 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

holds on a common resolution, with E exceptional over both models. The curve-through-a-general-point argument in Proposition 6.1 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

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 (6.6) implies

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

is relatively nef. Lemma 3.1 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 2.1 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 (6.6) 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

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 \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

which is impossible.□

Proof. Suppose it were infinite. Use Proposition 5.3 and choose W, then a sufficiently large i and W_i, as in Propositions 6.1 and 6.3. 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 2.6 in both directions gives

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 (6.8) show that p(C) is Q_{i,W_i}-trivial. Proposition 6.3(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 2.1 apply to r. There is a class \xi\in\BC(Z_i) with

We next form the ordinary canonical-and-boundary comparison. Using compatible canonical data, the difference

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 (6.9),

Hence F is numerically trivial over X_i. Lemma 2.3, applied to F and to -F, gives F=0. Pullback injectivity for s now yields

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.□

Proof. Set D=K_T+B and D'=K_{T'}+B'. The ordinary comparison in Proposition 3.4 and Lemma 2.3 gives, on a common smooth compact Kähler resolution p:V\to T, q:V\to T',

where F is an actual divisor exceptional over T'. If D' is pseudo-effective, its pullback plus [F] is pseudo-effective; descent along p shows that D is pseudo-effective. Conversely, if D is pseudo-effective, exceptional descent along q applied to q^*D'+[F] proves that D' is pseudo-effective. Both uses of pullback and descent are justified by Lemma 2.2.□

Proof. Suppose that c_1(D) contains a closed positive current T with local plurisubharmonic potentials. Choose a point y where such a potential is finite and which lies on a positive-dimensional component of a fibre of g. This is possible on the dense open locus of fibres of the expected dimension: the pole set of a plurisubharmonic potential contains no open subset. That fibre component is projective, so hyperplane sections through y give an integral curve C through y contained in the fibre. Let \nu:C^\nu\to Y denote its normalization followed by inclusion.

Local potentials pull back under \nu to subharmonic functions or to the constant -\infty. Finiteness at a preimage of y excludes the latter alternative on the connected curve. Thus they define a positive current \nu^*T in the class \nu^*c_1(D), and

This contradicts D\cdot C<0, which follows from relative ampleness. The choice of y is essential: no restriction of T to a curve contained in its pole set has been used.□

Proof of Theorem 1.2. Run the scaling construction on the given pair (X,\Delta). Proposition 3.4 supplies each ordinary negative step, and Proposition 6.4 proves that the construction stops after finitely many such steps. Write its endpoint as (Y,\Gamma). All intermediate varieties are normal compact Kähler fourfolds, all boundaries are the rational transforms of \Delta, and all pairs remain ordinary klt and globally Weil-divisor \Q-factorial with the stipulated canonical datum. The optional strong global property is preserved by the same Proposition. The auxiliary extractions used to prove finiteness do not change this program, which starts on X itself.

The stopping rule gives either a nef adjoint K_Y+\Gamma or a Mori contraction g:Y\to S. In the latter case Proposition 3.4 supplies a normal compact Kähler base, a projective surjective morphism with connected fibres, \dim S<4, relative numerical rank \rho(Y/S)=1, and relative ampleness of -(K_Y+\Gamma). These are precisely the required Mori fibre space conditions, with relative rank computed from global Cartier classes. The same Proposition gives relative Bott–Chern dimension one for every negative contraction, including g.

By Lemma 7.1, the adjoint at every stage has the same pseudo-effectivity status as K_X+\Delta. If the latter is pseudo-effective, Lemma 7.2 excludes the Mori alternative, so the endpoint is nef. If it is not pseudo-effective, the endpoint cannot be nef, since nef classes are pseudo-effective by Lemma 2.2; hence the endpoint has the asserted Mori fibre structure.

For completeness, the nef endpoint has the discrepancy properties of a log minimal model. No step extracts a divisor. On a common resolution of the finite sequence, with maps p:V\to X and q:V\to Y, the ordinary effective comparison divisors add to

with F exceptional over Y. Consequently, for every prime divisor E over these models,

If a prime divisor on X is contracted, its discrepancy increases strictly at the divisorial step that contracts its transform and never decreases thereafter. Thus the displayed inequality is strict for every such prime. Here a denotes the log discrepancy, as throughout the paper.

If the initial adjoint is nef, the construction takes no steps; conversely, a zero-step nef outcome requires the initial adjoint to be nef. In the non-pseudo-effective case a zero-step program is also allowed when the first selected contraction already gives a Mori fibre structure on X. This completes both endpoint assertions.□

Proof. Construction. Let b:B\to\Pbb^4 be the blowup of a point, let H=b^*\cO_{\Pbb^4}(1), and let F be the exceptional divisor. The graph of projection from that point embeds B in \Pbb^4\times\Pbb^3, and the restriction of \cO(1,1) is 2H-F. Thus 2H-F is very ample. Since H is globally generated,

is very ample as well. Take a general smooth member U\in|6H-2F|. Its image U_0\subset\Pbb^4 is smooth away from the blown-up point. The quadratic jets of the defining sextics at that point are arbitrary, so a general member has a nondegenerate quadratic initial term. The holomorphic Morse lemma identifies its unique singularity with

The restriction \pi:U\to U_0 resolves this node, with exceptional quadric

Adjunction gives

since K_B=-5H+3F.

Integral weak Lefschetz for the smooth ample threefold U\subset B [19, Section 7, Corollary 7.3] gives

This is the blowup calculation underlying the classical example in [20, Definition 5.1, Remark (b)]. Let C be a smooth elliptic curve and set X=U\times C. Its canonical divisor is the pullback of (A.1), so is effective. If \ell is either ruling in E\times\{c\}, then K_X\cdot\ell=-1; in particular K_X is not nef.

All negative contractions have the same target. Put D=E\times C and let h=\{e\}\times C be a horizontal curve. A curve not contained in D has nonnegative degree against both the effective divisor D and the nef hyperplane pullback. Thus every K_X-negative curve lies in D. The two rulings of E have the same class in H_2(U,\R): both have degrees 0,-1 against the displayed basis of H^2(U,\R). Künneth and H^1(U,\R)=0 now show that every integral curve \gamma\subset D has class

Here a is the sum of the projection degrees to the two factors of E=\Pbb^1\times\Pbb^1, and b is the projection degree to C. Moreover K_X\cdot\gamma=-a, so negativity means a>0.

Let f:X\to Z be any nontrivial projective birational contraction with connected fibres, normal compact Kähler target, and -K_X relatively ample. Choose a Kähler class \omega_Z and set \alpha=f^*\omega_Z. For a curve, vanishing of its degree against \alpha is equivalent to being contracted by f. In particular \alpha\cdot\ell\geq0 and \alpha\cdot h>0: the horizontal curve cannot be contracted because K_X\cdot h=0. A contracted curve \gamma has the form (A.2) with a>0, and

It follows that b=0 and \alpha\cdot\ell=0. Thus all rulings in every E\times\{c\} are contracted, and f is not small.

Set q=\pi\times\id_C:X\to U_0\times C. Its nontrivial fibres are the quadrics E\times\{c\}, which are connected by their rulings, so f is constant on every q-fibre. Conversely, every curve in an f-fibre has the form (A.2) with b=0, and is contracted by q. A connected projective fibre is connected by chains of curves, so q is constant on every f-fibre. The two targets are isomorphic: the reduced image of (q,f) in their product projects properly, bijectively and bimeromorphically to each normal target, hence finitely and isomorphically. Therefore q is the only possible first negative birational contraction. It is divisorial, and its relative numerical rank is one because both quadric rulings have the same nonzero numerical class.

Failure of analytic local \Q-factoriality. Near the singular locus of U_0\times C, consider the prime analytic divisor

On the blowup of the singular axis, its strict transform meets each exceptional quadric in a ruling line. If nP were Cartier near an axis point for some integer n>0, its pullback would have the form n\widetilde P+kE_{\mathrm{exc}}. The associated line bundle restricts trivially to the quadric over that point, because it is pulled back from a line bundle at a point. On the other hand its restriction is

up to interchanging the factors. Triviality forces k=0 and n=0, a contradiction. The target is not analytically locally \Q-factorial. Since K_X is not nef and there is no small negative contraction, there is no permissible first step under that local convention.

Compatibility with the global convention. The same example has a valid one-step global program. Indeed, -E is \pi-ample, so -K_X is q-ample, and adjunction on U_0 gives

which is nef. On the blowup of the node the exceptional divisor has log discrepancy two; its smooth product with C is a log resolution with the same log discrepancy. Hence the target is terminal, in particular klt.

It remains to check that the local divisor P is not a global factoriality obstruction. The displayed integral cohomology of U and Hodge decomposition give H^1(U,\cO_U)=H^2(U,\cO_U)=0 and \Pic(U)=\Z[H|_U]\oplus\Z[E]. The exponential sequence and Künneth therefore give

Strict transform and divisor pushforward identify

Every surviving class descends to a Cartier class, either from the hyperplane bundle on U_0 or from a line bundle on C. Chow’s theorem makes every global analytic Weil divisor on this projective target algebraic, so every such divisor is Cartier [21, Section 4, no. 19, Proposition 13]. GAGA identifies its global coherent rank-one reflexive sheaves with algebraic divisorial sheaves, which are therefore invertible as well [21, Section 3, no. 12, Theorems 2–3]. The plane P is defined only on an analytic neighbourhood; it need not extend to a divisor on the whole compact space.□