Termination for projective log canonical fourfolds with rational boundary

Proof. Every prime divisor on X^+ maps to a prime divisor on Z, since f^+ is small. Its strict transform on X^- has the same adjoint coefficient: use the compatible canonical divisors and the stated boundary transform. Thus v_*H=0. The normalization map from G to its closed graph in X^-\times_Z X^+ is finite. The sum of the pullbacks of the relatively ample divisors -D^- and D^+ is relatively ample on that product; its finite pullback is -H. In particular, -H is v-nef, and negativity gives H\geq0.

For a proper birational morphism onto a normal variety, its nonisomorphism locus on the target is exactly its positive-dimensional-fibre locus. Indeed, over the zero-dimensional-fibre locus the morphism is finite locally on the target, hence an isomorphism by normality. Consequently each graph fibre over \Sigma has positive dimension: it maps surjectively to each of the two fibres and at least one of those fibres has positive dimension. Graph fibres are connected, by Stein factorization for the proper birational map G\to Z.

Every closed point of a connected positive-dimensional projective fibre lies on a curve in that fibre. To see this, a point which lies on no positive-dimensional irreducible component would be an isolated zero-dimensional component, contradicting connectedness; a positive-dimensional projective component contains a curve through each of its points. If a point of the graph fibre were outside \operatorname{Supp}H, choose such a curve through it. That curve is not contained in the effective \mathbb Q-Cartier divisor H, so its intersection with H is nonnegative. This contradicts the relative ampleness of -H. Thus the whole inverse image of \Sigma lies in the support. On the common isomorphism open the adjoints agree, so the reverse inclusion holds as well.

Pull the divisor identity back to a smooth common model carrying \xi. Compatible canonical divisors give (2.3). The order of an effective \mathbb Q-Cartier divisor at a valuation is positive exactly when its center lies in the support. Since this support is the full inverse image of \Sigma, the two descriptions of strictness follow.

For a nontrivial step the support is nonempty and contains a prime divisor of the fourfold G. This divisor has dimension three. Its map into W^-\times W^+ is finite onto its image, giving (2.4).

The assertion about W^+ also follows from v_*H=0: a prime divisor in W^+ would have a divisorial strict transform on G lying in \operatorname{Supp}H, contrary to the zero pushforward. If f^- is small, codimension-one comparison similarly gives u_*H=0, so \dim W^-\leq2. Finally each component of \Sigma is a positive-dimensional-fibre locus for at least one of the two morphisms. Its full inverse image on that side has dimension at least one larger, so \dim\Sigma\leq1.□

Proof. Use (2.3). In a small step the same valuations are exceptional on both models. A lost prime divisor is contained in W^-.□

Proof. Every irreducible closed r-dimensional subvariety of the open complement is the restriction of its closure in V. Hence the restriction of cycle spans is onto. The localization sequence

is exact. Since \dim A\leq r, its first group is generated by the fundamental classes of its r-dimensional components. These images already belong to the cycle span, proving the kernel assertion. If such a component exists, its class in V is nonzero: intersection with the rth power of an ample divisor has positive degree.

In a step the complements of W^- and W^+ are isomorphic. Apply the first assertion with r=3. The positive bad set has dimension at most two, so the positive-side rank equals the rank on this common open. The negative-side rank is at least that rank, and is strictly larger if a divisor is lost. These nonnegative integer ranks cannot strictly decrease infinitely often. This proves the first finiteness assertion, including mixed steps.

After passing to a small tail, both bad sets have dimension at most two. If the positive sets have dimension at most one, repeat the argument with r=2. Each negative-side surface gives a strict rank decrease.□

Proof. After the two applications of Lemma 2.3, both bad sets have dimension at most one. Their dimensions then contradict (2.4) at every further nontrivial step.□

Proof. Put E=q_1^*D_1-q_2^*D_2=F_2-F_1. Its pushforward to V_1 is effective. On curves contracted by q_1, the divisor E is the negative of a nef divisor, so -E is q_1-nef. Negativity gives E\geq0. Interchanging the two ends gives -E\geq0, hence E=0.

If D_2 is relatively ample, every curve in a fibre of q_1 has zero degree against q_2^*D_2, and is therefore contracted by q_2. The image under q_2 of that projective connected fibre must be a point. Indeed a positive-dimensional projective image would contain a curve, and projective hyperplane sections of its inverse image supply a curve mapping onto it. The graph factorization lemma for a proper morphism with connected fibres onto a normal variety gives q_2=hq_1. Equality of pullbacks and birational pushforward yield D_1=h^*D_2.□

Proof. Near the center the crepant boundary is \sum_{j\in J}(1-a_j)T_j. Subtracting it from the reduced boundary gives (3.1). The reduced SNC pair is lc and has integral Cartier adjoint, so its log discrepancies are nonnegative integers. At the generic center choose c-|J| additional regular parameters cutting that center inside its stratum. Adjoining their divisors with coefficient one still gives an SNC pair locally. Log canonicity of this larger reduced boundary implies e(\xi)\geq\sum_\ell\xi(t_\ell)\geq c-|J|.

For completeness, the zero-error assertion can be seen by a toroidal modification. The orders along the crossing equations give a positive integral vector in the cone of the stratum. Make a regular fan subdivision containing its primitive ray; the resulting local toric modification, pulled back through the crossing coordinates, is log crepant for the reduced SNC structure. The lifted center lies in the open stratum of the divisor for this ray. If it were a proper subvariety of that stratum, the preceding transverse-parameter argument would give positive reduced-SNC discrepancy. Thus it is the generic point of that divisor, and the normalized valuation is its order. The toric construction is split over the residue field of the original stratum, so a primitive vector determines one divisor there. There are finitely many strata and finitely many bounded integral vectors.□

Proof. Choose the resolution of Lemma 3.1 with SNC exceptional divisor. If the lifted center meets that SNC divisor and contributes discrepancy at most one, positivity of its a_j forces e(\xi)=0 and bounds every positive order in (3.1). There are finitely many resulting monomial valuations. If the center avoids the SNC divisor, an exceptional valuation has center of codimension at least two on the smooth isomorphism open, and hence log discrepancy at least two. Divisors already on the resolution form a finite list of possible exceptionals. These cases exhaust the valuations.□

Proof. The underlying variety Y is terminal. Indeed \Theta is effective and \mathbb Q-Cartier, so its removal only increases discrepancies. In particular Y is smooth in codimension two. Choose a closed set A of codimension at least two containing \operatorname{Sing}Y, the nonsmooth loci of the labels, and every intersection of two distinct labels. On Y\setminus A the ambient space is smooth and at most one smooth label is present.

Resolve and principalize the ideal of A, simultaneously with the label divisors, by a morphism isomorphic over Y\setminus A. The full inverse image of A has SNC divisorial support, all of whose components are exceptional over Y. Their log discrepancies for (Y,\Theta) are strictly greater than one. Any valuation centered in A has its lifted center in one of these exceptional components. If its discrepancy is less than two, (3.1) therefore forces e(\xi)=0; all orders are bounded because all coefficients a_j are positive. Lemma 3.1 gives only finitely many such valuations.

On the smooth open away from all labels, an exceptional valuation has discrepancy at least two. At a center of codimension c in a single label of coefficient b_h, the same formula gives

Since 1-b_h>0, discrepancy less than two forces c=2. At its generic point choose regular parameters x,t, with x the label equation. Add t=0 as an auxiliary component of coefficient zero to the SNC calculation. Then

The strict bound and positivity force \xi(t)=1 and e'(\xi)=0. The zero-error characterization gives precisely (3.2). Conversely each such primitive vector defines the indicated divisorial order and discrepancy. Only finitely many j satisfy the displayed inequality. Combining this with the finite list over A proves every assertion.□

Proof. For klt input, [8, Corollary 1.4.3] extracts precisely any prescribed finite set of exceptional valuations with log discrepancies at most one by a projective birational morphism with \mathbb Q-factorial source. The restriction in that corollary on discrepancy-one centers is vacuous for a klt pair. Apply it to I_0. The boundary in (3.3) is effective with coefficients less than one. Any valuation exceptional over Y_i is exceptional over X_i and is not one of the extracted divisors; it therefore has discrepancy greater than one. This proves terminality.

Put

The new coefficients lie in [0,1) and are no larger than their old values. Since Y_i is \mathbb Q-factorial, the removed effective boundary is \mathbb Q-Cartier. Its order at every valuation is nonnegative, so this move preserves terminality and increases discrepancies weakly.

On a smooth common model W of Y_i and X_{i+1} over Z_i, with projections q and r, consider

The divisor A_i is the divisorial trace on Y_i of the pullback of D_{i+1}. At primes from X_i this follows from smallness downstairs. At each marked prime E_{i,s} it is exactly the definition of its new coefficient using a_{i+1}(s). Thus q_*E=0. Also -E is q-nef because D_{i+1} is ample over Z_i. Negativity gives E\geq0. Every prime on X_{i+1} corresponds to a prime on Y_i with the same coefficient, so E is exceptional over X_{i+1} as well.

Run a chosen finite klt MMP for A_i over Z_i. The precise existence input is [8, Theorem 1.2 and Corollary 1.4.2]: a \mathbb Q-factorial klt pair with relatively big boundary admits a terminating program with suitable scaling. Here Y_i\to Z_i is birational, so both its boundary and its adjoint are relatively big; their generic fibre is a point, as in [8, Definition 3.1.1(7)]. Choose a general effective rational scaling divisor with sufficiently positive class and sufficiently small component coefficients, so the augmented pair remains klt and its adjoint is relatively nef. A fibre-type output over Z_i is impossible by dimensions. Hence a finite choice reaches a relatively nef end (\widehat Y,\widehat\Theta).

For this finite program, the input pullback equals the nef-end pullback plus an effective end-exceptional divisor. We already have the comparison with the ample divisor D_{i+1} on X_{i+1}. Lemma 2.5 therefore gives a morphism \widehat p:\widehat Y\to X_{i+1} and the actual crepant identity

No marked divisor can have been contracted during the finite program: its discrepancy at the input after the coefficient decrease is a_{i+1}(s), and crepancy gives the same value at the end, whereas a contraction losing it would give strict improvement by Lemma 2.1. No other prime can have been lost either. Such a prime comes from X_i, hence from X_{i+1}, and a birational morphism to the normal X_{i+1} has a strict transform of that prime. The program extracts no divisors, so a lost prime could not reappear.

Every step was consequently small. Terminality persists by Corollary 2.2, and the end again extracts exactly I_0. Set it equal to Y_{i+1}. This constructs the junctions inductively and proves all discrepancy and coefficient assertions. Figure 1 summarizes the construction.□

Proof. Fix a model and a center V. Choose a projective log resolution which also principalizes the ideal of V. Its divisorial inverse image has finitely many components dominating V. The mld in question is the minimum of their log discrepancies. Indeed every valuation centered at \eta_V has its lifted center in at least one such component, and the positive-coefficient formula (3.1) bounds its discrepancy below by that component’s discrepancy. Conversely those component valuations have center V.

Cut the fourfold by two general sufficiently ample hyperplanes so that the resulting surface meets V at general points and avoids the proper closed images in V of resolution strata which do not dominate it. Use simultaneous Bertini for the finitely many relevant strata. The downstairs surface is normal; the two pulled-back cuts on the smooth resolution are smooth and transverse to the SNC data. They contain no exceptional divisor as a component, because neither general hyperplane contains an exceptional image. Their complete intersection has no component contained in the exceptional locus: each exceptional divisor has image of dimension at most two, and two general pulled-back cuts leave dimension at most one there.

Adjunction cancels the two normal determinant factors on the source and target of the restricted resolution. Thus the crepant coefficients on the surface resolution are the restrictions of the original coefficients. The boundary on the normal surface has the original finite coefficient set: generic hyperplane sections of each boundary prime are reduced, and no new boundary divisor is introduced. All sliced crepant coefficients have positive log discrepancy, so the surface pair is klt. Components of the principalized inverse image dominating V cut to divisors with the same discrepancies over each chosen general intersection point. Strata with proper image were avoided. Applying (3.1) on the slice now gives the same minimum as above. This proves the assertion.□

Proof. Suppose infinitely many steps have a bad surface with mld at most one on one specified side. A computing divisor is exceptional downstairs and belongs to the stabilized finite set I_0. Some fixed member s computes the mld at infinitely many such occurrences. Its discrepancies along this subsequence increase strictly. On the negative side, strict improvement occurs in the step immediately following the occurrence; on the positive side, it occurs immediately before the occurrence. Monotonicity between occurrences then gives strict increase in either case. By Lemma 3.5, these are surface mld values with a fixed finite coefficient set, contradicting ACC. Apply this to both sides and discard finitely many initial steps. Ambro’s bound gives generic ambient smoothness at the remaining bad surfaces.

If an extracted label with discrepancy at most one has a codimension-two center V downstairs at some remaining junction, V cannot be contained in the following negative bad locus. Otherwise its mld would be at most the discrepancy of that label, contradicting the first assertion. The birational step is therefore an isomorphism near the generic point of V, and the center and its discrepancy persist on the next model. The same argument repeats at every later step. This label’s coefficient is hence constant from that junction onward. A varying label cannot have such a center. It is exceptional downstairs, so its center has codimension at least three; in dimension four its image has dimension at most one.□

Proof. Since b_h<1, a summand in (4.1) can be positive only if j<1/(1-b_h). Thus the sum is finite. Each summand is nonincreasing when b_h decreases. At all subsequent stages the possible indices j are bounded by the bound at the initial stage under consideration. Consequently d_h is a nonincreasing sequence of nonnegative integers and is eventually constant. There are only finitely many labels. Coefficients change only at the coefficient-decrease move of a bridge, and stay unchanged during its small flips. Constancy at consecutive junctions therefore gives constancy throughout the bridge as well.□

Proof. The variety Y is terminal and hence smooth in codimension two. Its local ring at \eta_V is therefore regular of dimension two. Write m_h for the multiplicity there of S_h, taking m_h=0 if the label does not contain V. The ordinary blowup of the closed point of this local surface defines a valuation v_1 with

In particular \sum_hm_hb_h<1, and the weight supplied by this valuation is w(\sum_hm_hb_h).

We first justify the branch multiplicity inequality

For a label containing V, let R be the one-dimensional local domain defined by its prime equation in this regular local surface. Its finite normalization \overline R is semilocal. For a general parameter t,

where the sum runs over the branches of the normalization above the closed point. The middle equality follows by applying multiplication by t to the finite-length module \overline R/R: its kernel and cokernel have equal length. Each summand in the last expression is a positive integer. This proves (4.5), with the residue degrees included.

For fixed labels Nb_h is integral. Apply the elementary inequality \lceil x_1+\cdots+x_r\rceil\geq \sum_{\ell=1}^r\lceil x_\ell\rceil-(r-1) for r\geq1 to the varying branches, repeating Nb_h once for each branch of that label. When there are no varying branches the same comparison is an equality for the fixed part. Effectiveness and (4.5) give

Only the varying branches can cause this rounding error.

It remains to pay the terms with j\geq2 in the defaults. A positive such term requires b_h>1/2. By (4.4), at most one label through V has this property, and its multiplicity must be one. Its prime equation is therefore a regular parameter, which we denote by x; complete it to regular parameters (x,t). In particular this label has precisely one normalization branch at V. For each j\geq2 with positive default, the monomial valuation with orders v_j(x)=j, v_j(t)=1 is a normalized divisorial valuation centered at V. These valuations are pairwise distinct and different from v_1. If b_h is the coefficient of x=0, effectiveness of the other boundary components gives

Thus w_{v_j}\geq w(1-j(1-b_h)). These distinct valuations pay all higher default terms, while v_1 gives (4.6). Summing proves (4.3).□

Proof. For a reduced projective pure threefold R, let \nu_R\colon N_R\to R be its normalization. Choose a closed reduced subset A_R\subset R of dimension at most two such that \nu_R is an isomorphism away from A_R, and put F_R=(\nu_R^{-1}A_R)_{\mathrm{red}}. The normalization square is a proper excision square. Its homology sequence, compatible with mixed Hodge structures, contains

This follows by comparing the localization sequences for (R,A_R) and (N_R,F_R), whose open complements are isomorphic; the mixed Hodge structures and their functoriality are those of [11, Proposition 8.2.2 and Proposition 8.3.9].

Apply this construction first to R=P_1, taking A_{P_1} to contain the nonisomorphism locus of its normalization. For R=Q take A_Q=Q\cap A_{P_1}. The normalization N_Q=\coprod_j Q_j^\nu is a union of entire components of N_{P_1}, so these choices give a morphism between the two normalization squares.

Let V run over the surface components of A_Q. Write E_{V,1},\ldots,E_{V,r_V} for the surface components of F_Q over V, and let e_{V,\alpha}=[\mathbb C(E_{V,\alpha}):\mathbb C(V)]. On their fundamental classes, the first component of the map in (5.4) is

Its kernel has dimension r_V-1. Every surface with r_V>1 lies in A_Q, because normalization is an isomorphism elsewhere. Consequently the kernel B of H_4(F_Q)\to H_4(A_Q) satisfies

All these terms are finite in number. The image of B\to H_4(N_Q) lies in the span of divisor classes, whose dimension is \sum_j\rho_*(Q_j^\nu). Therefore

By exactness, U is the image of \partial_Q.

The map H_4(F_Q)\to H_4(F_{P_1}) is injective: each surface generator of the source is the same surface in one of the selected normalization components of P_1, and different such generators remain distinct. Naturality of the boundary maps now shows that \partial_{P_1} maps the image of H_5(Q)\to H_5(P_1) onto a space containing a copy of U. Since H_4(F_{P_1}) is pure of weight -4, strictness of morphisms of mixed Hodge structures [10, Théorème 2.3.5(iii)] gives

Finally the localization sequence in Borel–Moore homology contains

Its first term vanishes by the dimension bound, so its second arrow is injective. Duality between homology and cohomology, with weight sign reversed, proves (5.3).□

Proof. We use intersection complexes and direct images in the derived category of mixed Hodge modules, and use the same symbols for their underlying rational complexes. The intersection complex of the irreducible fourfold Y is pure of weight four. The projective decomposition theorem gives a finite direct sum

where each Z is a proper irreducible closed subset of X, and M_{Z,i} is a pure Hodge module of strict support Z and weight 4+i. Multiple summands with the same support and index are allowed. Birationality gives the unique full-support summand \mathrm{IC}_X in degree zero. The weight and decomposition statements used here are [35, §§1.6–1.8, 1.12–1.13].

We claim that every proper-support summand in (5.5) satisfies

Choose an algebraic stratification adapted to \mathrm{IC}_Y and refine it by p^{-1}Z. Also stratify the target so that over a dense smooth open subset Z^\circ of Z every source stratum meeting a fibre has constant fibre dimension; strata whose images are not dense in Z are avoided by shrinking Z^\circ. This is possible by constructibility and generic stratified triviality. In particular it includes the strata introduced by fibre-dimension jumps before shrinking.

Let B\subset p^{-1}Z be one of the refined strata dominating Z, and put u=\dim B. Since p is birational and Z is proper, p^{-1}Z is a proper closed subset of Y, and hence u\leq3. On B we have

Indeed, if B lies in the smooth open stratum, the only degree is -4\leq-u-1. Otherwise let t\geq u be the dimension of the original stratum containing it. The strict support condition for an intersection complex gives vanishing for q>-t-1, and this implies (5.7). This is the stalk support condition in the middle-extension axioms [18, §2.3 and §3.3, axiom [AX1](c)].

For x\in Z^\circ, the stratum B\cap p^{-1}(x) has dimension u-z. Compactly supported cohomology on this stratum vanishes above degree 2(u-z). By (5.7) its contribution to the cohomology of the fibre with coefficients in \mathrm{IC}_Y vanishes in degrees greater than

A finite filtration by strata and proper base change give the same upper bound for the stalk of Rp_*\mathrm{IC}_Y at x. On a sufficiently small Z^\circ, the nonzero stalk of M_{Z,i}[-i] is in degree i-z. Since it is a direct summand, i-z\leq2-2z, proving (5.6).

Consider now a support Z not contained in C. Stratify Z\cap C compatibly with M_{Z,i}, and let S be a stratum of dimension s. Then s\leq z-1. The stalks of M_{Z,i} on S vanish above degree -s-1: on the local-system open of Z their only degree is -z\leq-s-1, and elsewhere this follows from the strict support condition, including after refinement as above. Thus the largest possible degree of a contribution from S to the compactly supported cohomology of M_{Z,i}[-i]|_C is

Filtration by strata shows that this summand contributes nothing to degree one on C. For the full-support summand \mathrm{IC}_X, each stratum of C has dimension s\leq1 and the same stalk argument gives the upper bound -s-1+2s=s-1\leq0. It too contributes nothing in degree one.

It remains to consider Z\subset C. Such a support is projective, and its contribution is

which is pure of weight (4+i)+(1-i)=5 by projective direct-image purity. Proper base change identifies the degree-one cohomology of (5.5) restricted to C with \mathbb H^1(P,\mathrm{IC}_Y|_P). All its nonzero summands are therefore pure of weight five, as claimed.□

Proof. There is a comparison morphism, equal to the identity on Y_{\mathrm{reg}}, in the derived category of mixed Hodge modules [35, §1.14]. Let \mathcal T be its cone:

The same comparison complex is studied by Park–Popa [32, Definition 6.1] as their rational-homology-manifold defect complex shifted by one. Restricting to P yields the exact sequence of mixed Hodge structures

Lemma 5.3 makes the last term pure of weight five. Consequently

Naturality under Q\subset P implies that the image of W_4H^5(P) in H^5(Q) is contained in the image of

Strictness of the restriction map for the weight filtration therefore gives

The cone is supported on \operatorname{Sing}Y. On a stratum of this locus of dimension s\in\{0,1\}, the strict stalk bound for \mathrm{IC}_Y and the triangle (5.10) give

In particular a point stratum has no contribution in degree zero. On T^\circ the only possible contribution in total degree zero is

At the generic codimension-three stratum, the middle-extension construction retains the cohomology of Rj_*\mathbb Q through unshifted degree two. Thus, by the attaching truncation in [18, §3.1 and §3.3, axiom [AX1](d)],

The first equality also follows directly from (5.10), since the constant complex has stalk cohomology only in degree -4.

Take the T^\circ to be disjoint and to omit every intersection point of singular curves. The intersection of a singular curve not contained in Q with Q consists of finitely many points. Hence the complement in \operatorname{Sing}Y\cap Q of the T^\circ for curves T\subset Q is finite. The open–closed cohomology sequence and the point-stratum vanishing give a surjection

The compact-support spectral sequence on a curve, together with the stalk bound and (5.13), identifies its summands with H_c^2(T^\circ,\mathcal V_T). The dimension bound from this surjection and (5.12) prove the lemma.□

Proof of Theorem 5.1. Lemmas 5.2 and 5.4 give

Proposition 6.1, proved in the next section, constructs for each such curve T at least b_T distinct normalized divisorial valuations v satisfying

Valuations arising from different curves have different centers and are distinct. Lemma 3.3 ensures that the set of valuations of discrepancy below two with codimension-three center is finite. Substituting their number for the last sum in (5.14) gives (5.1).□

Proof. Use the perverse normalization for \mathrm{IC}_Y, so that its restriction to Y_{\mathrm{reg}} is \mathbb Q[4]. The decomposition theorem for the projective morphism \pi provides an inclusion of the full-support summand

whose restriction to Y_{\mathrm{reg}} is the identity; see [34, Theorem 5.3.1] and [35, §§1.7–1.13]. Restriction upstairs gives a natural morphism

Its composite with s is the canonical comparison \mathrm{IC}_Y\to Rj_*\mathbb Q[4]. In fact, adjunction identifies morphisms from \mathrm{IC}_Y to Rj_*\mathbb Q[4] with morphisms from j^*\mathrm{IC}_Y=\mathbb Q[4] to itself, and the indicated composite restricts to the identity.

At the generic curve stratum, which has complex codimension three, the middle-extension construction retains the ordinary unshifted cohomology through degree two. Consequently the comparison induces

This is the middle-perversity truncation description of intersection cohomology [18, §2]; subsequent extension across the zero-dimensional strata does not change this restriction. Taking degree -2 in the two morphisms above proves the assertion: the map induced by s, composed with the inverse of this comparison isomorphism, is a right inverse to r.□

Proof. The variety Y has rational singularities, since it is klt [26, Theorem 5.22]. In particular, R^2\pi_*\mathcal O_{\widetilde Y}=0. Write A=\mathcal O_{Y,y}, let \mathfrak m be its maximal ideal, and put

Every F_n has the same underlying topological space as F_0=F_y. In the exact sequence

the coherent kernel has H^3(J_n)=0, by the dimension bound. Therefore H^2(F_{n+1},\mathcal O_{F_{n+1}})\to H^2(F_n,\mathcal O_{F_n}) is surjective. Proper formal functions [39, Tag 02OD] gives

For an inverse sequence of surjective maps the limit maps surjectively to each term, by successive lifting. Hence every term, in particular H^2(F_y,\mathcal O_{F_y}), is zero. This argument applies also when y is the generic point of T; no flatness at y is required.

For closed y, the analytic exponential sequence on the projective complex scheme F_y gives

The exponential sequence remains exact in the presence of nilpotents: the exponential and logarithm are inverse on a nilpotent ideal, and the usual sequence is exact on the reduction. The scheme form of GAGA [33, Corollary 4.3 and Theorem 4.4], extending Serre’s projective comparison [36], identifies the last group with the group already proved to vanish and identifies analytic line bundles with algebraic ones, including on this possibly nonreduced proper scheme. Thus c_1 is surjective integrally, and tensoring with \mathbb Q proves the assertion.□

Proof. We give the parameter argument because the bundles are needed over K_T, rather than only over its algebraic closure. Fix a relatively ample line bundle \mathcal O_{\mathcal F}(1). Every line bundle on a closed fibre is a quotient of some \mathcal O_{F_t}(-m)^{\oplus N}. Let m, N, and the Hilbert polynomial vary. Grothendieck’s Quot-scheme theorem [20, Theorems 3.1–3.2] gives the corresponding countable collection of schemes of finite type over T^\circ. On each, take the open locus where the universal quotient is invertible on the whole pulled-back fibre family. These loci still parameterize every line bundle on every closed fibre. To justify openness, the family and the universal quotient are flat over the Quot scheme; a fibrewise locally free quotient of rank one is locally free near that fibre by the local criterion for flatness. Properness then removes the image of the complementary closed locus.

There are only countably many irreducible components of these parameter spaces. Choose a very general t\in T^\circ(\mathbb C) outside the closures of the images of all components which do not dominate T^\circ. Lemma 6.3 supplies finitely many line bundles on F_t whose classes c_1,\ldots,c_N span \mathcal W_t. Each is represented by a point of a dominating irreducible parameter component. The first Chern class of the universal line bundle gives a section of the pullback of \mathcal W to that component: this follows by restricting its class on the total family to the fibres and applying proper base change. The analytic space of this irreducible complex parameter component is path connected.

Choose a closed point of the generic fibre of this parameter component. Its residue field is finite over K_T. Spreading the point, normalizing, and shrinking the base yields a connected finite étale cover of a dense open of T^\circ, together with a line bundle on the pulled-back family. Choose a path in the parameter component from the original quotient point to a point on this cover. Flatness of the universal Chern class identifies the transported c_i with the class at that cover point. The finite-index subgroup fixing the chosen sheet fixes this class; conjugating by the projected path gives a finite-index subgroup fixing c_i at the original basepoint. Shrinking a smooth complex curve does not diminish the monodromy image of a local system originally defined on it: the induced map on fundamental groups is surjective. It follows that each c_i has a finite orbit under \Gamma=\pi_1(T^\circ,t). Intersecting the stabilizers of this finite spanning set gives a subgroup of finite index acting trivially on \mathcal W_t. The image G of \Gamma in \operatorname{GL}(\mathcal W_t) is therefore finite. Lemma 6.2 gives the same conclusion for \mathcal V_T. Take a common dense open where all these covers are finite étale, move the basepoint into it, and transport the c_i along one common path to the new basepoint, still denoted t. They still span \mathcal W_t. Each cover class is a monodromy translate of the corresponding transported c_i. Changing its sheet realizes every element of that orbit, so choose a sheet giving c_i itself.

We next take norms of actual line bundles, not merely invariant points of a Picard functor. For the connected cover associated with c_i, let H_i\subset\Gamma be the subgroup fixing its chosen sheet, and let d_i=[\Gamma:H_i]. The cover carries a line bundle \mathcal L_i whose class at that sheet is c_i and is fixed by H_i. Its norm along the finite locally free map of fibre families is a line bundle on the original family; norms commute with base change [39, Tags 0BCY and 0BD2]. At t its first Chern class is

For completeness, H_i need not contain the kernel of \Gamma\to G. Nevertheless its image fixes c_i, and each element of the finite G-orbit of c_i occurs equally often in the coset sum. Both expressions in (6.1) are d_i times the orbit average, which proves the equality with its stated factor.

The averages of a spanning set span \mathcal W_t^G. Moreover, the surjection r induces a surjection on invariants: averaging any lift of an invariant vector gives an invariant lift. Poincaré duality on the connected oriented real surface T^\circ identifies the dimension of H_c^2(T^\circ,\mathcal V_T) with the dimension of the coinvariants of its monodromy representation. For a finite group over \mathbb Q, invariants and coinvariants have the same dimension, since averaging identifies the latter with the former. Thus

The norm bundles have generic fibres defined over K_T itself, and their images span these invariants. Select b_T of them whose images form a basis and rename their generic fibres L_1,\ldots,L_{b_T}. On a common smaller open, their images are independent flat sections, proving the claim. Their invariant dimension, and hence b_T, is unchanged by further shrinking.□

Proof. Write A=\mathcal O_{Y,\eta_T}, \mathfrak m=\mathfrak m_A, and Z=\widetilde Y\times_Y\operatorname{Spec}A. The ring A is an excellent local ring essentially of finite type over \mathbb C, with residue field K_T. Let \widehat A be its completion, and write F_n=Z\times_A A/\mathfrak m^{n+1} as in the proof of Lemma 6.3.

For n\geq0, the square-zero ideal for F_n\subset F_{n+1} is

Multiplication gives a surjection

Its source is a finite direct sum of \mathcal O_{F_0}. The kernel is coherent and has vanishing third cohomology, since \dim F_0\leq2. Lemma 6.3 therefore implies H^2(J_n)=0. The exact sequence of units for a square-zero thickening, with 1+J_n\simeq J_n additively, shows that the obstruction to lifting a line bundle from F_n to F_{n+1} lies in this zero group [39, Tag 0C6R]. Each L_i consequently has compatible lifts to every F_n.

Grothendieck’s existence theorem [19], in the projective form [39, Tag 0885], for the scheme Z_{\widehat A} algebraizes these compatible systems of coherent sheaves and their morphisms. It produces invertible sheaves: algebraize the systems of inverse line bundles at the same time, and use full faithfulness to algebraize their tensor-product isomorphisms with \mathcal O. Denote the resulting line bundles on Z_{\widehat A} by \widehat L_i. Their restrictions to F_0 are the specified L_i, with the chosen identifications.

The finite-presentation limit property for invertible sheaves and their isomorphisms [39, Tags 0B8W and 01ZR] descends all \widehat L_i to line bundles on Z_R for a single finitely generated A-algebra R equipped with a map \phi:R\to\widehat A. One may include their inverses and the evaluation isomorphisms in these data. Since A is noetherian, R has a finite presentation. Applying Artin approximation to its finite system of equations gives a map

whose reduction in K_T agrees with that of \phi, where A^h is the ordinary henselization of A, whose completion is \widehat A. Only approximation modulo \mathfrak m is used. This is the approximation theorem for an essentially finite-type local ring over a field [4, Theorem 1.10]; the pointed étale formulation, including equality of residue fields, is [39, Tag 07QZ].

Pulling the descended line bundles back by \psi gives line bundles on Z_{A^h} with the required special fibres. More explicitly, the two maps R\to K_T are identical, so their restrictions are the same pullbacks of the line bundles on Z_R to Z_{K_T}=F_0. Thus the argument preserves bundles on the entire scheme fibre; it does not merely preserve their numerical classes.

The ordinary henselization is the filtered colimit of pointed étale neighbourhoods with unchanged residue field [39, Tags 0A02 and 0A03]. The same finite-presentation property descends our finite list of bundles to one such neighbourhood. Spreading it from \operatorname{Spec}A to an affine open of Y gives q:Y_e\to Y and \eta' as claimed, after shrinking around \eta'. Since Y is normal, its étale neighbourhood is normal. Its irreducible components are open and closed; take the one containing \eta'. It is integral and dominant over Y, since an étale map is open. We may take Y_e quasi-projective by the affine construction.

Finally \widetilde Y_e is smooth, since it is étale over the smooth variety \widetilde Y. Its irreducible components are disjoint and open. Every component has nonempty open image in \widetilde Y, and hence meets \pi^{-1}(Y_{\mathrm{reg}}). Over this dense open the base change is the nonempty integral open q^{-1}(Y_{\mathrm{reg}}) of Y_e. Every component meeting that integral open forces there to be just one component. Hence \widetilde Y_e is integral. Its map to Y_e is projective and is an isomorphism over q^{-1}(Y_{\mathrm{reg}}), so it is birational.□

Proof. Suppose that D'=\sum_i\lambda_iD_i', with \lambda_i\in\mathbb Q, has zero degree on every C_j. We will show that D' is numerically trivial over a neighbourhood of \eta' and that a multiple therefore descends to Y_e. A line bundle from Y_e has zero punctured Chern class, so this descent will contradict the independence of the original classes unless every \lambda_i is zero.

Choose an algebraic closure of K_T. The geometric irreducible components of an integral K_T-curve C_j are permuted transitively by the absolute Galois group. The divisor D' is defined over K_T, so its degrees on these components are equal. Degree is preserved under extension of the ground field and adds over the components of the fundamental cycle. The equality D'\cdot C_j=0 therefore implies zero degree on every geometric curve component. This argument also accounts for any multiplicities if the whole scheme fibre is nonreduced.

All the reduced geometric components are defined over one finite extension of K_T. Normalize a dense open of T' in that extension, and shrink until the resulting curve cover is finite étale. Take the closures of the geometric components in the pulled-back projective family. By generic flatness and openness of geometric integrality, we may make their curve families flat with geometrically integral fibres. Their degrees against a Cartier multiple of D' are then constant and equal to zero. Include any zero-dimensional generic components in this construction as well; after shrinking, their fibres remain zero-dimensional. The union of these closures accounts set-theoretically for the whole family after another shrinking. Indeed its complement has constructible image in the base curve, and this image does not contain the generic point, so its closure is a finite set. Thus every curve in every remaining fibre over T' has degree zero. There are no vertical curves away from T', where g is an isomorphism. Removing the finitely many discarded points from Y_e proves the desired relative numerical triviality.

Choose a positive integer m such that L=mD' is Cartier. It is g-nef. By (6.2), L-K_{Y'} is also g-nef. It is g-big: relative bigness is tested on the generic fibre, which is a point for this birational morphism. The relative base point free theorem for the effective klt pair (Y',0) therefore makes L semiample over Y_e; the saturation hypothesis in its sub-klt formulation is automatic here [17, Theorem 2.1 and Remark 3.16].

We explain why a multiple descends all the way to Y_e. Its semiample morphism, followed by Stein factorization, gives

where h has connected fibres, W is normal, k is projective, and a positive multiple of L is the pullback of a k-ample line bundle on W. Because L has degree zero on all g-vertical curves, h contracts every curve in every g-fibre. These fibres have dimension at most one and are connected, the latter because g is proper birational with normal target. Their images under h are consequently single points. Thus every fibre of k is zero-dimensional. The morphism k is finite and birational, and normality of Y_e makes it an isomorphism. We have proved that

for a positive integer m' and a line bundle M on Y_e, with m' also clearing all the \lambda_i.

On the smooth open of Y_e, both g and the resolution are isomorphisms. By construction of the pushforwards D_i, (6.3) identifies the restriction there of

with the pullback of M. At a general closed point of T', the bundle M is trivial on a neighbourhood in the base. Its first Chern class on the punctured smooth neighbourhood is therefore zero. Naturality of R\pi_{e*}\mathbb Q\to Rj_{e*}\mathbb Q says that the image of a resolution-fibre Chern class is precisely this punctured Chern class of the same bundle. Hence

along a dense open of T'. The independence established after Lemma 6.5 forces every \lambda_i to be zero. This proves the lemma.□

Proof. The function-field extension \mathbb C(Y_e)/\mathbb C(Y) is finite, because the dominant étale morphism is quasi-finite. The elementary finite-extension inequalities for valuations imply that the restriction of a nontrivial discrete valuation is nontrivial and discrete, that its value group has finite index in the original group, and that the residue-field extension is finite [39, Tag 0ASH]. Consequently, if w is divisorial, the normalized restriction v has residue transcendence degree \dim Y-1 and is divisorial. One can see the last assertion without any choice of a completion: choose \dim Y-1 value-zero rational functions with algebraically independent residues, resolve their maps to \mathbb P^1 on a proper birational model of Y, and then resolve that model. The centre of v has dimension at least \dim Y-1, by those residues, and is not the generic point, since v is nontrivial. Thus it is a divisor.

Choose a projective birational morphism \mu:Z\to Y with Z smooth and carrying the prime divisor E with v=\operatorname{ord}_E, and form

The scheme Z_e is smooth, because p is étale. It is also integral with function field \mathbb C(Y_e). Indeed, its irreducible components are disjoint and open, and each has nonempty open image in Z. Every component therefore meets the inverse image of a dense open U\subset Y over which \mu is an isomorphism. That inverse image is the nonempty integral open q^{-1}(U)\subset Y_e. There can be only one component. The projective map \mu_e is an isomorphism over q^{-1}(U), proving the function-field assertion.

By properness of \mu_e, the valuation w has a centre x on Z_e. Its image on Z is \eta_E, since its restriction is a positive multiple of v, and its image on Y_e is \eta'. The points lying above this specified pair are the points of

There is therefore exactly one possible point x. Since p is étale, its local ring \mathcal O_{Z_e,x} is a discrete valuation ring, and its maximal ideal is generated by a uniformizer of \mathcal O_{Z,\eta_E}. Thus the ramification index is one. Any normalized valuation of \mathbb C(Y_e) centred at this DVR equals its order: every nonzero element of the fraction field is a power of a uniformizer times a unit of the DVR, and a centred valuation is zero on such units. It follows that w is uniquely determined by v and \eta'. Its restriction is exactly v, not a larger integral multiple. This proves both injectivity and the normalization assertion.

Finally choose compatible canonical divisors in the displayed étale square. Then

The coefficient at the divisor with generic point x is the coefficient at E, because its ramification index is one. Adding one to these coefficients proves equality of log discrepancies.□

Proof of Proposition 6.1. If b_T=0 there is nothing to prove. Otherwise apply Lemmas 6.4 and 6.5 and use the small model g:Y'\to Y_e constructed above. Lemma 6.6 gives at least b_T distinct arithmetic curve components C_i of g^{-1}(\eta'). Their closures S_i\subset Y' are distinct surfaces. They have codimension two, so terminality makes Y' smooth at their generic points. The ordinary blowup of each smooth generic centre S_i has a unique exceptional prime there. Let w_i be its normalized divisorial valuation. The codimension-two blowup formula gives

Their centres S_i distinguish them. By (6.2), they have the same log discrepancy over Y_e, and their centres on Y_e are exactly T'.

Lemma 6.7 now gives distinct normalized divisorial valuations v_i=w_i|_{\mathbb C(Y)}, each with centre T and with

Choose a prime component D of \Theta with positive coefficient and containing T. Since Y is \mathbb Q-factorial, a positive multiple of D is Cartier near \eta_T, with a local equation in the maximal ideal of \mathcal O_{Y,\eta_T}. A valuation centred there has strictly positive value on this equation. Effectivity of \Theta consequently gives v_i(\Theta)>0, and hence

The centres distinguish the valuations arising from different curves. This completes the proof.□

Proof. Take a projective resolution r:\widetilde S\to S which is an isomorphism over U=S_{\mathrm{reg}}, and put E=r^{-1}(S\setminus U). Normality gives \dim(S\setminus U)\leq1, so \dim E\leq2. Suppose a rational combination of the contracted divisors has zero class in H_4(S,\mathbb Q). The corresponding combination of their strict transforms has zero restriction to H_4^{\mathrm{BM}}(U,\mathbb Q). The localization sequence

is exact. Its first group is generated by the fundamental classes of the irreducible surface components of E; components of dimension at most one contribute nothing in degree four. Consequently one can correct the strict-transform combination by a rational combination of resolution-exceptional divisors to obtain a divisor with zero homology class on \widetilde S.

Every divisor in this corrected combination is exceptional over T. This is true for the original strict transforms by assumption. For a resolution-exceptional divisor it follows because its image in S has dimension at most one, and hence so does its image in T. On the smooth projective threefold \widetilde S, a homologically trivial divisor is numerically trivial. Applying the negativity lemma to this exceptional divisor and to its negative shows that it is zero as a divisor. None of the strict transforms is resolution-exceptional, so all original coefficients vanish.□

Proof. At a crepant junction p:Y\to X_i, let Q be the reduced union of the varying labels. If this union is empty, Lemma 4.3 already makes every surface bracket nonnegative, and all other terms of (4.2) are nonnegative. Otherwise each component of Q has positive coefficient and, by Proposition 3.6, \dim p(Q)\leq1. For every surface in Q, its branch number in Theorem 5.1 equals r_V^{\mathrm{var}}. Surfaces outside Q have no negative allowance in (4.3).

For a varying label b_h>0, so d_h\geq w(b_h)\geq1. Moreover, every valuation in the right-hand count of Theorem 5.1 has a codimension-three center and weight at least one, and thus is counted in the first term of (4.2). It follows from that theorem that

Substitution in (4.7) proves (i). The valuations on the right have codimension-three centers, so none has been reused as a valuation in a surface bracket.

For a coefficient decrease the variety, all centers, all branch numbers, and all normalization ranks remain fixed. The boundary removed is effective and \mathbb Q-Cartier; hence every discrepancy is nondecreasing and every weight is nonincreasing. The d_h remain unchanged by our preparation. To check that this comparison is a finite one, take the union of the exceptional surface centers in Lemma 3.3 for the two boundaries. Outside this finite set a nonzero local contribution consists of the echoes on a single label. Each of their finitely many possibly positive weights is nonincreasing, while their sum is the unchanged d_h. Thus each such echo weight is unchanged. The codimension-at-least-three contributions, and the contributions at the finitely many exceptional surface centers, can therefore be compared as finite sums with unchanged corrections. Their weights only decrease; a strict decrease of any exceptional valuation strictly decreases the total. This proves (ii) and (iii) for a coefficient move.

For a small flip, all prime labels and all exceptional divisorial valuations correspond on the two sides. By Lemma 2.1, the set

is the same on both sides. Set

This is finite. Each bad set has dimension at most two and has finitely many surface components; a valuation with a surface center contained in it has one of these centers. Lemma 3.3 gives finiteness at each of them and finiteness of all positive-weight terms with centers of codimension at least three.

The generic centers not contained in the bad sets identify on the common open. Their discrepancies, local corrections, and labels therefore agree. Write w_v^\pm=w_v(Y^\pm,\Theta^\pm). Separating the finite bad-set contributions in (4.2) gives

The second equality is exactly (7.1). Discrepancy monotonicity makes every summand in the last expression nonnegative. A strict weight drop gives a positive summand and hence a strict drop of the difficulty. Valuations whose centers change from a surface to a curve, or conversely, are still counted once in the raw part on each side; they merely move between the first two displayed sums of (4.2). Positive weights becoming zero are included in \mathcal A_0. Thus neither kind of change is lost in the comparison. This completes (ii) and (iii).□

Proof. Suppose that such a program has infinitely many birational steps. Lemma 2.3, applied in cycle dimension three, removes all divisor-losing steps after a finite prefix. The remaining program is small. Construct its terminal chain by Proposition 3.4, and make the truncations in Proposition 3.6 and Lemma 4.1. By Proposition 7.2, the difficulties at its crepant junctions are nonnegative integers and are nonincreasing.

Consider a downstairs step with an irreducible surface V\subset W_i^+. Proposition 3.6 says that the ambient variety is smooth at \eta_V and that its minimal log discrepancy there exceeds one. The ordinary generic blowup of V defines a normalized valuation v whose positive-side discrepancy is

where m'_h are boundary multiplicities at \eta_V and b'_h are the original boundary coefficients carried by strict transform downstairs. Effectiveness gives a^+\leq2, and the choice of N gives

By the full bad-set strictness in Lemma 2.1, a^-:=a(v;X_i,B_i)<a^+. Consequently

In particular the argument includes a^+=2, when the positive-side weight is zero. Strict discrepancy increase alone would not imply this unit drop without the integral final deficit.

The valuation v is exceptional downstairs on both sides, since the step is small. It does not belong to the stabilized list I_0: its positive-side discrepancy exceeds one, whereas I_0 is the common list of exceptional valuations of discrepancy at most one. The terminal junctions extract exactly I_0, so v is exceptional on both junction models. Every intervening birational move is small, and a coefficient change does not alter the model. Hence v stays exceptional throughout this finite bridge. Crepancy at its endpoints identifies its junction weights with those in (7.3). At least one move in the bridge therefore strictly lowers its weight. By Proposition 7.2, the difficulty strictly decreases between these consecutive junctions.

A nonincreasing sequence of nonnegative integers has only finitely many strict decreases. Thus only finitely many downstairs steps have positive-side bad surfaces. Discard them. On the remaining small tail, Lemma 2.3 in cycle dimension two removes negative-side bad surfaces after finitely many further steps. Both bad loci of every subsequent step would then have dimension at most one, contradicting the inequality \dim W_i^-+\dim W_i^+\geq3 of Lemma 2.1. This contradiction proves termination.□

Proof. Put D=K_U+\Delta and F=\lfloor\Delta\rfloor. For a sufficiently small positive rational number \varepsilon, the pair

is klt, and D_\varepsilon=K_U+\Delta_\varepsilon is negative on R. Here F is \mathbb Q-Cartier by \mathbb Q-factoriality. Decreasing an effective \mathbb Q-Cartier part of a dlt boundary preserves dlt, and a dlt boundary with no coefficient one is klt; see [26, Corollary 2.39 and Proposition 2.41]. Negativity on R is an open condition on \varepsilon, since it can be checked on one nonzero numerical class in this ray.

The klt cone and contraction theorem gives the contraction f:U\to V of R, with normal projective target and connected fibres. It has relative Picard number one. If f is small, klt flip existence gives a projective small map f^+:U^+\to V on which D_\varepsilon^+=K_{U^+}+\Delta_\varepsilon^+ is relatively ample; see [8, Corollary 1.4.1]. The output is \mathbb Q-factorial; the elementary statements are also recorded in [8, Lemma 3.10.2].

We check the positive sign for the full boundary. In N^1(U/V) there is a positive rational number t such that

Indeed both classes have negative degree on R, this relative space has dimension one, and their degrees on an integral curve are rational. The \mathbb Q-Cartier divisor L=D-tD_\varepsilon is numerically trivial over V. A Cartier multiple of L is relatively nef, and its difference from D_\varepsilon is relatively ample. Relative klt base point freeness therefore makes a positive multiple of L semiample over V; the applicable statement is [8, Theorem 3.9.1]. Its associated morphism contracts every curve in every f-fibre. A projective image of positive dimension contains a curve, and a curve in such an image has a curve lift. Consequently that morphism is constant on each connected f-fibre. Normality and the usual factorization through a proper morphism with connected fibres show that it factors through V. Equivalently, after a further multiple its semiample model is finite and birational over V, hence is V. Thus

for a \mathbb Q-Cartier divisor M on V.

Since the flip is small on both sides, this identity of rational line bundles transforms into

The full adjoint is therefore ample over V. The graph comparison of Lemma 2.1 applies to these adjoints and their strict boundaries and proves discrepancy improvement; in particular the full positive pair is lc. The same comparison gives the usual dlt preservation. More explicitly, every valuation of log discrepancy zero on the positive side had discrepancy zero on the negative side, and its centre was not contained in the bad locus. Its generic centre is consequently in the unchanged snc neighbourhood supplied by the dlt property. The characterization of dlt by such snc neighbourhoods, or directly [8, Lemma 3.10.10(1)], gives dlt on U^+.

For an elementary divisorial contraction the target is \mathbb Q-factorial by the usual klt contraction theorem, so the pushforward full adjoint is \mathbb Q-Cartier. The same discrepancy argument gives dlt preservation; see also [8, Remark 3.10.5 and Lemma 3.10.10(1)].□

Proof. Suppose such a program were infinite. By Lemma 2.3, only finitely many of its steps lose a prime divisor. Discard that finite prefix, leaving a sequence of elementary dlt flips.

We use special termination in its ordinary dlt form: [15, Theorem 4.2.1 and Remark 4.2.2]. This theorem assumes the log MMP for \mathbb Q-factorial dlt pairs in dimensions at most one less than the dimension in question. For rational boundaries it needs only the rational-boundary forms of those lower-dimensional results. Here they are the classical at-most-three-dimensional log MMP, including flip existence and arbitrary termination, as in [25]. In particular we do not invoke termination for four-dimensional lc pairs. The required fourfold special-termination conclusion is also stated directly in [14, Theorem 1.2]: eventually both the flipping and the flipped loci are disjoint from the round-down of the boundary.

After this further truncation, replace each boundary \Delta_i by \Delta_i-\lfloor\Delta_i\rfloor. These pairs are klt. The removed divisors miss the surgery on both sides, so they have degree zero on all curves of the two elementary contractions. Thus the same negative rays and the same two relative signs remain valid for the decreased boundaries. Relative ampleness is unchanged by this relatively numerically trivial difference. We have obtained an infinite program from a projective klt fourfold pair with rational boundary, contrary to Theorem 7.4.□

Proof. Choose ample divisors A on U and H on Z. The cone

is a closed face of the absolute cone. A negative vertical curve exists by the failure of relative nefness. Hence the linear functional K_U+\Delta has negative minimum on the compact slice F_h\cap\{A\cdot\gamma=1\}. It has a negative extreme point there, which generates an extremal ray of F_h and therefore of \overline{\mathrm{NE}}(U). The lc cone theorem represents this negative ray by a curve. Its zero intersection with h^*H says that it is vertical, since H is ample.

The absolute contraction contracts only curves in this ray. It follows that h is constant on each of its connected fibres: otherwise a positive-dimensional projective image would contain a curve, which could be lifted to a curve in the fibre. The factorization through its normal target follows by the usual proper connected-fibre factorization. If h is birational, a fibre-type contraction would force the dimension of that target to be smaller than \dim Z=\dim U, although it dominates Z. This is impossible.□

Proof. Suppose the contrary. Lemma 2.3 discards only finitely many steps and leaves an infinite small tail

in which both morphisms are small. All choices in this tail are kept fixed.

Take a projective crepant \mathbb Q-factorial dlt modification

The dlt blowup theorem provides such a morphism, with

see [16, Theorem 10.4]. In particular every extracted prime has coefficient one and log discrepancy zero.

Inductively suppose that p_i:(U_i,\Delta_i)\to(X_i,B_i) is such a crepant dlt lift. Over Z_i run elementary dlt MMP steps for K_{U_i}+\Delta_i until it becomes relatively nef. At any non-nef intermediate stage, Lemma 8.3 gives a negative absolute extremal ray inside the vertical face. The face need not be entirely negative. The elementary step exists by Lemma 8.1, its contraction factors the map to Z_i, and it is birational because that map is birational. Every total space is projective and \mathbb Q-factorial dlt. An infinite choice of such relative steps would consequently be an infinite absolute elementary dlt program, excluded by Theorem 8.2. The chosen relative program therefore reaches a nef end, say (U_i',\Delta_i').

This finite bridge is nonempty. Choose a curve C contracted by X_i\to Z_i. It has D_i\cdot C<0. Projectivity of p_i provides a curve \widetilde C dominating C: take a component of the inverse image dominating C and cut by sufficiently many general ample divisors. Crepancy gives

Thus the starting lift was not nef over Z_i.

We identify the end with a crepant lift of the prescribed positive model X_{i+1}. On a common smooth model of U_i, U_i', and X_{i+1}, the pullback of the starting adjoint can be written in two ways. The relative MMP writes it as the pullback of K_{U_i'}+\Delta_i' plus an effective divisor exceptional over U_i'. Crepancy of p_i and Lemma 2.1 write it as the pullback of D_{i+1} plus an effective divisor exceptional over X_{i+1}. One end is nef over Z_i and the other is ample there. Lemma 2.5 gives a projective birational morphism

Every prime exceptional for p_{i+1} is the transform of a prime already exceptional for p_i. Indeed the downstairs map is small, every downstairs prime persists, and the elementary program upstairs extracts no primes. Each remaining extracted prime therefore still has coefficient one. Setting (U_{i+1},\Delta_{i+1})=(U_i',\Delta_i') supplies the next crepant dlt lift without any additional modification.

Concatenating these finite nonempty bridges produces an infinite absolute elementary dlt program from (U_{i_0},\Delta_{i_0}). This again contradicts Theorem 8.2. The argument has followed every prescribed downstairs step; it has not replaced their ray choices by a program with scaling.□

Proof. The lc cone and contraction theorem applies without \mathbb Q-factoriality; see [16, Theorem 1.1]. It gives, for the chosen negative extremal ray, the stated contraction f:X\to Z. Its target is projective. Relative Picard number one and the negative degree of D give relative ampleness of -D. The fibre-type case already has precisely the required endpoint properties, so assume f birational.

Choose a sufficiently divisible integer r>1 and an ample Cartier divisor H on Z such that

is generated by global sections. This is possible by relative ampleness and an ample base twist. A general member G gives an effective rational \mathbb Q-Cartier divisor A=G/r with

For the latter assertion, take a log resolution of (X,B) and choose G generally in the free system. Its pullback has no exceptional component and is transverse to the finitely many SNC strata on the resolution. Adding its coefficient 1/r keeps every coefficient at most one in the crepant log-smooth calculation. This proves lc, including when the original pair has coefficient-one components.

We apply [7, Theorem 1.1(1)–(2)] in its published rational complemented-pair form. Its hypotheses are that (X/Z,B+A) is lc, B,A are effective rational divisors, A is \mathbb Q-Cartier, the structural morphism is projective and surjective, and K_X+B+A\sim_{\mathbb Q}0/Z. It does not assume \mathbb Q-factoriality or ampleness of A. It gives a log minimal model or a Mori fibre space for (X/Z,B), and the adjoint on a nef log minimal model is semiample over Z. A Mori fibre space over Z is impossible here: its intermediate base would have dimension less than four and would dominate the four-dimensional birational base Z. We therefore obtain a good log minimal model (M,B_M) over Z.

Let q:M\to V be its relative semiample contraction. Since M\to Z is birational, this is a birational contraction, and V\to Z is projective and birational. There is an ample-over-Z \mathbb Q-Cartier divisor L on V with K_M+B_M\sim_{\mathbb Q}q^*L. Put B_V=q_*B_M. Pushing the linear equivalence forward shows that the actual divisor

is \mathbb Q-Cartier and \mathbb Q-linearly equivalent to L. Moreover

Indeed the difference is q-exceptional and relatively numerically trivial, so negativity in both signs makes it zero. Thus the ample divisor on V is its actual adjoint, not just a numerical substitute.

The log-minimal-model discrepancy comparison gives, on a smooth common resolution a:W\to X, b:W\to M,

see [7, Remark 2.6]. This also explains possible extractions in the definition of a log minimal model. A prime extracted on M is given coefficient one; its coefficient in E is then -a(v;X,B)\le0. Effectivity forces this coefficient to vanish, so that extraction is crepant. After composition with q, the same E is effective and exceptional over V.

We prove that V\to Z is small, although X\to Z need not be. Let G_0 be the normalized graph of X\dashrightarrow V, with projections u:G_0\to X and v:G_0\to V, and set

The pullback of H_0 to a smooth model dominating W and G_0 is E pulled back. Pushing down gives

Its negative is ample over Z: it is the sum of the pullbacks of -D and D_V from the two factors, and the graph maps finitely into their fibre product. As in the support argument of Lemma 2.1, every graph fibre over a point where one of the two maps to Z is nonisomorphic is contained in \operatorname{Supp}H_0. Indeed that fibre is connected and positive-dimensional, every closed point lies on a contracted curve, and anti-ampleness contradicts effectivity on a curve not contained in the support.

If a prime divisor on V were exceptional over Z, the preceding support property would force a component of H_0 to dominate it. This contradicts v_*H_0=0. Hence V\to Z is small. Every prime on V consequently corresponds to a prime on X through their common codimension-one open over Z. Thus B_V is exactly the strict transform of f_*B; there are no additional boundary primes on V. The effective discrepancy difference gives

for every divisorial valuation, so (V,B_V) is lc. The diagram X\to Z\leftarrow V is the required step, including a mixed step when f loses a divisor and V\to Z is a nontrivial small modification. No positive-side relative Picard-number assertion has been used.□

Proof. Injectivity on numerical divisors follows by testing against curves defined over k_0. For surjectivity, a line bundle \mathcal L on X_{0,K} spreads to a line bundle on X_0\times S, where S is an integral finite-type k_0-scheme and the given map \operatorname{Spec}K\to S is dominant. Shrink S if necessary and choose a closed point s\in S(k_0). The line bundles \mathcal L and (\mathcal L_s)_K have the same numerical class on X_{0,K}. Indeed they are fibres of the same line-bundle family on the fixed variety X_{0,K}. Since k_0 is algebraically closed, S is geometrically integral; hence the parameter scheme S_K is connected. The family’s degree on any fixed integral curve in X_{0,K} is constant. This last assertion is the constancy of degree in a line-bundle family on a proper curve, or follows from constancy of its Euler characteristic. Thus every numerical divisor class comes from k_0.

Similarly, an integral curve on X_{0,K} spreads to a flat projective family of geometrically integral curves in X_0\times S after shrinking an integral finite-type parameter space. Geometric integrality is available since the given extension field is algebraically closed. Specialize to a k_0-point of this open parameter space. Degrees against every line bundle on X_0 are unchanged by flatness, so the resulting integral curve represents the same numerical class. The already established statement for numerical divisors makes these precisely all necessary degree tests. Conversely, an integral curve over an algebraically closed field stays integral on algebraically closed extension. The effective curve classes, their positive spans, and their closures therefore identify. The usual nondegenerate numerical pairings give the asserted identification of curve spaces.

Pullback commutes with base extension. For a projective morphism f_0:X_0\to Z_0 with projective target, choose an ample line bundle H on Z_0. An integral curve is contracted precisely when its degree against f_0^*H is zero. If a curve on the extension is contracted, its integral numerical representative over k_0 has the same zero degree and is also contracted. Conversely, every contracted integral curve over k_0 stays integral and contracted on extension. The identifications therefore preserve the span of contracted curve classes in N_1(X_0)_{\mathbb R}. The relative numerical divisor space is dual to this span, so its dimension, the relative Picard number, is preserved.□

Proof. At each index the varieties, maps, boundaries, chosen rational canonical divisors, and their compatibility involve only finitely many coefficients. Include equations for projective embeddings, ample line bundles, relative ample multiples of the two adjoints, the Cartier multiples defining their \mathbb Q-Cartier property, and the relevant bundle isomorphisms. Include a log resolution at each index, with its SNC divisors and crepant pullback coefficients. There are only countably many such data. They are therefore all defined over one countably generated field over \mathbb Q. Its algebraic closure inside k is a countable algebraically closed field k_0 over which the whole diagram is defined. It admits an embedding into \mathbb C.

These algebraically closed extensions preserve normality, integrality, dimension, projectivity, birationality, smallness, and whether a morphism is an isomorphism. Relative ampleness is preserved and is detected by faithful field extension. The chosen canonical divisors and Cartier data preserve the actual adjoints and the strict-transform or pushforward boundary identities. The descended log resolutions are smooth with the same SNC data. Their crepant coefficients preserve lc and klt wherever asserted. For lc, the calculation in Lemma 3.1 uses only the identity a(v)=\sum_jv(T_j)a_j+e(v) with a_j\geq0 and e(v)\geq0; the coefficients a_j may vanish. No finiteness assertion requiring their positivity is used.

Lemma 9.1, applied at every index, preserves extremality, negativity, and the relative Picard number of each chosen contraction. An ample polarization on its target identifies the contracted integral curves by degree zero against its pullback, so these curve classes are preserved as well. Connected fibres of the birational maps follow from properness and normality. For a prescribed fibre-type endpoint, the identity f_*\mathcal O_X=\mathcal O_Z is detected over k_0 by proper flat base change along the faithful field extension k_0\subset k, and preserved by the same theorem along k_0\subset\mathbb C. This keeps the prescribed endpoint itself. Each nontrivial birational step remains nontrivial. Consequently all indices of an infinite proposed program survive under this single embedding.

Only the original countable program and the data testing its properties are descended here. The auxiliary terminalizations, valuation constructions, and Hodge-theoretic objects in the contradiction may then be constructed over \mathbb C.□

Proof. Let (X,B) be a pair over k and let R be a negative extremal ray. Choose a countable algebraically closed field of definition k_0\subset k for (X,B) and embed k_0 into \mathbb C. By Lemma 9.1, R identifies with a negative extremal ray on both X_0 and X_{0,\mathbb C}. Proposition 8.5 produces the required complex contraction and, in the birational case, its lc positive model.

These outputs are finite-presentation data. Spread them over an integral finite-type k_0-scheme S whose function field embeds in \mathbb C, keeping the input equal to the fixed product X_0\times S. Include projective embeddings and polarizations of the targets, Cartier multiples and relative ample bundles for the adjoints, the boundary divisors and strict-transform identities, and log resolutions with their crepant coefficients. After shrinking S, the geometric fibres of the targets are normal and integral with the same dimensions, and the required morphisms are surjective and projective. In the birational case they are birational, and the positive map is small. To see the latter assertions directly, spread the isomorphisms on dense opens together with their inverses. The complements on the positive source can be kept of dimension at most two; this excludes an exceptional prime in every remaining fibre. Smoothness and the SNC conditions on the finitely many resolutions, and the exact crepant pullback formulas, can likewise be retained after shrinking. The resulting boundaries are the prescribed transforms and the resulting pairs are lc.

The relative signs are retained by spreading ample line-bundle data. For example, choose a relatively very ample power of each relevant adjoint multiple and a sufficiently positive base twist making it generated. Spread the resulting projective embedding. Its restriction preserves relative ampleness. All these requirements concern a finite diagram and hold on a nonempty open subset of S.

We check that the particular ray is preserved, including its relative Picard number. Write f_{\mathbb C}:X_{0,\mathbb C}\to Z_{\mathbb C} for the complex contraction. Choose an ample line bundle H on its target and include it in the spread. Its pullback is a line bundle on the fixed product X_0\times S, so its numerical class is constant on the connected parameter space, by the argument of Lemma 9.1. Thus

is the same at every specialization.

Choose an integral curve C_0 over k_0 with nonzero class in R, using the integral-curve part of Lemma 9.1 and a curve spanning the complex negative ray. Its degree against a fixed ample line bundle on X_0 is positive. For any closed point s\in S(k_0) in the chosen open and every integral curve C\subset X_0,

The first two equivalences follow from the projection formula, ampleness of the target polarizations, and constancy of \alpha. The last is the defining property of the original complex contraction. Hence f_s contracts precisely the given ray. Since it contracts C_0, the span of its contracted curve classes is the nonzero one-dimensional span of R. Its relative Picard number is therefore one. The fixed class \alpha thus controls the contracted curves at every specialization.

For a birational output, connected fibres follow from proper birationality and normality of the targets. In the fibre-type case, one may take the Stein factorization of f_s. Its intermediate target is normal because X_0 is normal, is finite over Z_s, and has the same dimension. The pullback of H_s to that target remains ample. Consequently the contracted integral curves and the span of their classes are unchanged, so the relative Picard number remains one. Relative ampleness of -D persists under this factorization. We now have connected fibres and the required fibre-type endpoint.

Choose a k_0-point of the nonempty open parameter space; such a point exists because k_0 is algebraically closed. The preceding construction gives the desired finite diagram over k_0. Extend it to k. Normality, lc singularities, the relative signs, the exact contracted ray, the relative Picard number, and the stipulated smallness are preserved. This is the required output for the original chosen ray.□

Proof of Theorem 1.2. If an allowed program over k had infinitely many birational steps, Proposition 9.2 would transfer that entire prescribed program to \mathbb C. This contradicts Proposition 8.4. Thus every allowed program has finitely many birational steps.

At any stage with non-nef adjoint, there is a negative extremal ray. This follows from the lc cone theorem, or from its complex form using Lemma 9.1. Every selected such ray has the contraction and output of Lemma 9.3. A finite prefix at which neither stopping condition holds therefore extends. A maximal program can stop only at a nef pair or at a fibre-type contraction. The latter has connected fibres, smaller-dimensional normal projective target, relative Picard number one, and relatively ample negative adjoint, exactly as required for the Mori fibre-space endpoint in Definition 1.1. An initially nef pair needs no operation.

These arguments prove continuation, finiteness for every sequence of permitted choices, and the asserted alternatives for a maximal program.□

Proof. Fix a maximal program X_0\dashrightarrow\cdots\dashrightarrow X_r, finite by Theorem 1.2, and put D_i=K_{X_i}+B_i. The section-space comparison in [30, proof of Corollary 12.1(i)] identifies the sections of the pseudo-effective D_0 with those of a semiample birational adjoint. Hence D_0 has a nonzero section in some positive Cartier degree. Choose a common positive multiple N of that degree and the Cartier indices of D_0,\ldots,D_r. A suitable power of the section is represented by a rational function s satisfying \operatorname{div}_{X_0}(s)+ND_0\geq0. At a birational step, push the effective divisor \operatorname{div}_{X_i}(s)+ND_i through f_i and take its strict transform through f_i^+. The boundary rule and compatible canonical divisors, together with smallness of f_i^+, identify the result with \operatorname{div}_{X_{i+1}}(s)+ND_{i+1}\geq0. Thus the section passes through every divisorial, flipping or mixed step.

A Mori fibre endpoint has a positive-dimensional fibre meeting the complement of this effective Cartier zero divisor; choose an integral curve C in that fibre through such a point. Its intersection with the zero divisor is nonnegative, but equals ND_r\cdot C<0 by relative ampleness of -D_r, a contradiction. The endpoint is therefore nef, and log abundance [30, Theorem 1.1] makes its rational adjoint semiample.□

Proof. Theorem 1.2 makes the program finite. Lemma 2.1 shows that log discrepancies do not decrease, including at mixed steps, while the boundary rule and smallness of the positive maps retain or delete the original coefficients. On a common smooth projective model p:W\to X, q:W\to Y, the graph comparisons give p^*(K_X+B)=q^*(K_Y+B_Y)+E with E\geq0. If the final adjoint were nef, this identity and birational invariance of pseudo-effectivity for rational Cartier divisors would make K_X+B pseudo-effective. The endpoint is therefore Mori. It remains \epsilon-lc, hence klt, and B_Y itself witnesses that Y is of Fano type over S, since -(K_Y+B_Y) is h-ample. The varieties are projective, so [29, Corollary 1.2] applies and gives the claimed index, independent of the program and its endpoint.□