Background and the role of the present argument

The generalized-pair formalism retains a nef divisor on a higher birational model as part of the adjoint data. Birkar–Zhang [4] developed effective birationality for these polarized pairs, extending the tools used to study Iitaka fibrations. The canonical bundle formula of Ambro [1] is one source of such nef data. In this paper the original nef summand descends to \(X\), but a minimal-model program forces us to keep track of it as a fixed nef divisor on a higher model.

Lazić–Peternell [22] formulated generalized abundance as a numerical extension of abundance and related it to semiampleness on varieties with trivial canonical class. Their Theorem B, assuming termination of flips and abundance in dimensions at most \(n\), proves numerical semiampleness when the ordinary adjoint has positive numerical dimension. Its numerical-dimension-zero case also assumes their Semiampleness Conjecture. Their Corollaries C and D give unconditional results for surfaces and for threefolds with ordinary adjoint of positive numerical dimension. These results identify the main remaining positivity issue: an arbitrary nef summand on a variety with trivial canonical class need not supply sections directly.

Subsequent work has addressed numerical nonvanishing, which asks for an effective numerical representative, and refinements of numerical semiampleness. Lazić–Peternell [23] obtained numerical nonvanishing under additional numerical or metric hypotheses and, in the metric case, conjectural MMP and abundance assumptions. For a Cartier nef summand, Chaudhuri [8] proved generalized abundance when the nef-reduction base of \(D\) has dimension two and \(K_X+B\) is effective, or when that base has dimension three and \(\kappa(K_X+B)>0\). Here nef reduction contracts the curves of degree zero through very general points; its precise properties are recalled in Section 2. On surfaces, Fontanari [10] proved actual semiampleness when the nef summand is Cartier and its numerical class is not proportional to that of the ordinary adjoint.

The birational reductions below follow [22]. The companion results [28, 29] supply ordinary log abundance and the existence of the particular terminating programs required in those reductions. The further ingredient proved here is that a nef divisor of full nef dimension on a klt Calabi–Yau pair is big. Full nef dimension means that every curve through a very general point has positive degree. Bigness is stronger: it requires the volume of the divisor to be positive. The gap between these two conditions is the geometric issue addressed by the proof.

Our treatment adapts the moving-jet and Frobenius arguments of [28]. The moving-point method has antecedents in the local-positivity work of Ein–Küchle–Lazarsfeld [9]. The finite-data comparison uses Frobenius jets, as studied by Mustaţă–Schwede [26], the canonical filtrations of Sun and Kitadai–Sumihiro [30, 18], and Langer’s control of Frobenius slopes [20]. We reproduce the comparison proof with its uniformity conditions. The adaptation to an arbitrary nef summand occurs in the geometric exclusions preceding it: the polarized birationality theorem of Birkar–Zhang gives curves of bounded degree, contradicting a growing integral lower bound for their degrees. We give explicit proofs of those exclusions, including the case of correspondences between finite covers.

Structure of the proof

There are two simultaneous induction statements. The first says that \(K_X+B+cM\) is big for every rational \(c>0\) whenever one positive combination of \(K_X+B\) and \(M\) has positive degree on all curves through a very general point. The second says that the positive part of \(K_X+B+M\) on a suitable smooth birational model is rational and numerically semiample. The positive part is the divisor left after removing the asymptotically fixed prime divisors. When the original adjoint is nef, it has no such negative part, and the second assertion descends to Theorem 1.

Section 3 sets up this induction. A pair of compatible minimal-model programs retains both a semiample ordinary adjoint and a nearby nef generalized adjoint. Positive Iitaka dimension then permits induction on the fibres. The remaining full-dimension case is a klt Calabi–Yau pair.

Sections 4 and 5 treat the essential terminal case with trivial canonical bundle. If a nef Cartier divisor \(L\) of full nef dimension were not big, its ample approximations could be rescaled to bounded volume while their degrees on very general curves tend to infinity. The bounded-degree obstruction controls the base loci of moving jet kernels. It produces a small vanishing-order bound on the variety itself and a large jet system on a projective bundle over its square. Section 6 shows that these two conclusions are incompatible. The comparison is made on fixed complex data before reduction to positive characteristic.

Section 7 passes through an index-one cover and a boundary perturbation to obtain the full-dimension assertion for all klt Calabi–Yau pairs. Section 8 proves the positive-part assertion by nef reduction and the canonical bundle formula, then descends numerical semiampleness to the original variety and to any algebraically closed field of characteristic zero.

The induction statements

For a projective klt \(\mathbb Q\)-pair \((X,B)\) and a nef \(\mathbb Q\)-Cartier divisor \(M\) on \(X\), write \[T=K_X+B.\] Suppose that \(T\) is pseudo-effective. We will use the condition that, for some \(a\in\mathbb Q_{>0}\), \[ (T+aM)\cdot C>0 \quad\text{for every integral curve $C$ through a very general point of $X$.} \tag{1}\] The divisor \(T+aM\) need not be nef. By Lemma 3, Condition (1) holds whenever \(M\) has full nef dimension. When \(T+aM\) is nef, the condition says exactly that its nef dimension is \(\dim X\).

We prove the following two assertions together, by induction on \(n\). The notation \(P_\sigma\) denotes the positive part of the divisorial Zariski decomposition; its properties used in the proof are recalled in Section 8.

\((P_n)\)

If \((X,B)\) is a projective klt \(\mathbb Q\)-pair of dimension \(n\), \(T=K_X+B\) is pseudo-effective, \(M\) is a nef \(\mathbb Q\)-Cartier divisor on \(X\), and (1) holds, then \(T+cM\) is big for every \(c\in\mathbb Q_{>0}\).

\((Z_n)\)

If \((X,B)\) and \(M\) have the same hypotheses, without (1), then there is a projective birational morphism \(w\colon W\to X\) from a smooth projective variety such that \[P_\sigma\bigl(w^*(T+M)\bigr)\] is a rational divisor numerically equivalent to a semiample \(\mathbb Q\)-divisor on \(W\).

Both assertions hold in dimensions zero and one. In dimension one, pseudo-effectivity of \(T\) and nefness of \(M\) mean that their degrees are nonnegative. Condition (1) makes every positive combination have positive degree, hence be ample. A divisor of degree zero is numerically equivalent to zero, and a divisor of positive degree is ample. These observations also give \((Z_1)\), since a nef divisor has zero negative part. On a point the assertions use the usual conventions.

Fix \(n\geq2\), and assume \((P_j)\) and \((Z_j)\) for \(j<n\). At any stage we may replace \(X\) by a small projective \(\mathbb Q\)-factorialization. The ordinary adjoint pulls back crepantly, \(M\) pulls back to a nef divisor, and (1) persists by using points in the isomorphism locus. Bigness descends under this birational pullback. A smooth model proving \((Z_n)\) for the \(\mathbb Q\)-factorialization also proves it for \(X\). We therefore impose \(\mathbb Q\)-factoriality when running the programs below.

Keeping a semiample adjoint fixed

We first record the elementary ray comparison used in the second program. The nef part of a generalized pair in this argument is the fixed nef \(\mathbb Q\)-Cartier b-divisor determined by \(M\) on the original variety. Its trace on a later model need not be nef.

Positive Iitaka dimension and the remaining case

The preceding construction leaves only one obstruction to bigness: the ordinary semiample adjoint may define a map to a point. If it defines a positive-dimensional base, the induction hypothesis handles its fibres.

Consequently, the remaining work for \((P_n)\) is the full-nef-dimension statement on klt Calabi–Yau pairs. Theorem 17 establishes its terminal, zero-boundary form. We then pass from that form to all klt Calabi–Yau pairs in Section 7.

The bounded-degree obstruction

In particular, the hypotheses of Lemma 10 cannot hold along a sequence with \(r\to\infty\) and with \(C_0,C_1\) bounded, even if the varieties and divisors vary. The following observation explains how full nef dimension and pseudo-effectivity of the canonical divisor pass to the moving subvarieties used below.

Subadjunction and high-multiplicity components

We use the following five results from the section Tools for moving base components of [28]. Their hypotheses distinguish generic information along a center from global information about the ambient variety. Intersections on an integral subvariety are computed after pullback to its normalization and a smooth projective resolution. A base ideal is isolated at the generic point of a subvariety when its localization there is primary for the maximal ideal.

Only generic log canonicity is required. The proof in [28] uses the dlt modification of [13], whose truncated boundary leaves an effective surplus supported over the non-lc locus. On a minimal stratum over \(V\), adjunction gives a pair that is klt over the generic point of its connected-fiber base. This suffices for the klt-trivial canonical bundle formula and its b-nef moduli divisor [12]. The effective discriminant, the moduli divisor, and finite ramification have nonnegative intersections with the pulled-back nef class. The comparison uses canonical Weil cycles, so it requires no \(\mathbb Q\)-Gorenstein assumption on the normalization of \(V\).

The degree estimate is the local multiplicity bound \(h^{d-f}\) combined with successive general cuts by the given linear series. The boundary is obtained by averaging general members with total coefficient equal to the generic log canonical threshold of \(\mathfrak b\). Thus it is an actual divisor to which Lemma 12 applies.

Full jet kernels and restriction to a fiber

The use of the full kernels matters: differentiation in the parameter lowers vanishing at the moving mark and still gives a global section in the lower kernel. Incidence dominance makes these parameter directions span the normal directions. The vanishing of all derivatives of order less than \(h\) then gives the ordinary-power containment, which can be restricted to general fibers as follows.

Here the fiber is chosen through a general incidence point. The statement does not assert the same conclusion for an arbitrary special fiber. Smoothness over the actual image identifies the restricted ideal of \(V_t\) with the reduced ideal of \(F\) locally; this is what preserves ordinary powers.

The \(\mathbb Q\)-factoriality needed for the base-component estimates can also be supplied by a small projective \(\mathbb Q\)-factorialization \(\sigma:Z'\to Z\) of a klt ambient variety. This passage is made only for components that meet the isomorphism open of \(\sigma\). Their strict transforms retain the generic base-ideal and multiplicity conditions for the pulled-back series, and the testing class \(\sigma^*R\) is nef. Since \(K_{Z'}=\sigma^*K_Z\), projection formula gives the same degree and canonical-intersection bounds on a common resolution. The pullback testing class need not be ample; nefness is the property used in these estimates. Passing to higher resolutions does not change the canonical intersections, because their additional exceptional divisors have zero intersection with the pulled-back testing class to the required power.

Finite covers with a fixed branch complement

A bound on the degree of a branched cover does not make the possible covers finite. The final companion input supplies the needed fixed branch complement from a family whose branch divisors do not sweep the base.

To see the role of the hypothesis, remove \(Y_{\mathrm{sing}}\) and the finitely many proper image closures \(\overline{g(\mathcal R_j)}\) for \(j\in I\). Every branch prime of a general Stein cover is accounted for by one of these relative Jacobian components. Purity therefore makes the restricted covers étale. The fundamental group of the resulting smooth complex variety is finitely generated, so it has only finitely many covers of degree at most \(e\). Finally, a finite normal cover of \(Y\) is determined by its function field, hence by its restriction to this fixed open.

Normalization and the curve obstruction

Throughout this section suppose that \(X,L\) form a counterexample to Theorem 17. Fix an ample Cartier divisor \(A\) on \(X\) and a positive rational number \(\epsilon\) such that \[ (14\epsilon)^n<\frac14, \qquad 80\epsilon<\frac34. \tag{10}\] Let \(t>0\) tend to zero along a sequence of rational numbers. Since \(L\) is nef and not big, \(L^n=0\). We may therefore choose positive integers \(r=r(t)\) satisfying \[ P=P_t=r(L+tA),\qquad r\longrightarrow\infty,\qquad P^n\longrightarrow\epsilon^n. \tag{11}\] For example, round \(\epsilon/((L+tA)^n)^{1/n}\) to the nearest positive integer. The divisor \(P\) is ample and rational Cartier, and \(P-rL\) is nef. After discarding finitely many terms, we have \[ \frac{\epsilon^n}{2}\le P^n\le2\epsilon^n. \tag{12}\] Every curve through a very general point of \(X\) has positive integral \(L\)-degree, hence \(P\)-degree at least \(r\). Thus a covering family of curves of bounded \(P\)-degree will give a contradiction.

We specify how the same obstruction applies to subvarieties and covers. A resolution \(W\to X\) has an effective canonical divisor: pull back a nowhere-zero canonical form on the smooth locus of \(X\) and use terminality. Hence \(W\), and therefore \(X\), is not uniruled [6]. By Lemma 11, if a smooth projective variety \(Y\) maps generically finitely to a positive-dimensional subvariety containing a sufficiently very general point of \(X\), then \(Y\) is not uniruled and the pulled-back Cartier divisor \(L_Y\) has full nef dimension. In particular, \(K_Y\) is pseudo-effective. If \(f=\dim Y<n\), the induction hypothesis \((P_f)\) applied to \((Y,0)\) gives \[ K_Y+L_Y\ \text{big}. \tag{13}\] All occurrences of “very general” below include the fixed conditions of Lemma 11.

Consequently, for \(f<n\) it suffices to find on such a \(Y\) a nef testing divisor \(P_Y\) with \[0<P_Y^f\le C_0,\qquad P_Y-rL_Y\ \text{nef},\qquad K_YP_Y^{f-1}\le C P_Y^f,\] where \(C_0,C\) are independent of \(t\). Indeed \(L_YP_Y^{f-1}\le P_Y^f/r\), so Lemma 10 applies. We will also use that lemma in dimension \(n\), after proving the required bigness separately. For the rest of this section, a bound is called uniform if it is independent of \(t\); it may depend on parameters fixed before \(t\) tends to zero. Since the dimensions lie in a finite range, Lemma 10 excludes all such configurations once \(r\) is sufficiently large.

The scalar jet estimate

Put \(\rho=(P^n)^{1/n}\). The first estimate bounds the vanishing order of every section at a very general point; it is stronger than a bound on a single fixed linear system.

The projective bundle and its volume

We now seek a jet system whose order is much larger than the scalar bound. The additional direction comes from the fibres of a projective bundle; its two summands keep track of the two copies of \(X\). For a divisor on \(X\), subscripts \(1,2\) denote pullback from the two factors of \(X\times X\). Set \[ Z=\mathbb P\bigl(\mathcal O_X(L)_1\oplus\mathcal O_X(L)_2\bigr) \xrightarrow{\pi}X\times X, \qquad \xi=c_1(\mathcal O_Z(1)), \qquad R=P_1+P_2+q\xi, \tag{15}\] where \(q\) is a positive integer to be fixed shortly. We use the convention \(\pi_*\mathcal O_Z(a)=\operatorname{Sym}^a(\mathcal O_X(L)_1\oplus\mathcal O_X(L)_2)\) for \(a\ge0\). The tautological class \(\xi\) is nef, and \(R\) is ample. The product and projective-bundle descriptions show that \(Z\) is klt. The two natural sections, called the axes, have classes \(\xi-L_1\) and \(\xi-L_2\). The canonical bundle formula gives \[ K_Z\sim-\text{(sum of the two axes)}. \tag{16}\] We call the complement of the axes the torus open.

The fibre of either projection \(Z\to X\) over a smooth point is, up to a constant one-dimensional factor, \[Z_{\mathrm{sl}}=\mathbb P(\mathcal O_X\oplus\mathcal O_X(L)), \qquad R_{\mathrm{sl}}=P+q\xi.\] Its axes have classes \(\xi\) and \(\xi-L\), and its canonical divisor is again minus their sum. The slice is projective and klt. In particular it has the singularity hypotheses needed to apply the base-component estimates, using a small \(\mathbb Q\)-factorialization if necessary as explained after Lemma 14.

Write \(d=2n+1\). There are positive constants depending only on \(n\) such that, for each fixed \(q\) and all sufficiently small \(t\), \[ c_nq\epsilon^{2n}\le R^d\le C_nq\epsilon^{2n}, \qquad R_{\mathrm{sl}}^{n+1}\le C_{\mathrm{sl},n}q\epsilon^n. \tag{17}\] To see this, expand in nef classes and use \[\pi_*(\xi^j)=\sum_{a=0}^{j-1}L_1^aL_2^{j-1-a} \quad(j\ge1).\] The term containing exactly one \(\xi\) is \((2n+1)\binom{2n}{n}q(P^n)^2\), giving the lower bound. For the upper bound, every occurrence of \(L_i\) can be bounded in mixed nef intersections by \(P_i/r\). After factoring out \(q\), the remaining powers of \(q/r\) are bounded by one once \(r\ge q\). There are only finitely many terms depending on \(n\). The slice calculation is the same, using \(\pi_*(\xi^j)=L^{j-1}\). Together with (12), this proves (17).

For a nef rational Cartier divisor \(R\) and a smooth point \(z\), its Seshadri constant is \[\varepsilon(R;z)=\inf_{C\ni z}\frac{R\cdot C}{\mathop{\mathrm{mult}}_z C},\] where \(C\) ranges over integral curves through \(z\). Thus a lower bound for this constant controls degree relative to multiplicity at the point.

We prove the proposition in the remainder of this section. Choose a positive rational gap \(\delta\) such that \[ (2/\delta)^nC_{\mathrm{sl},n}\epsilon^n<1. \tag{19}\] Next choose \(q\) so large that \(d+1\) increasing levels of gap \(\delta\), with lowest level strictly greater than \(2n+2\), lie strictly below \((c_nq\epsilon^{2n})^{1/d}\). These choices precede the limit \(t\to0\).

Fix the prescribed opens \(U_1,U_2\), if any. Suppose that (18) fails throughout their smooth torus preimage for arbitrarily small \(t\). For any such \(t\), asymptotic Riemann–Roch and the elementary jet count at a smooth point show that all the full kernels at the chosen levels are nonzero for sufficiently large divisible \(k\). Indeed the highest level has \(d\)-th power smaller than \(R^d\). Failure of (18) supplies, through each very general point, a curve with degree divided by multiplicity smaller than the lowest level. Every section in the lowest kernel vanishes identically on that curve: otherwise its intersection with the curve is at least its vanishing order times the curve’s multiplicity, contradicting the degree inequality. Thus the lowest base locus has a positive-dimensional component through the mark. Take \(k\delta\ge4\) and apply Lemma 14.

We obtain a family of proper integral components \(V\subset Z\) whose incidence dominates \(Z\). For its general members, of dimension \(f\), the estimates are \[ \begin{split} 0<R^f\cdot V&\le(2/\delta)^{d-f}R^d,\\ K_{V^*}R^{f-1}&\le(K_Z+cR)|_V R^{f-1}, \qquad 0<c\le2(d-f)/\delta. \end{split} \tag{20}\] Here and below classes on a resolution are understood by pullback. The positivity follows from ampleness of \(R\). General members meet the torus open, so (16) makes the ambient canonical contribution nonpositive. All constants in these bounds are uniform now that \(q,\epsilon,\delta\) have been fixed.

Excluding positive-dimensional slice fibres

We first constrain the two projections of \(V\) to \(X\). For either projection, consider the component \(F\) of a general fibre of \(V\to\operatorname{im}(V\to X)\) through a general incidence point. Choose that incidence point in the opens of Lemma 15, over very general points of both factors of \(X\), and in the smooth torus open. We claim that \[ \dim F\notin\{1,\ldots,n\}. \tag{21}\]

Suppose instead that \(1\le f_{\mathrm{sl}}=\dim F\le n\). The ambient fibre is \(Z_{\mathrm{sl}}\), of dimension \(n+1\). The restriction lemma gives a nonzero restricted subseries with \(F\) an isolated base component and base ideal contained generically in \(\mathcal I_F^h\), where \(h=\lceil k\delta/2\rceil\). Generic smoothness is imposed over the actual image of \(V\); this is why a general fibre through the incidence point was chosen. The point is also smooth in the ambient slice. Applying Lemma 13 on that slice gives \[ \begin{split} 0<R_{\mathrm{sl}}^{f_{\mathrm{sl}}}\cdot F &\le(2/\delta)^{n+1-f_{\mathrm{sl}}} R_{\mathrm{sl}}^{n+1},\\ K_{F^*}R_{\mathrm{sl}}^{f_{\mathrm{sl}}-1} &\le(K_{Z_{\mathrm{sl}}}+cR_{\mathrm{sl}})|_F R_{\mathrm{sl}}^{f_{\mathrm{sl}}-1}, \end{split} \tag{22}\] with \(c\le2(n+1-f_{\mathrm{sl}})/\delta\). Cartier multiples are chosen before restriction, so the constant line from the fixed factor causes no change in this calculation.

Let \(W_0\subset X\) be the image of \(F\) in the other factor. There are three cases: \(F\) is generically finite over a proper subvariety of \(X\); \(F\) contains the projective-line fibres over its image; or \(F\) is a multisection over all of \(X\).

Correspondences and their branch divisors

For these remaining \(V\), (20) gives uniform bounds \[ R^n\cdot V\le C, \qquad K_{V^*}R^{n-1}\le cR^n\cdot V\le C, \tag{28}\] after increasing \(C\). Each projection has bounded degree: \(R-P_i\) is nef, and \(P^n\) is bounded below.

For each fixed \(t\), resolve the general members in their algebraic family and shrink the parameter space so that the resolutions form a smooth projective family with integral fibres and morphisms to \(X\) in both slots. The dominant incidence and all numerical bounds persist. Finite extensions of the parameter space are allowed. Over \(X_{\mathrm{sm}}\), name the components of the relative Jacobian divisors as in Lemma 16. Consider a component whose image on a general member is a divisor \(J\subset X\).

Such divisors \(J\) cannot sweep \(X\). If they did, choose a general one through a very general point. Let \(E_J\) be a ramification divisor over it on the corresponding \(V^*\). Pullback of a canonical form from \(X\) gives an effective canonical divisor on \(V^*\). Regularity follows by factoring through a resolution of \(X\) and using its canonical singularities; the coefficient of \(E_J\) is positive because its image meets \(X_{\mathrm{sm}}\) generically. The difference \(R-P_i\) is nef, so (28) and projection formula bound \[0<d_J:=P^{n-1}\cdot J\le C.\] On a resolution \(J^*\), Lemma 12, applied to \((X,J)\) at the generic point of \(J\), gives \[ \begin{split} K_{J^*}P^{n-2} &\le(K_X+J)\cdot J\cdot P^{n-2}\\ &=J^2P^{n-2} \le\frac{(JP^{n-1})^2}{P^n} \le C' d_J. \end{split} \tag{29}\] The pair is lc at that generic point because \(X\) is normal and hence smooth there. The middle inequality is the Hodge index inequality, applied on a resolution and obtained by ample approximation of the pulled-back testing class. It does not require \(J\) to be nef. The final constant is uniform by the volume lower bound and the bound for \(d_J\).

Since \(J\) sweeps \(X\), its resolution satisfies the very general point conditions used in (13). With testing class \(P|_{J^*}\), whose top self-intersection is \(d_J\), Equation (29) gives precisely the hypotheses of Lemma 10 in dimension \(n-1\). This contradiction excludes sweeping branch divisors.

Lemma 16 now gives, for each slot and this fixed \(t\), only finitely many possible finite normal Stein covers of \(X\), up to isomorphism over \(X\). The point of the preceding step is that degree bounds alone would not give this conclusion: non-sweeping branch images provide a fixed smooth branch complement for the family. Neither that complement nor the resulting finite lists need be uniform in \(t\).

Fixed covers and an abelian model

Each \(V^*\) maps birationally to both of its normal Stein covers, and hence determines a graph of a birational map between a pair of covers in the finite lists. We explain why some such fixed pair has a family of graphs dominating its product. Restrict the finite set of pairs to those realized by at least one original member \(V^*\) satisfying (28). The original incidence dominates \(Z\), so the projected incidences for these pairs cover a dense open of \(X\times X\). For a fixed pair, graphs of birational maps lie in countably many Hilbert schemes. Stratify these schemes so that the fibres are integral and flat and both projections are birational; the latter conditions may be tested by dominance and degree one after shrinking. Because \(\mathbb C\) is uncountable, a dense open of \(X\times X\) cannot be covered by countably many proper closed subsets. Thus, for one fixed pair \(T_1,T_2\), some integral graph family has dominant evaluation to \(X\times X\). Since \(T_1\times T_2\) is finite over \(X\times X\) and has the same dimension, this evaluation is dominant onto \(T_1\times T_2\) as well.

Both covers are non-uniruled because they are finite over \(X\). The birational-group theorem of Hanamura [17], in the form recalled in [5], implies that they are birational to an abelian variety. Here are the details of the application. Identify the two covers birationally by one map in the family. On a suitable smooth projective birational model of the resulting non-uniruled variety, the reduced birational group is a group scheme locally of finite type, with identity component an abelian variety acting regularly. Conjugate the graph family to this model and shrink again to obtain a flat family of integral graphs. By the flat-graph representability statement, its reduced integral parameter space maps to the reduced birational group. Its connected image lies in one translate of the identity component. Product-dominant evaluation therefore gives a dense orbit for that abelian identity component. The orbit is a quotient of an abelian variety by a stabilizer, hence is itself an abelian variety. The fixed cover is birational to it. This is the birational-group part of the fixed-cover argument in [28]; no canonical nonvanishing conclusion from that proposition is being used.

This structural conclusion depends only on the two covers. We may therefore return to an original member \(V^*\) belonging to this pair, which retains (28), and resolve further to obtain a birational morphism \[p_0:V^*\longrightarrow A_0\] to an abelian variety. Higher resolutions do not change the canonical intersection with the pulled-back \((n-1)\)-st power of \(R\). Let \(L_0\) be the pullback of \(L\) from either slot. It is nef, Cartier, and of full nef dimension. We claim \[ K_{V^*}+L_0\ \text{big}. \tag{30}\]

The pushforward \(N_0=(p_0)_*L_0\) is pseudo-effective on \(A_0\), and is therefore nef. Indeed translations make the pseudo-effective and nef cones of an abelian variety coincide. If \(N_0\) were not big, the standard description of nef line bundles on abelian varieties would give a positive-dimensional abelian subvariety on whose translates \(N_0\) is numerically trivial. One can see this from the semipositive Hermitian form of the numerical class: its kernel is rational for the lattice, since the alternating form of the integral Cartier class is integral, and therefore defines an abelian subvariety. A general translate meets the image of the exceptional locus of \(p_0\) in codimension at least two, if at all. Complete-intersection curves in that translate can thus be chosen through very general points and avoiding this image. Their lifts have \(L_0\)-degree zero, contradicting full nef dimension. Hence \(N_0\) is big.

By the negativity lemma, \[p_0^*N_0-L_0=E_0\ge0\] is exceptional. The canonical divisor of \(V^*\) has an effective representative \(E_K\) with positive coefficient along every \(p_0\)-exceptional divisor, because the target is smooth with trivial canonical bundle. Choose a rational \(0<u<1\) sufficiently small that \(E_K-uE_0\ge0\). Then \[K_{V^*}+uL_0\sim_\mathbb Qu p_0^*N_0+(E_K-uE_0)\] is big. Adding the nef divisor \((1-u)L_0\) proves (30).

We may now apply Lemma 10 in dimension \(n\) on \(V^*\), with testing class \(R\) and empty boundary. The bounds (28) give both the top intersection bound and the canonical ratio, while \(R-rL_0\) is nef and (30) supplies bigness. This final contradiction excludes the correspondence case.

The finite-data statement

For a vector bundle on a smooth projective curve, its slope is degree divided by rank. The least and greatest Harder–Narasimhan slopes are denoted by \(\mu_{\min}\) and \(\mu_{\max}\). For a line bundle on a smooth variety, to generate jets through order \(a\) at a point means surjectivity to its stalk modulo the \((a+1)\)-st power of the maximal ideal.

The point \(x\) is independent of the two base coordinates of \(z_*\). No nefness is required of \(D,S\), or \(K_W\), and the theorem does not assume that \(K_W\) is pseudo-effective. Its hypotheses specify the flag and the movable curve class themselves. All these data will be fixed before the residual characteristic tends to infinity.

Reduction and Frobenius jets

We begin the proof of Theorem 20 by spreading its hypotheses and converting the fixed ordinary jets into jets of order proportional to the residual characteristic. Put \(\Lambda=\prod_{i=1}^{n-1}l_i\). The flag satisfies \[ [C]=\Lambda H^{n-1},\qquad h_i\cdot C=l_i\Lambda H^n. \tag{33}\] Choose one sufficiently ample integral Cartier divisor \(T_0\) so that every \(T_0+aS\), for integers \(0\le a\le(k_0-1)q\), has a section nonvanishing at each of the two base coordinates of \(z_*\). Fix these finitely many sections. All models, morphisms, divisors, points, the flag, the complete-intersection witness for \(\gamma\), and the jet and filler sections spread over an integral finitely generated \(\mathbb Z\)-algebra of characteristic zero. Shrink its spectrum so that the fibers and flag are smooth projective and geometrically integral, the birational morphisms remain birational, the fixed ample bundles remain ample, and the finite jet surjection and nonvanishings persist. Birationality is preserved by spreading an isomorphism between dense opens. All displayed intersection numbers are constant.

We also preserve \(\mu_{\min}(\Omega_W^1|_C)\ge0\). One way is to spread its characteristic-zero Harder–Narasimhan filtration, make all its factors locally free, and use openness of their semistability on curves; their degrees and ranks are fixed. Equivalently, a single relative ample twist globally generates the curve bundle. Every quotient of rank \(a\) then has degree bounded below by \(-ad\) for a fixed integer \(d\). A negative-degree quotient therefore has one of finitely many ranks and degrees. The proper relative Quot schemes for those Hilbert polynomials have images missing the generic point: after removal of torsion, a negative-degree coherent quotient there would give a negative-degree vector-bundle quotient. Removing those finitely many images proves the same openness assertion directly.

This base has closed points in arbitrarily large characteristics. Take algebraic closures of their residue fields and exclude denominator primes. Retain the notation for the reductions. The curve \(\gamma\) continues to pair nonnegatively with every effective divisor: pull that divisor back under the fixed birational morphism and intersect with its fixed ample divisors. This tests effective divisors on the reduction itself. No specialization of a characteristic-zero effective cone is used.

Full rank from the two-slot jets

Let \(F:W\to W'\) be relative Frobenius to the base-field twist. It is finite flat because \(W\) is smooth. Write \(S'\) for the twist of \(S\), so \(F^*S'=pS\), and define \[\mathcal E=F_*\mathcal O_W(B_p),\qquad s=\mathop{\mathrm{rk}}\mathcal E=p^n, \qquad R=s^2.\] For integers \(0\le a\le q\), put \(U_a=H^0(W,\mathcal O_W(B_p+paS))\). Projection formula gives an evaluation map on \(W'\times W'\): \[ \bigoplus_{\substack{a+b=q\\a,b\ge0}} U_a\otimes U_b\otimes \mathcal O(-aS'_1-bS'_2) \longrightarrow \mathcal E\boxtimes\mathcal E. \tag{34}\]

The jet calculation at \(z_*\) proves generic full rank; the diagonal estimate below will be tested at \((x',x')\), obtained from the separately fixed point \(x\).

Sections on the Frobenius diagonal

The Frobenius fiber product has the cartesian square \[ \begin{tikzcd} \Delta_F=W\times_{W'}W \arrow[r,"\mathrm{pr}_2"] \arrow[d,"\mathrm{pr}_1"'] & W \arrow[d,"F"]\\ W \arrow[r,"F"'] & W'. \end{tikzcd} \tag{35}\] Its reduced subscheme is the diagonal \(W\). Put \[\mathcal L_p= \mathcal O_{\Delta_F}(B_{p,1}+B_{p,2}+pqS_1).\] Write \(h=F\circ\mathrm{pr}_1=F\circ\mathrm{pr}_2: \Delta_F\to W'\). On this fiber product, \[\mathcal O(pS_1)\simeq h^*\mathcal O_{W'}(S')\simeq\mathcal O(pS_2).\] Consequently every pair of weights \(a+b=q\) gives the same line bundle: \[\mathcal O_{\Delta_F}(B_{p,1}+B_{p,2}+paS_1+pbS_2) \simeq\mathcal L_p.\] On the diagonal of \(W'\times W'\), every source twist of (34) becomes \(-qS'\). Finite base change and projection formula identify the twisted target as \[(\mathcal E\otimes\mathcal E)(qS')\simeq h_*\mathcal L_p.\] The product sections defining the columns restrict to global sections of this one bundle \(\mathcal L_p\). Their values at a diagonal point therefore lie in the image of \(H^0(\Delta_F,\mathcal L_p)\) in the corresponding fiber of \(h_*\mathcal L_p\). This proves that the rank at every diagonal point is at most \[ h^0(\Delta_F,\mathcal L_p). \tag{36}\]

For a rank-\(n\) bundle \(V\) in characteristic \(p\), denote by \(A_j(V)\) the degree-\(j\) part of its symmetric algebra modulo the \(p\)-th powers of local generators. This construction is independent of frame: the \(p\)-th power of a linear combination is the sum of the \(p\)-th powers in characteristic \(p\). Each \(A_j(V)\) is locally free and a quotient of \(V^{\otimes j}\). Set \[u=n(p-1),\qquad e_j=\mathop{\mathrm{rk}}A_j(V),\qquad \sum_{j=0}^u e_j=p^n.\] Filtering \(\mathcal L_p\) by powers of the ideal of the reduced diagonal in \(\Delta_F\) gives the graded bundles \[ \mathcal G_j=\mathcal O_W(2B_p+pqS)\otimes A_j(\Omega_W^1), \qquad 0\le j\le u. \tag{37}\] Indeed, in smooth local coordinates the algebra is generated by parameter differences with their \(p\)-th powers zero, and its degree-one conormal is \(\Omega_W^1\). This coordinate calculation can be made étale locally or in completions and identifies the global associated graded. This is the diagonal-ideal form of the canonical Frobenius filtration [18]. The same calculation, using only the bundle from the second factor, filters \(F^*\mathcal E\) with pieces [30] \[ \mathcal O_W(B_p)\otimes A_j(\Omega_W^1). \tag{38}\]

Complementary multiplication is a perfect pairing, giving \[ A_{u-j}(V)\simeq A_j(V)^\vee\otimes(\det V)^{p-1}. \tag{39}\] On monomials, each exponent vector pairs with its complement to \((p-1,\ldots,p-1)\). The top line transforms by the character \((\det V)^{p-1}\): this is immediate on diagonal matrices and hence identifies the character of \(\mathrm{GL}_n\) on that line.

A uniform bound for the diagonal rank

We bound the sections of every graded bundle in (37) by its rank times the same scalar polynomial in \(p\). This uniformity lets us sum the ranks of the graded pieces without an additional factor for their number. We begin with a slope bound on the fixed curve; here slope means ordinary degree divided by rank.

The divisors \(B_p-pD=T_0-(p-k_0N_p)D\) range over a fixed finite list. Combining (31), (33), and Lemma 23 therefore gives, uniformly for \(0\le j\le u\), \[\begin{align*} \mu_{\max}(\mathcal G_j|_C) &\le p(4+2/r)\Lambda H^n+O(1)\\ &\le M_p:=7p\Lambda H^n+c_0, \tag{42}\end{align*}\] where \(c_0\ge0\) is fixed. The coefficient \(7\) provides harmless room in this common bound.

The two orders of the determinant

Choose \(R\) generically independent columns of (34) from bases of its source vector spaces. Their determinant is a nonzero section \[0\ne\sigma\in H^0(W'\times W',\mathcal Q_1\boxtimes\mathcal Q_2)\] of an actual external-product line bundle. To specify its factors, let \(m_i\) be the sum of the weights of the \(R\) chosen columns in slot \(i\). The target determinant is \((\det\mathcal E)^s\boxtimes(\det\mathcal E)^s\), and the chosen source determinant is \(\mathcal O(-m_1S')\boxtimes\mathcal O(-m_2S')\). Thus \[\mathcal Q_i=(\det\mathcal E)^{\otimes s} \otimes\mathcal O_{W'}(m_iS'),\qquad \lambda_i=m_i/R.\] Every column weight lies between \(0\) and \(q\), so \(0\le\lambda_i\le q\) independently of the columns chosen.

Under the numerical identification with the base-field twist, \[ \frac{c_1(\mathcal Q_i)}R =\frac{c_1(\mathcal E)}s+\lambda_iS =\frac{B_p}p+\frac{p-1}{2p}K_W+\lambda_iS, \qquad 0\le\lambda_i\le q. \tag{45}\] The Frobenius first-Chern-class identity is also [30]; it is an identity of rational divisor classes. For the second equality, use (38). Complementary degrees in (39) have total first Chern class \(e_j(p-1)K_W\). Summing gives \[\sum_jc_1(A_j(\Omega_W^1))=\frac{s(p-1)}2K_W.\] Thus \(c_1(F^*\mathcal E)=sB_p+s(p-1)K_W/2\), and Frobenius pullback of twisted divisor classes multiplies by \(p\), as required.

For \(0\le\lambda\le q\), the fixed movable inequality gives \[(D+K_W/2+\lambda S)\cdot\gamma\le40\epsilon J\cdot\gamma,\] where pullback by \(\pi_x\) is understood. This concerns only intersections of fixed divisors and the fixed complete-intersection witness, so it remains true on the reduction. The class in (45) differs from its left-hand divisor at \(\lambda=\lambda_i\) by \[\frac1p\bigl(B_p-pD-K_W/2\bigr).\] Its numerator belongs to the fixed finite list \(T_0-aD-K_W/2\), \(a\in\{0,\ldots,k_0-1\}\). The resulting error after intersection with \(\gamma\) is therefore \(O(p^{-1})\), uniformly over the selected determinant columns.

Let \(x'\) be the twisted point and let \(0\ne\tau_i\in H^0(W',\mathcal Q_i)\). The strict transform of its effective divisor on the blowup at \(x'\) pairs nonnegatively with the twist of \(\gamma\). Since \(J\cdot\gamma>0\), it follows that \[ \frac{\mathop{\mathrm{ord}}_{x'}(\tau_i)}R\le40\epsilon+O(1/p). \tag{46}\] The constant is uniform in the columns and in \(\tau_i\). This argument tests sections on the reduction itself; it does not require them to lift to characteristic zero.

Künneth identifies the space containing \(\sigma\) with \(H^0(W',\mathcal Q_1)\otimes H^0(W',\mathcal Q_2)\). Choose bases adapted to the orders of vanishing at \(x'\) in each factor. In each degree their leading terms are linearly independent. Tensor products of these leading terms remain independent in each bidegree of the two disjoint sets of parameters; terms of different bidegrees cannot cancel. Every nonzero tensor consequently has order at most the sum of the two maximum basis orders. Applying (46), we obtain \[ \mathop{\mathrm{ord}}_{(x',x')}\sigma\le R\bigl(80\epsilon+O(1/p)\bigr). \tag{47}\]

On the other hand, (36) and Lemma 24 bound the matrix rank at \((x',x')\) by a number strictly smaller than \(R/4\). Trivialize its line bundles near that point. Invertible constant row operations make more than \(3R/4\) rows vanish at the point. Each entry of those rows belongs to the maximal ideal, so every term of the determinant has order at least \(3R/4\). Therefore \[ \mathop{\mathrm{ord}}_{(x',x')}\sigma\ge3R/4. \tag{48}\] Equations (47) and (48) contradict \(80\epsilon<3/4\) for sufficiently large \(p\).

This contradiction proves Theorem 20. The full-rank calculation used \(z_*\), while the diagonal estimate holds at every point and was tested at the separately chosen \(x\). The errors were controlled by one fixed finite list of divisor classes and one fixed flag polynomial. Thus their constants do not depend on \(p\), the filtration index, or the chosen columns.

Application to the terminal case

We return to Theorem 17. Thus \(X\) is a projective terminal \(\mathbb Q\)-factorial variety of dimension \(n\ge2\) with \(K_X\sim0\), and \(L\) is a nef Cartier divisor of full nef dimension. Assume that \(L\) is not big. Recall the ample rational divisors \(P=r(L+tA)\), where \(A\) is a fixed ample Cartier divisor and \(t>0\) tends to zero, with \[r\longrightarrow\infty,\qquad P^n\longrightarrow\epsilon^n, \qquad \rho=(P^n)^{1/n}.\] The positive rational number \(\epsilon\) was chosen so that \((14\epsilon)^n<1/4\) and \(80\epsilon<3/4\). Choose the integer \(q\) supplied by Proposition 19, and then choose \(t\) sufficiently small that this proposition and Lemma 18 apply, that \(r>q+1\), and that \(P^n<(2\epsilon)^n\). We now fix \(t\), and therefore also \(P,r,\rho\) and the projective bundle in (15). Every remaining choice is made over \(\mathbb C\) before reducing to positive characteristic.

Positive parts under the required comparisons

We will repeatedly use the following properties of Nakayama’s divisorial Zariski decomposition [27].

In particular, a positive part that is rational and numerically semiample is nef, so these properties persist on higher smooth models. Suppose that on a common smooth resolution of a birational contraction one has \[ p^*D\equiv q^*D'+E, \tag{52}\] where \(D,D'\) are pseudo-effective rational Cartier divisors and \(E\) is effective and \(q\)-exceptional. If a smooth model over the target has rational numerically semiample positive part for \(D'\), take a higher common resolution dominating that model. The second part of Lemma 27 pulls that positive part to the common resolution. Its first and third parts, applied to (52), give the same conclusion for \(D\). This is the comparison that will be used at each birational step.

An MMP trivial on the nef divisor

Apply the lemma after a small \(\mathbb Q\)-factorialization, with \(L=m_0M\) for an integer \(m_0>0\) making \(L\) Cartier. The program is \(M\)-trivial, so on a common resolution its ordinary adjoint comparison also gives \[p^*(T+M)=q^*(T'+M')+E,\] where \(E\) is effective and exceptional over the output. The transformed ordinary adjoint \(T'\) remains pseudo-effective, and \(M'\) is nef. By (52), it suffices to prove \((Z_n)\) on that output. We may consequently retain the notation \(X,B,T,M,L\) and assume in addition that \[ T+mL\text{ is nef}. \tag{53}\]

The case of full nef dimension

Suppose first that \(T+mL\) has full nef dimension. Then (1) holds, and \((P_n)\) shows that \(T+M\) is big. Write, by Kodaira’s lemma, \[T+M\sim_\mathbb QA_1+G_1,\] where \(A_1\) is ample and \(G_1\geq0\) is rational. Choose a rational \(v>0\) sufficiently small that \((X,B+vG_1)\) is klt. Since \(M+vA_1\) is ample, a general effective rational representative \(A'\) of this divisor can be chosen with sufficiently small coefficients that \((X,B+vG_1+A')\) is klt. Its adjoint satisfies \[ K_X+B+vG_1+A'\sim_\mathbb Q(1+v)(T+M). \tag{54}\] It is big and has a good minimal model: one may use the big-case theorem of [3], or combine Theorems 5 and 4. On a common resolution, the left side of (54) is the pullback of a semiample divisor plus an effective exceptional divisor. Its positive part is therefore the semiample pullback. Dividing by \(1+v\) proves \((Z_n)\) in this case.

Descent along a nef reduction

We treat the remaining case, where the nef dimension of \(T+mL\) is strictly smaller than \(n\). Let \[X\dashrightarrow Z\] be its nef reduction, with \(Z\) normal and projective. A very general compact fibre \(F\) is contained in the morphism locus and has positive dimension. On this fibre, \[(T+mL)|_F\equiv0.\] The restriction \(T|_F\) is pseudo-effective. To see this, choose effective rational approximations to \(T\) after adding positive ample perturbations tending to zero, and choose \(F\) outside the countably many exceptional choices for these approximations. Their restrictions are effective, and their classes approach \(T|_F\). After a resolution of \(F\) the same argument applies to the pullbacks. As \(L|_F\) is nef and \(T|_F\equiv-mL|_F\), both divisors are numerically trivial. Indeed, intersect this equality with an ample complete-intersection curve on a smooth resolution of \(F\). Pseudo-effectivity makes the left side nonnegative and nefness makes the right side nonpositive. Their common value is zero, and a nef divisor with zero intersection against an ample complete-intersection curve is numerically trivial.

Restriction to a sufficiently general fibre gives a klt pair \((F,B|_F)\) with adjoint \(T|_F\). By Theorem 4, its numerically trivial adjoint is semiample, hence \[ K_F+B|_F\sim_\mathbb Q0. \tag{55}\] These restrictions are made where the almost holomorphic nef reduction is a morphism. Normality and the klt restriction statement hold after shrinking the base; alternatively they follow by restricting a log resolution over a general smooth open. Numerical triviality, initially obtained for very general fibres, also follows on a general fibre by the constancy of the relevant intersection numbers in this family.

We next use the birational descent of nef divisors [22]. After resolving the nef reduction and modifying its base birationally, it supplies a diagram \[X\xleftarrow{p}X_1\xrightarrow{f}Z_1\] in which \(X_1,Z_1\) are smooth and projective, \(p\) is birational, \(f\) is a contraction, and there is a rational divisor \(M_1\) on \(Z_1\) with \[ p^*M\equiv f^*M_1. \tag{56}\] The divisor \(M_1\) is nef, since its pullback is numerically equivalent to the nef divisor \(p^*M\) and nefness descends under a surjective morphism. We may take a further log resolution of the pair on top without affecting (56).

Write the crepant boundary as \(B_1^+-B_1^-\), with effective rational divisors of disjoint support, so that \[K_{X_1}+B_1^+-B_1^-=p^*(K_X+B).\] Set \(B_1=B_1^+\) and \(E=B_1^-\). Since the original boundary is effective, \(E\) is \(p\)-exceptional. Klt singularities make all coefficients of \(B_1\) less than one, and its support is simple normal crossings. Thus \((X_1,B_1)\) is klt and \[ K_{X_1}+B_1=p^*T+E,\qquad E\geq0. \tag{57}\] In particular its adjoint is pseudo-effective.

If \(F_1\) is a general fibre of \(f\), it maps birationally onto the corresponding compact fibre \(F\) of the nef reduction. Every component of \(E\) that meets \(F_1\) restricts to an exceptional divisor over \(F\): for a centre dominating the base, codimension at least two is preserved on the general fibre, and the other centres are avoided. Equations (55) and (57) therefore show that \[ \kappa\bigl(F_1,(K_{X_1}+B_1)|_{F_1}\bigr)=0. \tag{58}\] The restriction is an effective exceptional divisor up to rational linear equivalence. Equivalently, its section spaces are those of the trivial divisor on the normal fibre \(F\). In particular the general fibre pair has pseudo-effective adjoint and has a good minimal model by Theorems 5 and 4.

A relative good model and a lower-dimensional adjoint

At this point the nef summand comes from \(Z_1\), while the ordinary adjoint has Iitaka dimension zero on its general fibres. We use a relative good model to express that ordinary adjoint as a pullback from a variety of dimension less than \(n\).

The relative log canonical ring of the rational klt pair \((X_1,B_1)\) over \(Z_1\) is finitely generated. More precisely, for a positive integer \(r\) making \(r(K_{X_1}+B_1)\) Cartier, its Veronese algebra \[\bigoplus_{\ell\geq0} f_*\mathcal O_{X_1}\bigl(\ell r(K_{X_1}+B_1)\bigr)\] is a finitely generated \(\mathcal O_{Z_1}\)-algebra. This is the relative finite-generation theorem for rational klt pairs, in the form [11], obtained from the finite-generation theorem of [3] and the canonical-bundle reduction of Fujino–Mori [14]. Together with the good minimal models of the very general closed fibres established above, this verifies both hypotheses of [16]. That theorem gives a good minimal model over \(Z_1\); the termination statement for an MMP with scaling once a good model exists, [16], supplies a relative program \[(X_1,B_1)\dashrightarrow(X_2,B_2).\] Thus \(X_2\) is projective and \(\mathbb Q\)-factorial, \((X_2,B_2)\) is klt, and \(K_{X_2}+B_2\) is semiample over \(Z_1\). This application uses very general closed fibres over \(\mathbb C\) and requires no change of the ground field to a geometric generic field.

By (56), we may replace the nef term numerically by \(f^*M_1\) before making the positive-part comparisons. This pullback is trivial on every step of the relative program and is carried unchanged from the base. The ordinary adjoint remains pseudo-effective by numerical pushforward.

Let \[X_2\xrightarrow{g}Z_2\xrightarrow{j}Z_1\] be the relative semiample fibration of \(K_{X_2}+B_2\), with \(g\) a contraction. Its construction gives a \(j\)-ample \(\mathbb Q\)-Cartier divisor \(A_2\) on \(Z_2\) such that \[ K_{X_2}+B_2\sim_\mathbb Qg^*A_2. \tag{59}\] The relative program preserves Iitaka dimension on the general fibres, by its effective exceptional pullback comparisons. From (58), the relative Iitaka dimension is zero. Hence \[\dim Z_2=\dim Z_1<n.\]

Apply Ambro’s canonical bundle formula consequence [1] to (59). Its hypotheses hold: \((X_2,B_2)\) is a projective klt pair with effective rational boundary, \(g\) is a contraction to a normal projective variety, and the ordinary adjoint is rationally linearly pulled back from a \(\mathbb Q\)-Cartier divisor. It gives an effective rational boundary \(B_{Z_2}\) such that \((Z_2,B_{Z_2})\) is klt and \[ K_{X_2}+B_2\sim_\mathbb Qg^*(K_{Z_2}+B_{Z_2}). \tag{60}\] The base adjoint is pseudo-effective. One direct verification, using only the inputs already in force, is as follows. The pseudo-effective ordinary adjoint on \(X_2\) has a good minimal model by Theorems 5 and 4; it therefore has a nonzero section in a sufficiently divisible positive multiple. Since \(g_*\mathcal O_{X_2}=\mathcal O_{Z_2}\), the projection formula and (60) identify this section space with the corresponding section space on \(Z_2\). Thus \(K_{Z_2}+B_{Z_2}\) even has an effective rationally linearly equivalent divisor.

We are now in the setting of \((Z_{\dim Z_2})\): the pair on the base is klt with pseudo-effective adjoint, and \(j^*M_1\) is a nef rational Cartier divisor. There is consequently a smooth projective birational model \(w_2\colon W_2\to Z_2\) on which \[P_\sigma\bigl(w_2^*(K_{Z_2}+B_{Z_2}+j^*M_1)\bigr)\] is rational and numerically semiample. Resolve the main component of \(X_2\times_{Z_2}W_2\) to obtain a smooth projective variety \(V\) mapping birationally to \(X_2\) and surjectively to \(W_2\). Since the displayed positive part is nef, the pullback formula of Lemma 27, together with (60), proves the same positive-part assertion on \(V\) for \[K_{X_2}+B_2+g^*j^*M_1.\]

Take a common resolution dominating both \(V\) and \(X_1\). The relative MMP compares \(K_{X_1}+B_1\) with \(K_{X_2}+B_2\) by adding an effective divisor exceptional over \(X_2\). The \(M_1\) terms have equal pullbacks, since they come from the relative base. The comparison (52) therefore proves the positive-part assertion on a smooth model over \(X_1\) for \(K_{X_1}+B_1+f^*M_1\). Finally, (56) and (57) give \[K_{X_1}+B_1+f^*M_1\equiv p^*(T+M)+E,\] where \(E\) is effective and \(p\)-exceptional. Another application of Lemma 27 proves \((Z_n)\) on a smooth model over the original \(X\).

Descent to the original variety

The positive-part statement is birational, whereas the main theorem requires a semiample representative on the original variety. The following descent uses the rational singularities of klt pairs; it is the numerical semiampleness statement of [22].

We can now prove Theorem 1 over \(\mathbb C\). Its divisor \(D=K_X+B+M\) is nef, so \[P_\sigma(w^*D)=w^*D\] on every smooth birational model. Proposition 26 makes this pullback numerically semiample. A klt variety has rational singularities, so Lemma 29 gives the required semiample rational Cartier divisor on \(X\).

Algebraically closed fields of characteristic zero

We finish by transferring the result to the ground fields in the statement of Theorem 1. The very general points used in the proof were needed only over \(\mathbb C\); numerical semiampleness itself descends by a finite-type parameter argument.