Numerical Semiampleness of Nef Adjoint Divisors

1. Preserve an ordinary semiample adjoint while making a generalized adjoint nef

Choose a terminating ordinary $T$-MMP and apply log abundance to its nef endpoint $U_0$. Next choose an MMP for $U_0+s_0N_0$ with sufficiently small $s_0>0$. If $dU_0$ is free Cartier, the extremal-ray length bound and the discrete inequality $U_i\cdot C\ge1/d$ force every step of this second program to be $U_i$-trivial. Thus the ordinary adjoint remains semiample while the generalized adjoint becomes nef.

The two effective exceptional pullback comparisons transport the positivity on very general curves to the endpoint. When $\kappa(T)>0$, the fibres of the ordinary semiample fibration admit the lower-dimensional $(P_j)$. When $\kappa(T)=0$, one has $U\sim_{\mathbb Q}0$ and reduces to full nef dimension on a Calabi–Yau pair. The input is the existence of a terminating choice, not termination of every sequence of flips.

Lemma 3.2 · pp. 6–7

Propositions 3.3–3.4 · p. 8

[OpenAI26c, Theorem 1.1; §3.3, pp. 2, 14–15]

[OpenAI26LA, Theorem 1.1, p. 2]

2. Obtain a numerically semiample positive part and descend to X

Statement $(P_n)$ and the lower-dimensional $(Z_j)$ yield Proposition 8.1. Since the main theorem assumes $D$ nef, $P_\sigma(w^*D)=w^*D$ on every smooth birational model. Choose a semiample $A_W\equiv w^*D$. Clear denominators and the torsion in numerically trivial line bundles modulo $\operatorname{Pic}^0$ to obtain

Klt singularities are rational, so pullback identifies $\operatorname{Pic}^0(X)$ with $\operatorname{Pic}^0(W)$ and $P_W=w^*P_X$. The right side is now the pullback of a line bundle on $X$. Using $w_*\mathcal O_W=\mathcal O_X$ and the projection formula descends global generation. The resulting divisor twists $rD$ by $P_X$, so the conclusion remains numerical equivalence.

Proposition 8.1 · p. 30

Lemma 8.4 · p. 35

[LP18, Lemma 2.14, p. 12]

3. Pass to every algebraically closed field of characteristic zero

Descend the variety, boundary and Cartier data to a finitely generated field, and embed its algebraic closure $k_0$ into $\mathbb C$. The complex conclusion gives a fixed $r>0$ and $P\in\operatorname{Pic}^0(X_{\mathbb C})$ for which $\mathcal O(rD)\otimes P$ is globally generated. In a Poincaré family over the Picard variety, global generation for this fixed $r$ defines a constructible set of parameters. Its nonemptiness is reflected by algebraically closed field extensions. A $k_0$-point therefore gives the desired representative, which can be extended to the original $k$. The argument with very general points itself need not be transferred to every ground field.

Lemma 8.5; §8.7 · pp. 35–36

1. Compare positive integral degrees with uniformly bounded curve degrees

Suppose $L$ is not big and fix an ample Cartier divisor $A$. As $t\downarrow0$, normalize

Every curve through a very general point has $L\cdot C\ge1$, hence $P\cdot C\ge r$. In the opposite direction, Lemma 4.1 constructs curves of uniformly bounded degree when the testing volume and adjoint intersection ratio are bounded. Its Birkar–Zhang input uses a reduced SNC boundary, Cartier index one for the nef summand, and a big adjoint. The lower-dimensional $(P_j)$ supply this bigness when needed.

Lemma 4.1 · pp. 9–10

[BZ15, Theorem 1.3, p. 3]

2. Exclude moving base components to obtain the two estimates

On the scalar side, an excessively vanishing section produces a moving base component. Its volume and canonical-intersection estimates feed into Lemma 4.1, giving $\operatorname{ord}_x(s)\le4k(P^n)^{1/n}$ for every nonzero $s\in H^0(X,kP)$ at a very general $x$.

The two-slot construction uses

Fix $q$ before making $t$ small. The proof excludes positive-dimensional slice fibres, lower-dimensional components and sweeping branch divisors, then treats finite correspondences dominant over both factors. Non-sweeping branch images reduce the possibilities to finitely many Stein covers for each fixed $t$. A product-dominant family of birational graphs between fixed covers leads to a model birational to an abelian variety; the bounded-degree obstruction then excludes the last case. Consequently there is a point outside both axes with $\varepsilon(R;z)>2n+2$. The finite lists of covers themselves are not asserted to be uniform in $t$.

Lemma 5.2 · p. 14

Proposition 5.3 · p. 15

§§5.4–5.6 · pp. 16–20

[OpenAI26LA, Lemmas 7.1–7.5, pp. 45–52]

3. Fix all complex data before applying the Frobenius comparison

On a resolution $w:W\to X$, set $D=w^*P$ and $S=w^*L$, and perturb $w^*P$ slightly to an ample $H$. Pseudo-effectivity of $K_W$ supplies generic semipositivity of the cotangent bundle and the flag required by the comparison. The two-slot Seshadri bound produces finitely many sections generating jets through order $(2n+2)k_0$. The scalar bound implies that $\pi_x^*(2D+E)-40\epsilon J$ on the blowup of a point is not pseudo-effective. Duality with movable curves supplies one fixed strongly movable class witnessing all the weight inequalities.

Theorem 6.1 rules out coexistence of these fixed data by opposite bounds on the vanishing order of a Frobenius-evaluation determinant. Thus the assumption that $L$ is not big is contradictory. The manuscript reproves the comparison theorem; this overview checks its statement and the application in §6.7. The full slope and determinant calculation in §§6.2–6.6, the external Hanamura/Blanc sources used in §5.6, and the restriction theorem producing the flag were not independently checked.

§6.7 · pp. 27–28

[CP15, Theorem 2.1, print p. 844 / PDF p. 11]

[BDPP13, Theorem 2.2, p. 6]

[OpenAI26LA, Theorem 9.1, p. 62]

1. Transfer the zero-boundary case to Theorem 5.1

The preceding reductions allow us to focus on $K_X+B\equiv0$ and $M$ of full nef dimension. If $B=0$, log abundance gives $K_X\sim_{\mathbb Q}0$. The index-one cover is klt and canonical; after crepant terminalization it is terminal and $\mathbb Q$-factorial with $K\sim0$. Apply Theorem 5.1 to a Cartier multiple of the pulled-back $M$, then descend bigness, which is detected by a dominant generically finite pullback.

[BCHM10, Corollary 1.4.3, p. 10]

2. Enlarge the boundary slightly and realize M as an ordinary adjoint

For $B\ne0$, set $T_*:=K_X+(1+\alpha)B\sim_{\mathbb Q}\alpha B$. If $\kappa(T_*)>0$, Proposition 3.3 gives bigness of $T_*+M$. If it is zero, the two-stage MMP has $B_Y=0$ at its endpoint, so the zero-boundary case just established gives the same conclusion. This avoids using Proposition 7.1 before completing its proof.

Choose an effective $G\sim_{\mathbb Q}\alpha B+M$. For a sufficiently small positive rational $u$,

and $(X,\Delta_u)$ is an ordinary klt pair. Its adjoint is nef, so log abundance now makes $M$ itself semiample. Full nef dimension forces the associated morphism to be generically finite, and hence $M$ is big. Finally, the positive-combination argument of Proposition 3.4 gives bigness of $T+cM$ for every $c>0$. The stronger intermediate semiampleness conclusion belongs to this specially constructed ordinary-adjoint situation.

§7 · pp. 29–30

[OpenAI26LA, Theorem 1.1, p. 2]

1. Prepare the alternatives by an MMP trivial on the nef divisor

Make $L=m_0M$ Cartier and choose $m>2n$. On a $(T+mL)$-negative ray, the inequality $L\cdot C>0$ would imply $(T+mL)\cdot C\ge m-2n>0$, using integrality and the ordinary ray-length bound. Thus the chosen terminating MMP is $L$-trivial and also $T$-negative. Its endpoint has $T+mL$ nef and retains a nef transform of $M$.

If $T+mL$ has full nef dimension, $(P_n)$ makes $T+M$ big. A decomposition $T+M\sim_{\mathbb Q}A+G$ realizes a positive rational multiple $(1+v)(T+M)$ as an ordinary big klt adjoint. The pullback of the semiample adjoint on its good minimal model gives its positive part.

Lemma 8.3 · pp. 31–32

§§8.3–8.4 · pp. 32–33

2. Reduce to an ordinary adjoint and a nef divisor on a smaller base

If the nef dimension is smaller, a very general compact nef-reduction fibre $F$ has $T|_F$ pseudo-effective, $M|_F$ nef, and $(T+mL)|_F\equiv0$. These signs make both restrictions numerically trivial. Ordinary log abundance then gives $K_F+B|_F\sim_{\mathbb Q}0$.

Birational descent of nef divisors produces $f:X_1\to Z_1$ after resolution, with $p^*M\equiv f^*M_1$ and $M_1$ nef. This is numerical equivalence. Write $K_{X_1}+B_1=p^*T+E$, where $E\ge0$ is $p$-exceptional. The ordinary adjoint has Iitaka dimension zero on the general fibre. Relative finite generation of its log canonical ring and good minimal models of very general fibres give a relative good model. Its semiample fibration $g:X_2\to Z_2$ satisfies

Apply Ambro's result to this Q-linear identity for an ordinary klt adjoint, replacing $A_2$ by $K_{Z_2}+B_{Z_2}$ with effective klt boundary. Pseudo-effectivity of the ordinary adjoint upstairs and sections supplied by its good model descend to show that the base adjoint is pseudo-effective. Statement $(Z_{\dim Z_2})$ therefore applies.

[LP18, Lemma 3.1, p. 13]

[Fuj09b, Theorem 1.4, p. 2]

[HX11, Theorem 2.12, p. 12]

[HX11, Corollary 2.9, p. 9]

[Amb03, Theorem 0.2, p. 2]

§8.5 · pp. 33–34

3. Use positive-part comparisons to return to the original model

The lower-dimensional positive part is numerically semiample, hence nef. Use the pullback formula for a nef positive part, its invariance under adding effective exceptional divisors, and the numerical invariance of its rationality and numerical semiampleness. These rules allow the numerical descent of $M$ and the Q-linear ordinary-adjoint formula to enter the same comparison. Reverse the relative MMP from $X_2$ and then remove the $p$-exceptional $E$, obtaining the asserted positive part of $w^*(T+M)$. This proves $(Z_n)$ from $(P_n)$ and the lower-dimensional $(Z_j)$.

Lemma 8.2 · p. 31

[LP18, Lemmas 2.4–2.5, pp. 7–8]

1. Check the hypotheses of the main theorem

If $K_X+B\equiv0$, the ordinary adjoint is pseudo-effective. For every nef $M$, the divisor $D=K_X+B+M\equiv M$ is nef. The semiample divisor $L\equiv D$ supplied by Theorem 1.1 is therefore numerically equivalent to $M$. This replaces a numerical representative; it does not produce sections of a nontorsion numerically trivial line bundle.

Corollary 1.2 · p. 2