The exceptional case of the Generalized Nonvanishing Conjecture for threefold pairs with $\kappa=0$, $q=0$, and maximal nef dimension, and the residual case of the Iitaka conjecture $C_{7,3}$

For projective klt pairs of dimension three, Chang (Math. Z. 2025) proved the Generalized Nonvanishing Conjecture — numerical effectivity of $K_X+\Delta+tL$ for all $t\ge 0$ — in every configuration except the single case where $\kappa(K_X+\Delta)=0$, the irregularity $q(X)=0$, and the nef divisor $L$ has maximal nef dimension $n(L)=3$. As a consequence, the classical Iitaka conjecture $C_{n,m}$ on additivity of Kodaira dimension is proved for $n\le 7$ except for the single configuration $C_{7,3}$ with $\kappa(F)=\kappa(Y)=0$ and all (augmented) irregularities vanishing, where Fujino–Mori canonical bundle formula data reduce the problem to showing $\kappa(X')\ge 0$ on a modified total space. The newest treatment (Ammar, 2026) further narrows but does not close these residual cases, which remain open. The problem asks to resolve the exceptional threefold nonvanishing case and, through it, to prove $\kappa(X)\ge 0$ in the residual $C_{7,3}$ configuration, thereby completing the Iitaka conjecture in dimension seven.

Background

Let $f:X\to Y$ be an algebraic fiber space, i.e. a surjective morphism with connected fibers between smooth complex projective varieties, and let $F$ be a very general fiber. The Kodaira dimension $\kappa(X)\in\{-\infty,0,1,\dots,\dim X\}$ measures the growth of the spaces of pluricanonical sections $H^0(X,mK_X)$; it is the fundamental birational invariant of a projective variety. The Iitaka conjecture (1970s) predicts additivity of Kodaira dimension along a fibration: $C_{n,m}$ states that if $\dim X=n$ and $\dim Y=m$, then $\kappa(X)\ge\kappa(F)+\kappa(Y)$. It is known when $\dim Y\le 1$ (Kawamata, Viehweg), when $\dim Y=2$ (Cao), when $Y$ is of general type, when $F$ is of general type, when $F$ has a good minimal model, i.e. a minimal model on which some positive multiple of the canonical divisor is basepoint-free (Kawamata; in particular $\dim F\le 3$), and when $Y$ has maximal Albanese dimension, i.e. when the Albanese map of $Y$ — to the abelian variety dual to its space of holomorphic one-forms — is generically finite onto its image (Cao–Păun); Birkar proved $C_{n,m}$ for $\dim X\le 6$. The general case remains one of the central open problems of birational geometry.

A pair $(X,\Delta)$ consisting of a normal projective variety and an effective $\mathbb{Q}$-divisor $\Delta$ is klt (Kawamata log terminal) if it has mild singularities in the sense of the minimal model program; the irregularity is $q(X)=h^1(X,\mathcal{O}_X)$, and the augmented irregularity $\tilde q(X)$ is the supremum of $q$ over all quasi-étale covers of $X$, i.e. finite covers that are étale in codimension one. A $\mathbb{Q}$-divisor is nef if it has non-negative intersection with every curve; the nef dimension $n(L)\le\dim X$ is the dimension of the image of the nef reduction map, a morphism with connected fibers determined (up to birational equivalence) by the condition that $L$ pulls back from the image and contracts exactly the curves on which $L$ is numerically trivial; $n(L)=\dim X$ is the maximal case. The numerical dimension $\nu(D)$ is an intersection-theoretic analogue of the Kodaira dimension, defined for pseudo-effective divisors — for nef $D$ it is the largest integer $\nu$ such that $D^{\nu}$ is not numerically trivial — and it satisfies $\kappa(D)\le\nu(D)$.

Lazić and Peternell introduced, in their work on generalized abundance, the Generalized Nonvanishing Conjecture (GNVC): if $(X,\Delta)$ is a klt pair with $K_X+\Delta$ pseudo-effective, i.e. of class lying in the closed cone spanned by effective divisors, and $L$ is a nef $\mathbb{Q}$-divisor, then for every rational $t\ge 0$ the class of $K_X+\Delta+tL$ is numerically equivalent to an effective $\mathbb{Q}$-Cartier divisor. They proved it for $\dim X\le 2$ and for $\dim X=3$ with $\nu(K_X+\Delta)\ge 1$. Writing $N_{d,k,q,n}$ for the conjecture in dimension $d$ with $\kappa(K_X+\Delta)=k$, $q(X)=q$, and $n(L)=n$, Chang proved that $N_{d,k,q,1}$ holds for $k\ge 0$, that $N_{d,k,q,2}$ holds for $k\ge 0$ and $d\le 5$, that $N_{3,k,q,3}$ holds for $q>0$, and that $N_{d,0,q,d}$ holds for $q\ge d-2$. Combining these with Lazić–Peternell's result yields his Corollary 1.6: $N_{3,k,q,n}$ is true except possibly when $k=0$, $q=0$, and $n=3$ — the unique surviving threefold obstruction.

The bridge from nonvanishing to the Iitaka conjecture goes through the canonical bundle formula of Fujino and Mori: for a fibration whose general fiber has $\kappa(F)=0$, after a birational modification $f':X'\to Y'$ there are an effective $\mathbb{Q}$-divisor $B$ (the discriminant) and a nef $\mathbb{Q}$-divisor $L$ (the moduli part) on $Y'$ with $pK_{X'}=pf'^*(K_{Y'}+B+L)+R$ for suitable $p>0$ and a controlled remainder $R$. Chang's Theorem 1.7 shows, following Birkar's strategy, that if $N_{d,k,q}$ holds with $d=\dim Y$, $k=\kappa(Y)$, $q=q(Y)$, then any such fibration satisfies $\kappa(X)\ge\kappa(Y)$. Combining all previously known cases with his new ones, his Theorem 1.8 establishes $C_{n,m}$ for all $n\le 7$ except possibly $n=7$, $m=3$, $\kappa(F)=\kappa(Y)=0$, and $q(X)=q(F)=q(Y)=\tilde q(Y)=0$; his Remark 5.5 records that in this remaining configuration the Fujino–Mori data necessarily satisfy $\kappa(Y',K_{Y'}+B)=\nu(Y',K_{Y'}+B)=0$ and $n(L)=3$, and that proving $\kappa(X')\ge 0$ under these explicit hypotheses would complete $C_{7,3}$, hence $C_{n,m}$ for all $n\le 7$. The two residual items form one coherent objective, since establishing the exceptional nonvanishing case $N_{3,0,0,3}$ supplies exactly the input Theorem 1.7 needs.

Problem Statement

Resolve the two linked residual configurations left open by Chang's "Generalized nonvanishing conjecture and Iitaka conjecture" (Math. Z. 310 (2025), Paper No. 30):

(1) The exceptional case of the Generalized Nonvanishing Conjecture in dimension three ($N_{3,0,0,3}$). Let $(X,\Delta)$ be a projective klt pair of dimension $3$ such that $K_X+\Delta$ is pseudo-effective with $\kappa(X,K_X+\Delta)=0$ and irregularity $q(X)=0$, and let $L$ be a nef $\mathbb{Q}$-divisor on $X$ of maximal nef dimension $n(L)=3$. Prove, or disprove by an explicit example, that for every rational $t\ge 0$ the divisor $K_X+\Delta+tL$ is numerically equivalent to an effective $\mathbb{Q}$-Cartier divisor. This is the unique configuration not covered by Chang's Corollary 1.6; settling it completes the Generalized Nonvanishing Conjecture for all klt threefold pairs.

(2) The residual case of the Iitaka conjecture $C_{7,3}$. Let $f:X\to Y$ be an algebraic fiber space between smooth projective varieties with $\dim X=7$, $\dim Y=3$, and general fiber $F$, such that $\kappa(F)=\kappa(Y)=0$ and $q(X)=q(F)=q(Y)=\tilde q(Y)=0$ — the single configuration not covered by Chang's Theorem 1.8. By Chang's Remark 5.5, in this remaining configuration the Fujino–Mori canonical bundle formula decomposition $pK_{X'}=pf'^*(K_{Y'}+B+L)+R$ (after the birational modification $f':X'\to Y'$ of the fibration, with discriminant $B$, moduli part $L$, and controlled remainder $R$) necessarily satisfies $\kappa(Y',K_{Y'}+B)=\nu(Y',K_{Y'}+B)=0$ and nef dimension $n(L)=3$. Prove that $\kappa(X)\ge 0$, equivalently $\kappa(X')\ge 0$, under these hypotheses; by Chang's Remark 5.5 this completes the proof of the Iitaka conjecture $C_{7,3}$, and hence of $C_{n,m}$ for all $n\le 7$.

The two items form a single objective: item (1), through the mechanism of Chang's Theorem 1.7, is precisely the input that yields item (2). Item (1) is required for all projective klt threefold pairs $(X,\Delta)$ with $\kappa(K_X+\Delta)=0$, $q(X)=0$, and nef $\mathbb{Q}$-divisor $L$ of maximal nef dimension $n(L)=3$; proving it only for the instance consisting of the pair $(Y',B)$ with the nef moduli part $L$ of the Fujino–Mori decomposition in item (2) resolves item (2) alone, not item (1). Item (2) can also be resolved directly, without first resolving item (1).

References

1. Chi-Kang Chang, "Generalized nonvanishing conjecture and Iitaka conjecture", Mathematische Zeitschrift 310 (2025), no. 2, Paper No. 30. DOI: 10.1007/s00209-025-03727-5; arXiv:2312.04015; https://arxiv.org/abs/2312.04015
2. Houari Benammar Ammar, "Kodaira dimension of algebraic fiber spaces over threefolds: Part I", preprint (2026). arXiv:2605.08598; https://arxiv.org/abs/2605.08598
3. Houari Benammar Ammar, "Note on Iitaka Conjecture $C_{n,m}$", preprint (2025). arXiv:2510.06412; https://arxiv.org/abs/2510.06412
4. Vladimir Lazić and Thomas Peternell, "On Generalised Abundance, I", Publications of the Research Institute for Mathematical Sciences 56 (2020), no. 2, 353–389. DOI: 10.4171/prims/56-2-3; https://doi.org/10.4171/prims/56-2-3
5. Osamu Fujino and Shigefumi Mori, "A Canonical Bundle Formula", Journal of Differential Geometry 56 (2000), no. 1, 167–188. DOI: 10.4310/jdg/1090347529; https://doi.org/10.4310/jdg/1090347529
6. Caucher Birkar, "The Iitaka conjecture $C_{n,m}$ in dimension six", Compositio Mathematica 145 (2009), no. 6, 1442–1446. DOI: 10.1112/S0010437X09004187; https://doi.org/10.1112/S0010437X09004187
7. Claudio Fontanari, "A question about generalized nonvanishing", Rendiconti del Circolo Matematico di Palermo, Serie II, 73 (2024), no. 7, 2599–2602. DOI: 10.1007/s12215-024-01054-y; arXiv:2312.09441; https://arxiv.org/abs/2312.09441
8. Andreas Höring, Vladimir Lazić, and Christian Lehn, "Nonvanishing results for Kähler varieties", preprint (2025). arXiv:2508.14634; https://arxiv.org/abs/2508.14634
9. Yongpan Zou, "On the Kodaira dimension of some algebraic fiber spaces", preprint (2024). arXiv:2409.19981; https://arxiv.org/abs/2409.19981