The Iitaka conjecture $C_{n,m}$: subadditivity of the Kodaira dimension for algebraic fiber spaces

The Iitaka conjecture $C_{n,m}$ predicts that for an algebraic fiber space $f:X\to Y$ between smooth complex projective varieties with general fiber $F$, the Kodaira dimensions satisfy $\kappa(X)\ge\kappa(F)+\kappa(Y)$. It is settled when the base is of general type, when the base has dimension at most 2, when the general fiber admits a good minimal model, when the total space has dimension at most 6, and when the base has maximal Albanese dimension; a 2025 preprint additionally claims the case of bases with Albanese dimension at least $\dim Y-2$. The conjecture remains open for bases of Albanese dimension at most $\dim Y-3$ — notably bases with non-negative Kodaira dimension and irregularity zero, such as Calabi–Yau threefold bases — where the known routes do not settle the general case: the Hodge-theoretic positivity methods behind the maximal-Albanese results do not apply, and the canonical bundle formula route is obstructed by the unknown semiampleness of its moduli part.

Background

The Kodaira dimension $\kappa(V)$ of a smooth projective variety $V$ is the birational invariant measuring the asymptotic growth of the spaces of pluricanonical sections, $\kappa(V)=\limsup_{m\to\infty}\log h^0(V,mK_V)/\log m$, with values in $\{-\infty,0,\dots,\dim V\}$; it is the basic coarse measure of position in the birational classification, from rationally connected-like ($\kappa=-\infty$) to general type ($\kappa=\dim V$). An algebraic fiber space is a surjective morphism $f:X\to Y$ with connected fibers between smooth projective varieties; write $n=\dim X$, $m=\dim Y$, and $F$ for a general fiber.

For a fibration, the natural question is how $\kappa(X)$ relates to $\kappa(F)$ and $\kappa(Y)$. Mori's easy addition formula gives the universal upper bound $\kappa(X)\le\kappa(F)+m$. The conjectured lower bound, posed by Iitaka in the 1970s and denoted $C_{n,m}$, predicts $\kappa(X)\ge\kappa(F)+\kappa(Y)$. It is a cornerstone of the Iitaka–Mori classification program: subadditivity would let the classification of a variety be assembled from that of its fiber and base along fibrations. The inequality is trivial when $\kappa(F)=-\infty$ or $\kappa(Y)=-\infty$ (the right-hand side is $-\infty$), so the essential case is $\kappa(F)\ge0$ and $\kappa(Y)\ge0$.

The Albanese variety $\mathrm{Alb}(Y)$ is the abelian variety dual to $H^0(Y,\Omega_Y^1)/H^1(Y,\mathcal{O}_Y)$, and the Albanese map $\mathrm{alb}_Y:Y\to\mathrm{Alb}(Y)$ is the universal morphism from $Y$ to an abelian variety; the Albanese dimension is $\alpha(Y)=\dim\,\mathrm{alb}_Y(Y)$ and the irregularity is $q(Y)=\dim\mathrm{Alb}(Y)=h^1(Y,\mathcal{O}_Y)$. Maximal Albanese dimension means $\alpha(Y)=\dim Y$.

The settled regimes of $C_{n,m}$ are: $Y$ of general type (Viehweg's weak positivity theorem, 1983); $m=1$ (Kawamata, 1982); the case where the general fiber admits a good minimal model, meaning a minimal model whose canonical class is semi-ample, i.e. some positive multiple of it is basepoint-free (Kawamata, 1985 — this subsumes Kollár's fibers-of-general-type case and Fujino's maximal-Albanese-fiber case, since those fibers are known to have good minimal models); $n\le6$ (Birkar, 2009); $n=7$ under additional hypotheses — non-negative Kodaira dimension of the source, positive Kodaira dimension of the fiber, or base not a threefold with $\kappa=q=0$ (Chang, 2025); $m=2$ (Cao, 2018); and bases of maximal Albanese dimension (Cao–Păun, 2017, with a simplified proof by Hacon–Popa–Schnell, 2018).

The candidate arises from a 2025 preprint of H. Benammar Ammar claiming $C_{n,m}$ for bases with $\alpha(Y)\ge m-2$, i.e. bases whose Albanese image has codimension at most 2; the proof composes the Cao–Păun inequality over the Albanese image with Kawamata's $C_{n,1}$ and Cao's $C_{n,2}$ over the Albanese fiber (a curve or surface), then runs an induction on $\dim X$. For $\kappa(Y)>0$ the same preprint derives the range alternatively from an effectivity theorem of Schnell (2025), obtained from the Campana–Peternell conjecture on singular metrics, whose hypothesis is that the base is not uniruled (not covered by rational curves); applied through the Iitaka fibration $I:Y\to I(Y)$ of $Y$ — the fibration defined by sufficiently divisible multiples of $K_Y$, whose base $I(Y)$, the Iitaka variety, has dimension $\kappa(Y)$ — this yields the statement only when $I(Y)$ is not uniruled. That hypothesis is the origin of the auxiliary limitation recorded in the source. The preprint has no journal version and no indexed citations, so its claimed regime should be treated as provisional.

What remains outside all of this are bases with $\alpha(Y)\le m-3$ — the Albanese image has codimension at least 3 — including every base with $\kappa(Y)\ge0$ and $q(Y)=0$, such as Calabi–Yau varieties (smooth projective varieties whose canonical class is torsion, in particular trivial, and whose irregularity is zero, so that $\kappa=0$ and $\alpha=0$). The inequality remains open there in general — partial results reach into the region, for instance $C_{n,3}$ for certain Calabi–Yau threefold bases (Benammar Ammar, 2026) — and neither standard route applies: the Cao–Păun machinery exploits Hodge-theoretic structure (positivity of direct image sheaves via singular hermitian metrics, cohomology jumping loci) that requires near-maximal Albanese dimension on the base, while the canonical bundle formula route — expressing $K_X$ modulo the fibration as the pullback of $K_Y$ plus a moduli part $L$ that is currently known only to be nef (numerically effective, i.e. it has non-negative intersection number with every curve) — is blocked by the unknown semiampleness and numerical $\mathbb{Q}$-effectivity of $L$. Schnell's theorem shows the Campana–Peternell conjecture would reduce subadditivity to the special case $\kappa(Y)=0$, but that reduction itself is conditional on an open conjecture.

Problem Statement

Does the inequality $\kappa(X)\ \ge\ \kappa(F)+\kappa(Y)$ hold for every algebraic fiber space $f:X\to Y$ between smooth complex projective varieties, where $F$ is a general fiber? Prove the Iitaka conjecture $C_{n,m}$ in the cases that remain open after the settled regimes (base of general type; base of dimension at most 2; general fiber admitting a good minimal model; total space of dimension at most 6; base of maximal Albanese dimension; and, per the cited preprint and taken as given rather than independently re-proved, bases with Albanese dimension at least $m-2$) — in particular for bases whose Albanese dimension satisfies $\alpha(Y)\le m-3$, including bases with $\kappa(Y)\ge0$ and irregularity $q(Y)=0$ such as Calabi–Yau threefolds — thereby establishing the conjecture in full generality; or refute the conjecture by exhibiting an explicit algebraic fiber space with $\kappa(X)<\kappa(F)+\kappa(Y)$.

References

1. Houari Benammar Ammar, "Note on Iitaka Conjecture $C_{n,m}$", arXiv:2510.06412 (2025), DOI 10.48550/arXiv.2510.06412, https://arxiv.org/abs/2510.06412
2. Houari Benammar Ammar, "Kodaira dimension of algebraic fiber spaces over threefolds: Part 1", arXiv:2605.08598 (2026), DOI 10.48550/arXiv.2605.08598, https://arxiv.org/abs/2605.08598
3. Junyan Cao and Mihai Păun, "Kodaira dimension of algebraic fiber spaces over abelian varieties", Inventiones mathematicae 207 (2017), 345–387, DOI 10.1007/s00222-016-0672-6, https://arxiv.org/abs/1504.01095
4. Christopher Hacon, Mihnea Popa, and Christian Schnell, "Algebraic fiber spaces over abelian varieties: Around a recent theorem by Cao and Păun", Contemporary Mathematics 712 (2018), 143–195, DOI 10.1090/conm/712/14346, https://arxiv.org/abs/1611.08768
5. Junyan Cao, "Kodaira dimension of algebraic fiber spaces over surfaces", Algebraic Geometry 5 (2018), no. 6, 728–741, DOI 10.14231/AG-2018-020, https://arxiv.org/abs/1511.07048
6. Yujiro Kawamata, "Kodaira dimension of algebraic fiber spaces over curves", Inventiones Mathematicae 66 (1982), 57–71, DOI 10.1007/BF01404756, https://doi.org/10.1007/BF01404756
7. Yujiro Kawamata, "Minimal models and the Kodaira dimension of algebraic fiber spaces", Journal für die reine und angewandte Mathematik 363 (1985), 1–46, DOI 10.1515/crll.1985.363.1, https://doi.org/10.1515/crll.1985.363.1
8. Eckart Viehweg, "Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces", Advanced Studies in Pure Mathematics 1 (1983), 329–353, DOI 10.2969/aspm/00110329, https://doi.org/10.2969/aspm/00110329
9. Caucher Birkar, "The Iitaka conjecture $C_{n,m}$ in dimension six", Compositio Mathematica 145 (2009), no. 6, 1442–1446, DOI 10.1112/S0010437X09004187, https://arxiv.org/abs/0806.4413
10. Fanjun Meng and Mihnea Popa, "Kodaira dimension of fibrations over abelian varieties", International Mathematics Research Notices 2024 (2024), no. 9, 7466–7487, DOI 10.1093/imrn/rnad209, https://arxiv.org/abs/2111.14165
11. Christian Schnell, "Singular metrics and a conjecture by Campana and Peternell", Pure and Applied Mathematics Quarterly 21 (2025), no. 3, 1269–1281, DOI 10.4310/pamq.250116023222, https://arxiv.org/abs/2202.01295
12. Chi-Kang Chang, "Generalized nonvanishing conjecture and Iitaka conjecture", Mathematische Zeitschrift 310 (2025), no. 2, DOI 10.1007/s00209-025-03727-5, https://arxiv.org/abs/2312.04015