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

The Iitaka conjecture $C_{n,m}$ predicts that for an algebraic fibre space $f\colon X\to Y$ between smooth projective varieties over an algebraically closed field of characteristic zero, with general fibre $F$, the Kodaira dimensions satisfy $\kappa(X)\ge\kappa(F)+\kappa(Y)$. Formulated in Iitaka's classification program in the 1970s, it remains one of the central open problems of birational geometry. Birkar's 'The Iitaka conjecture $C_{n,m}$ in dimension six' (Compositio Mathematica, 2009) settled all cases with $\dim X\le 6$ and identified the two obstructions that persist today: the absence of good minimal models (abundance) for fibres of dimension at least four with $0\le\kappa(F)<\dim F$, and the Campana–Peternell conjecture on decompositions $K_X\equiv A+M$. As of September 2026, the conjecture remains open in general: bases of dimension at most two are settled (Kawamata; Junyan Cao), bases of maximal Albanese dimension are settled (Chen–Hacon; Cao–Păun), and Chang settled $\dim X\le 7$ except for one degenerate configuration over a threefold base; but fibrations over threefold bases with trivial invariants (for example Calabi–Yau bases with vanishing irregularity) and all higher-dimensional cases outside the known criteria remain unresolved, as do abundance in dimension at least four and the Campana–Peternell conjecture, which Schnell showed to be almost equivalent to the nonvanishing conjecture.

Background

Let $X$ be a smooth projective variety over an algebraically closed field of characteristic zero and let $K_X$ be its canonical divisor. The Kodaira dimension $\kappa(X)\in\{-\infty,0,1,\dots,\dim X\}$ is the growth rate of the pluricanonical spaces of global sections $H^0(X,mK_X)$: for divisible $m\gg 0$ one has $\dim H^0(X,mK_X)\sim m^{\kappa(X)}$, with $\kappa(X)=-\infty$ if all these spaces vanish. It is the fundamental birational invariant measuring pluricanonical growth, and $\kappa(X)=\dim X$ means that $X$ is of general type. An algebraic fibre space is a surjective morphism $f\colon X\to Y$ of normal projective varieties with connected fibres; write $n=\dim X$, $m=\dim Y$, and let $F$ be a general fibre, so $\dim F=n-m$. The easy addition formula (Mori 1987, Corollary 1.7) gives the upper bound $\kappa(X)\le\kappa(F)+\dim Y$, and the Iitaka conjecture, formulated in Iitaka's classification program in the 1970s, is the conjectural matching lower bound, denoted $C_{n,m}$:

$$\kappa(X)\ge\kappa(F)+\kappa(Y).$$

(The statement is vacuous unless $\kappa(F)\ge 0$ and $\kappa(Y)\ge 0$.) Viehweg proposed the stronger Iitaka–Viehweg form $C^+_{n,m}$, in which $\kappa(Y)$ is replaced by $\max\{\kappa(Y),\operatorname{var}(f)\}$, where $\operatorname{var}(f)$ is the variation of the family, that is the dimension of the image of the base under the moduli map of the fibres.

The main unconditional route to $C_{n,m}$ passes through the minimal model program (MMP), the program of algorithmically improving a variety by contracting and flipping curves on which $K_X$ is negative. A minimal model of $X$ is a birational model $X_{\min}$ with terminal singularities (the mildest singular class under which the MMP is known to behave) and nef canonical divisor $K_{X_{\min}}$, where nef means non-negative on every curve; it is a good minimal model if moreover some multiple of $K_{X_{\min}}$ is base-point free (semiample). The minimal model conjecture predicts that every smooth projective variety of non-negative Kodaira dimension has a minimal model, and the abundance conjecture predicts that every minimal model is good. Kawamata (1985) proved that $C_{n,m}$ holds whenever the general fibre $F$ has a good minimal model. Since abundance is known in dimension at most three (Kawamata 1992), and since varieties of general type have good minimal models in all dimensions (Birkar–Cascini–Hacon–McKernan 2010), this covers the cases of fibre dimension at most three and of general-type fibres (the latter already settled directly by Kollár 1987). Independently, $C_{n,m}$ is known when the base is a curve (Kawamata 1982) and, in the stronger $C^+$ form, when the base is of general type (Viehweg 1983).

Birkar's 'The Iitaka conjecture $C_{n,m}$ in dimension six', Compositio Mathematica 145 (2009), 1442–1446, pushed the total-dimension frontier to $n\le 6$. The only case not covered by Kawamata's criterion was $n=6$, $m=2$, $\dim F=4$ with $\kappa(F)=0$: a minimal model of $F$ exists, but abundance for it is unknown. Birkar bypassed this with a theorem for base dimension two and $\kappa(F)=0$ (his Theorem 1.4), combining the Fujino–Mori canonical bundle formula (which, for a fibration with $\kappa(F)=0$, rewrites pluricanonical divisors upstairs in terms of $K_{Y'}+B+L$ on a birational model $Y'$ of the base, with $B$ effective and $L$ nef), the nef reduction map (a rational map through which a nef divisor becomes numerically trivial on its fibres), and a theorem of Campana–Peternell asserting that $\kappa(X)\ge\kappa(A)$ whenever $K_X\equiv A$ with $A$ effective, where numerical equivalence $\equiv$ means equality of intersection numbers with all curves. Campana and Peternell conjectured the extension in which $K_X\equiv A+M$ with $M$ merely pseudo-effective, that is numerically in the closure of the cone of effective divisors; this is Conjecture 2.2 of Birkar's paper and remains a second bottleneck.

After 2009 the frontier moved along three axes. Junyan Cao (2018) settled the log version of the conjecture for all fibrations over bases of dimension at most two, with no restriction on the fibre (the log version allows an auxiliary boundary divisor on $X$), superseding Birkar's Theorem 1.4; combined with Kawamata's criterion this makes Birkar's $n\le 6$ theorem a formal consequence of later work. Cao–Păun (2017), building on Chen–Hacon (2011) and extended by Hacon–Popa–Schnell (2017), settled bases of maximal Albanese dimension — the Albanese variety is the abelian variety dual to $H^0(X,\Omega_X^1)$, and maximal Albanese dimension means that the Albanese map is generically finite onto its image — and Benammar Ammar (2025) extended this to bases of Albanese dimension at least $m-2$, while Popa–Schnell (2022) settled smooth morphisms whose fibres are canonically polarised (in particular of general type). Chang (2025) proved the generalized nonvanishing conjecture for threefolds of positive Kodaira dimension or positive irregularity $q=h^1(X,\mathcal O_X)$ and deduced $C_{n,m}$ for $n\le 7$ except in one degenerate configuration: $m=3$, $\kappa(F)=\kappa(Y)=0$, with all irregularities $q(X)=q(F)=q(Y)=\tilde q(Y)=0$, where $\tilde q$ is the augmented irregularity (the supremum of the irregularities over all finite étale covers). Threefold bases remain under active attack — Benammar Ammar (2026) proved some cases of $C_{n,3}$, including certain Calabi–Yau threefold bases, where a Calabi–Yau threefold is a smooth projective threefold with $\kappa=0$, $q=0$ and trivial canonical class — and the conjecture has sharp known boundaries: it fails for compact complex manifolds that are not Kähler (Magnússon 2012) and for anticanonical analogues in characteristics two and three (Benozzo 2026), while in positive characteristic it is known only in restricted settings such as threefolds over finite fields (Birkar–Chen–Zhang 2018). Fujino's monograph (2020) surveys the conjecture and its partial results.

Problem Statement

Prove or disprove the Iitaka conjecture $C_{n,m}$ for algebraic fibre spaces in its remaining open range. Precisely: let $f\colon X\to Y$ be a surjective morphism with connected fibres between smooth projective varieties of dimensions $n$ and $m$ over an algebraically closed field of characteristic zero, and let $F$ be a general fibre. Prove, or refute by an explicit example, the inequality

$$\kappa(X)\ge\kappa(F)+\kappa(Y).$$

The conjecture is already known when $m\le 2$; when the general fibre has a good minimal model (in particular when $\dim F\le 3$ or $F$ is of general type); when $Y$ is of general type; when $Y$ has Albanese dimension at least $m-2$; and when $n\le 7$, except for the configuration $n=7$, $m=3$, $\kappa(F)=\kappa(Y)=0$, $q(X)=q(F)=q(Y)=\tilde q(Y)=0$. The conjecture is open precisely outside these known cases: fibrations over a Calabi–Yau threefold base with vanishing irregularity whose general fibre has $\kappa(F)=0$ (open already for $n=7$), and higher-dimensional cases with fibre dimension at least four and $0\le\kappa(F)<\dim F$ over bases of dimension at least three with small Albanese dimension. The stronger Iitaka–Viehweg form $C^+_{n,m}$ implies $C_{n,m}$; a counterexample to the inequality necessarily has $\kappa(F)\ge 0$ and $\kappa(Y)\ge 0$. The linked Campana–Peternell conjecture and the abundance conjecture in dimension at least four are recognised bottlenecks.

References

1. 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
2. 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
3. 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
4. János Kollár, 'Subadditivity of the Kodaira Dimension: Fibers of General Type', Advanced Studies in Pure Mathematics 10 (1987), 361–398. DOI: 10.2969/aspm/01010361, https://doi.org/10.2969/aspm/01010361
5. Shigefumi Mori, 'Classification of higher-dimensional varieties', Proceedings of Symposia in Pure Mathematics 46, Part 1 (1987), 269–331. DOI: 10.1090/pspum/046.1/927961, https://doi.org/10.1090/pspum/046.1/927961
6. Yujiro Kawamata, 'Abundance theorem for minimal threefolds', Inventiones Mathematicae 108 (1992), 229–246. DOI: 10.1007/BF02100604, https://doi.org/10.1007/BF02100604
7. Osamu Fujino and Shigefumi Mori, 'A Canonical Bundle Formula', Journal of Differential Geometry 56 (2000), 167–188. DOI: 10.4310/jdg/1090347529, https://doi.org/10.4310/jdg/1090347529
8. Caucher Birkar, 'The Iitaka conjecture $C_{n,m}$ in dimension six', Compositio Mathematica 145 (2009), 1442–1446. DOI: 10.1112/s0010437x09004187; arXiv:0806.4413, https://doi.org/10.1112/s0010437x09004187
9. Caucher Birkar, Paolo Cascini, Christopher D. Hacon and James McKernan, 'Existence of minimal models for varieties of log general type', Journal of the American Mathematical Society 23 (2010), 405–468. DOI: 10.1090/S0894-0347-09-00649-3, https://doi.org/10.1090/S0894-0347-09-00649-3
10. Christopher D. Hacon and James McKernan, 'Existence of minimal models for varieties of log general type II', Journal of the American Mathematical Society 23 (2010), 469–490. DOI: 10.1090/S0894-0347-09-00651-1, https://doi.org/10.1090/S0894-0347-09-00651-1
11. Caucher Birkar, 'On existence of log minimal models', Compositio Mathematica 146 (2010), 919–928. DOI: 10.1112/S0010437X09004564, https://doi.org/10.1112/S0010437X09004564
12. Jungkai Alfred Chen and Christopher D. Hacon, 'Kodaira dimension of irregular varieties', Inventiones Mathematicae 186 (2011), 481–500. DOI: 10.1007/s00222-011-0323-x, https://doi.org/10.1007/s00222-011-0323-x
13. Frédéric Campana and Thomas Peternell, 'Geometric stability of the cotangent bundle and the universal cover of a projective manifold', Bulletin de la Société Mathématique de France 139 (2011), 41–74. DOI: 10.24033/bsmf.2599, https://doi.org/10.24033/bsmf.2599
14. Gunnar Þór Magnússon, 'Automorphisms and examples of compact non Kähler manifolds', arXiv:1204.3165 (2012), https://arxiv.org/abs/1204.3165
15. Christopher Hacon, Mihnea Popa and Christian Schnell, 'Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Paun', arXiv:1611.08768 (2016; revised v2, 2017), https://arxiv.org/abs/1611.08768
16. Sándor J. Kovács and Zsolt Patakfalvi, 'Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension', Journal of the American Mathematical Society 30 (2017), 959–1021. DOI: 10.1090/jams/871, https://doi.org/10.1090/jams/871
17. 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://doi.org/10.1007/s00222-016-0672-6
18. Junyan Cao, 'Kodaira dimension of algebraic fiber spaces over surfaces', Algebraic Geometry 5 (2018), 728–741. DOI: 10.14231/AG-2018-020; arXiv:1511.07048, https://doi.org/10.14231/AG-2018-020
19. Caucher Birkar, Yifei Chen and Lei Zhang, 'Iitaka conjecture for 3-folds over finite fields', Nagoya Mathematical Journal 229 (2018), 21–51. DOI: 10.1017/nmj.2016.61, https://doi.org/10.1017/nmj.2016.61
20. Osamu Fujino, 'Iitaka Conjecture: An Introduction', SpringerBriefs in Mathematics, Springer, Singapore (2020). DOI: 10.1007/978-981-15-3347-1, https://doi.org/10.1007/978-981-15-3347-1
21. Vladimir Lazić and Thomas Peternell, 'On Generalised Abundance, I', Publications of the Research Institute for Mathematical Sciences 56 (2020), 353–389. DOI: 10.4171/prims/56-2-3, https://doi.org/10.4171/prims/56-2-3
22. Kenta Hashizume, 'Log Iitaka conjecture for abundant log canonical fibrations', Proceedings of the Japan Academy, Series A 96 (2020). DOI: 10.3792/pjaa.96.017; arXiv:1902.10923, https://doi.org/10.3792/pjaa.96.017
23. Juanyong Wang, 'On the Iitaka Conjecture $C_{n,m}$ for Kähler Fibre Spaces', Annales de la Faculté des Sciences de Toulouse, Série 6, 30 (2021), 813–897. DOI: 10.5802/afst.1690; arXiv:1907.06705, https://doi.org/10.5802/afst.1690
24. Sung Rak Choi and Jinhyung Park, 'On subadditivity of Okounkov bodies for algebraic fiber spaces', arXiv:2111.01386 (2021), https://arxiv.org/abs/2111.01386
25. Mihnea Popa and Christian Schnell, 'On the behavior of Kodaira dimension under smooth morphisms', arXiv:2202.02825 (2022), https://arxiv.org/abs/2202.02825
26. Hyunsuk Kim, 'Canonical bundle formula and a conjecture on certain algebraic fiber spaces by Schnell', arXiv:2412.19769 (2024), https://arxiv.org/abs/2412.19769
27. Christian Schnell, 'Singular metrics and a conjecture by Campana and Peternell', Pure and Applied Mathematics Quarterly 21 (2025), 1269–1281. DOI: 10.4310/pamq.250116023222; arXiv:2202.01295, https://doi.org/10.4310/pamq.250116023222
28. Chi-Kang Chang, 'Generalized Nonvanishing Conjecture and Iitaka Conjecture', Mathematische Zeitschrift 310, Article 30 (2025). DOI: 10.1007/s00209-025-03727-5; arXiv:2312.04015, https://doi.org/10.1007/s00209-025-03727-5
29. Houari Benammar Ammar, 'Note on Iitaka Conjecture $C_{n,m}$', arXiv:2510.06412 (2025), https://arxiv.org/abs/2510.06412
30. Marta Benozzo, 'On the Iitaka conjecture for anticanonical divisors in positive characteristic', Michigan Mathematical Journal 76 (2026), 3–23. DOI: 10.1307/mmj/20226272; arXiv:2204.02045, https://doi.org/10.1307/mmj/20226272
31. Houari Benammar Ammar, 'Kodaira dimension of algebraic fiber spaces over threefolds: Part 1', arXiv:2605.08598 (2026), https://arxiv.org/abs/2605.08598
32. Zhiyuan Jiang, 'An approach to the abundance conjecture for Kähler varieties via algebraic reduction', arXiv:2604.03879 (2026), https://arxiv.org/abs/2604.03879