Iitaka's $C_{n,m}$ conjecture in positive characteristic: the remaining regimes (higher dimensions, small characteristics, and wild geometric generic fibres)

The Iitaka conjecture $C_{n,m}$ predicts that for a fibration $f\colon X\to Z$ between smooth projective varieties over an algebraically closed field, with geometric generic fibre $X_{\bar\eta}$, the Kodaira dimensions satisfy $\kappa(X)\ge\kappa(Z)+\kappa(X_{\bar\eta})$. Birkar-Chen-Zhang (Nagoya Math. J. 229 (2018), 21-51) proved $C_{3,m}$ over the algebraic closure of a finite field in characteristic $p>5$ and recorded the limitations of its proof: the restriction to $\overline{\mathbb{F}}_p$ (used to make numerically trivial divisors torsion and to obtain semi-ampleness of nef and big divisors), the hypothesis $p>5$ (the threefold minimal model program), the exceptional case of restricted Kodaira dimension one in its relative good minimal model theorem over general algebraically closed fields, and the extension to higher dimensions or wild fibres. The first and third limitations were subsequently removed by Ejiri-Zhang, who proved $C_{3,m}$ over every algebraically closed field of characteristic $p>5$ when the geometric generic fibre is smooth, while Cascini-Ejiri-Kollar-Zhang refuted the naive extension to wild fibres by constructing, over any algebraically closed field of positive characteristic, fibrations violating the strong form of subadditivity. The still-open problem is the positive-characteristic Iitaka conjecture in the remaining regimes: (i) dimension at least 4; (ii) dimension 3 in characteristics 2, 3 and 5; and (iii) the weak form for fibrations with singular or non-reduced geometric generic fibre.

Background

Kodaira dimension and fibrations. For a normal projective variety $X$ over a field and a Cartier divisor $D$, the Iitaka dimension (Kodaira dimension when $D=K_X$) is $\kappa(X,D)=-\infty$ if $|mD|=\emptyset$ for every $m>0$, and otherwise the maximal dimension of the images of the rational maps defined by the pluricanonical linear systems $|mD|$; $\kappa(X):=\kappa(X,K_X)$ is the basic birational invariant measuring the asymptotic space of pluricanonical forms, and it stratifies the birational classification of varieties. A fibration is a projective surjective morphism $f\colon X\to Z$ with $f_*\mathcal{O}_X\simeq\mathcal{O}_Z$. Its generic fibre $X_\eta$ is the fibre over the generic point of $Z$, a variety over the function field $k(Z)$, and the geometric generic fibre $X_{\bar\eta}$ is its base change to an algebraic closure of $k(Z)$. In characteristic zero, $X_{\bar\eta}$ is smooth whenever $X$ and $Z$ are; in characteristic $p>0$ the function-field extension need not be separable, and $X_{\bar\eta}$ can be singular or even non-reduced (a wild fibre) already for smooth $X$ and $Z$. For a possibly singular or non-reduced geometric generic fibre, $\kappa(X_{\bar\eta})$ is understood in the resolution-independent sense used in the literature cited here (Cascini-Ejiri-Kollar-Zhang, Definition 1.3): the minimum of $\kappa(Z,K_Z)$ over projective normal birational models $Z$ of the reduction of $X_{\bar\eta}$; this agrees with the classical definition whenever $X_{\bar\eta}$ has a smooth model.

The conjecture. Iitaka conjectured, for fibrations between smooth projective varieties of dimensions $n$ and $m$ over an algebraically closed field, the subadditivity

$$\kappa(X)\ \ge\ \kappa(Z)+\kappa(X_{\bar\eta}),$$

denoted $C_{n,m}$: the total space is required to be at least as general as base and fibre together. Over the complex numbers many cases are known; over algebraically closed fields of positive characteristic the conjecture, read with smooth geometric generic fibre, is settled only in restricted regimes: fibrations of relative dimension one in every characteristic (Chen-Zhang), bases of general type (bases of maximal Kodaira dimension) under Frobenius conditions (Patakfalvi; the relevant condition is that the Hasse-Witt matrix, i.e. the matrix of the Frobenius $p$-power map acting on the top differential forms of the generic fibre, is not nilpotent), and dimension three in characteristic $p>5$ (Birkar-Chen-Zhang). A further dimension-free case for fibrations over abelian varieties under Frobenius-stability conditions (Frobenius stability means that the pluricanonical section spaces of the generic fibre are generated under Frobenius pullback) was announced by Baudin but is not currently established: the revised preprint carries the author's note that a lemma error invalidates the whole proof.

The source paper and its limitations. Birkar-Chen-Zhang proved $C_{3,m}$ for $k=\overline{\mathbb{F}}_p$ and $p>5$. Two special properties of $\overline{\mathbb{F}}_p$ were essential. First, a numerically trivial divisor (one with zero intersection numbers with all curves, i.e. trivial only up to numerical equivalence) on a projective variety over $\overline{\mathbb{F}}_p$ is torsion, so numerical and $\mathbb{Q}$-linear triviality essentially coincide there. Second, basepoint-freeness results of Keel for nef and big divisors in positive characteristic (nef: non-negative intersection with every curve; big: positive asymptotic self-intersection volume; semi-ample: a positive multiple is base-point-free and defines a morphism) apply without extra hypotheses over $\overline{\mathbb{F}}_p$. The hypothesis $p>5$ enters through the log minimal model program (MMP) for threefolds, the algorithm that, for a pair $(X,B)$ with mild (Kawamata-Log-Terminal, klt) singularities, replaces $X$ by a birational model on which $K_X+B$ is nef (a minimal model), ideally a good minimal model on which it is semi-ample; this was available only for $p>5$ (Hacon-Xu and later work), as were auxiliary base-point-freeness and $F$-regularity inputs (F-regularity is a singularity condition defined via Frobenius and tight-closure techniques). Their proof reduces $C_{3,1}$ to relative good minimal models over the base curve (their Theorem 1.6), whose proof, as the authors note, works over any algebraically closed field of characteristic $p>5$ except when the restricted Kodaira dimension on the generic fibre (the Iitaka dimension of the restriction of $K_X+B$ to the generic fibre) equals one, the subcase handled using semi-ampleness of nef and big divisors on surfaces over $\overline{\mathbb{F}}_p$.

After the source paper. Ejiri-Zhang removed the field restriction, proving $C_{3,m}$ over every algebraically closed field of characteristic $p>5$ when $X_{\bar\eta}$ is smooth; the elliptic-base case with $\kappa(X_{\bar\eta})=1$, precisely the exceptional subcase above, is circumvented using the classification of vector bundles on elliptic curves and finite flat base changes. On the other hand, Cascini-Ejiri-Kollar-Zhang showed that the strong form of subadditivity with $\kappa(X_\eta)$ in place of $\kappa(X_{\bar\eta})$ (the generic fibre over the non-closed field $k(Z)$ can carry more pluricanonical sections than its geometric counterpart) fails over any algebraically closed field of positive characteristic, via explicit fibrations with wild generic fibres. Consequently the open problem consists of the remaining regimes: dimension at least 4; dimension 3 in characteristics 2, 3, 5; and the weak form $\kappa(X)\ge\kappa(Z)+\kappa(X_{\bar\eta})$ for wild geometric generic fibres. For the small-characteristic regimes, the threefold MMP was subsequently established in characteristic 5 (Hacon-Witaszek) and the relative dlt (divisorially log terminal) MMP over $\mathbb{Q}$-factorial threefold bases in all characteristics, but Frobenius-purity of rational double points (the mildest surface singularities; purity can fail in characteristics 2, 3 and 5) and the base-point-freeness inputs remain available only for $p>5$, and in characteristics 2 and 3 even the MMP is open. For the wild-fibre regime, over $\overline{\mathbb{F}}_p$ the weak form for threefolds over curves holds unconditionally (the source paper's Corollary 1.3, which passes to a smooth model of the geometric generic fibre), but over general algebraically closed fields only conditional results are known (conditional on Frobenius-stable canonical rings of the generic fibre, i.e. the subrings of its pluricanonical section rings spanned by the images of pluricanonical sections under pullback by iterates of the Frobenius morphism; on relative semi-ampleness of $K_X$; or on a big generic canonical divisor).

Problem Statement

Determine the validity of Iitaka subadditivity for fibrations in positive characteristic outside the settled regime (dimension at most 3, characteristic $p>5$, smooth geometric generic fibre, algebraically closed base field). Let $k$ be an algebraically closed field of characteristic $p>0$ and let $f\colon X\to Z$ be a fibration between smooth projective varieties over $k$, with generic fibre $X_\eta$ and geometric generic fibre $X_{\bar\eta}$.

(1) Higher dimensions. Prove or disprove

$$\kappa(X)\ \ge\ \kappa(Z)+\kappa(X_{\bar\eta})$$

when $X_{\bar\eta}$ is smooth and $\dim X\ge 4$. The special case $C_{4,1}$ — fibrations from smooth projective fourfolds to smooth projective curves — is the first open instance: is $C_{4,1}$ true in every positive characteristic?

(2) Small characteristics. Prove or disprove the same inequality for $\dim X=3$ over algebraically closed fields of characteristic $p\in\{2,3,5\}$ with $X_{\bar\eta}$ smooth. The threefold minimal model program is known in characteristic 5 (Hacon-Witaszek), but the remaining inputs used in the settled characteristic-$p>5$ proof (base-point-freeness, Frobenius-purity of rational double points, semi-positivity, i.e. nefness of the direct images of the relative pluricanonical sheaves) are not available there, and in characteristics 2 and 3 even the program itself is open.

(3) Wild geometric generic fibres. Determine the boundary between validity and failure of the weak form

$$\kappa(X)\ \ge\ \kappa(Z)+\kappa(X_{\bar\eta})$$

when $X_{\bar\eta}$ is singular or non-reduced. Over $\overline{\mathbb{F}}_p$ the weak form holds unconditionally for threefolds over curves (the source paper's Corollary 1.3, which passes to a smooth model of the geometric generic fibre); outside that case no unconditional result is known. The strong form with $\kappa(X_\eta)$ is false over any algebraically closed field of positive characteristic (Cascini-Ejiri-Kollar-Zhang); the target here is the weak form.

The three components are independent open problems. The common objective is the map of dimension, characteristic and generic-fibre-hypothesis regimes under which Iitaka subadditivity holds and fails in positive characteristic. The base field is assumed algebraically closed; the extension to non-algebraically-closed base fields, raised in the source paper, is an additional open direction but not a component here. Known regimes: relative dimension one in all characteristics (Chen-Zhang); dimension 3 in characteristic $p>5$ with smooth geometric generic fibre (Birkar-Chen-Zhang over $\overline{\mathbb{F}}_p$; Ejiri-Zhang over all algebraically closed fields); and the dimension-free conditional cases with hypotheses on the base, such as a base of general type with non-nilpotent Hasse-Witt matrix on the generic fibre (Patakfalvi).

References

1. Caucher Birkar, Yifei Chen, Lei Zhang, 'Iitaka's $C_{n,m}$ 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 (arXiv:1507.08760, https://arxiv.org/abs/1507.08760)
2. Sho Ejiri, Lei Zhang, 'Iitaka's $C_{n,m}$ conjecture for 3-folds in positive characteristic', Mathematical Research Letters 25 (2018), no. 3, 783-802. DOI: 10.4310/mrl.2018.v25.n3.a3, https://doi.org/10.4310/mrl.2018.v25.n3.a3 (arXiv:1604.01856, https://arxiv.org/abs/1604.01856)
3. Paolo Cascini, Sho Ejiri, Janos Kollar, Lei Zhang, 'Subadditivity of Kodaira dimension does not hold in positive characteristic', Commentarii Mathematici Helvetici 96 (2021), no. 3, 465-481. DOI: 10.4171/cmh/517, https://doi.org/10.4171/cmh/517 (arXiv:2003.13206, https://arxiv.org/abs/2003.13206)
4. Yifei Chen, Lei Zhang, 'The subadditivity of the Kodaira dimension for fibrations of relative dimension one in positive characteristics', Mathematical Research Letters 22 (2015), no. 3, 675-696. DOI: 10.4310/mrl.2015.v22.n3.a3, https://doi.org/10.4310/mrl.2015.v22.n3.a3 (arXiv:1305.6024, https://arxiv.org/abs/1305.6024)
5. Zsolt Patakfalvi, 'On subadditivity of Kodaira dimension in positive characteristic over a general type base', Journal of Algebraic Geometry 27 (2018), 21-53. DOI: 10.1090/jag/688, https://doi.org/10.1090/jag/688 (arXiv:1308.5371, https://arxiv.org/abs/1308.5371)
6. Sho Ejiri, 'Weak positivity theorem and Frobenius stable canonical rings of geometric generic fibers', Journal of Algebraic Geometry 26 (2017), no. 4, 691-734. DOI: 10.1090/jag/698, https://doi.org/10.1090/jag/698 (arXiv:1508.00484, https://arxiv.org/abs/1508.00484)
7. Lei Zhang, 'Subadditivity of Kodaira dimensions for fibrations of three-folds in positive characteristics', Advances in Mathematics 354 (2019), 106741. DOI: 10.1016/j.aim.2019.106741, https://doi.org/10.1016/j.aim.2019.106741 (arXiv:1601.06907, https://arxiv.org/abs/1601.06907)
8. Lei Zhang, 'A note on Iitaka's conjecture $C_{3,1}$ in positive characteristics', Taiwanese Journal of Mathematics 21 (2017), no. 3. DOI: 10.11650/TJM/7931, https://doi.org/10.11650/TJM/7931 (arXiv:1609.09592, https://arxiv.org/abs/1609.09592)
9. Sho Ejiri, 'Direct images of pluricanonical bundles and Frobenius stable canonical rings of fibers', Algebraic Geometry 11 (2024), no. 1, 71-110. DOI: 10.14231/ag-2024-003, https://doi.org/10.14231/ag-2024-003 (arXiv:1909.07000, https://arxiv.org/abs/1909.07000)
10. Sho Ejiri, 'Numerical Kodaira dimension of algebraic fiber spaces in positive characteristic', preprint (2022, revised 2023). arXiv:2212.09245, https://arxiv.org/abs/2212.09245
11. Jefferson Baudin, 'On the Albanese morphism of varieties with Frobenius stable Kodaira dimension zero', preprint (2024; the 2025 revision carries the author's note that a lemma error invalidates the whole proof). arXiv:2401.06074, https://arxiv.org/abs/2401.06074
12. Christopher Hacon, Chenyang Xu, 'On the three dimensional minimal model program in positive characteristic', Journal of the American Mathematical Society 28 (2015), no. 3, 711-744. DOI: 10.1090/s0894-0347-2014-00809-2, https://doi.org/10.1090/s0894-0347-2014-00809-2
13. Christopher Hacon, Jakub Witaszek, 'The minimal model program for threefolds in characteristic 5', Duke Mathematical Journal 171 (2022), no. 11. DOI: 10.1215/00127094-2022-0024, https://doi.org/10.1215/00127094-2022-0024 (arXiv:1911.12895, https://arxiv.org/abs/1911.12895)
14. Christopher Hacon, Jakub Witaszek, 'On the relative minimal model program for threefolds in low characteristics', Peking Mathematical Journal 5 (2022), no. 2, 365-382. DOI: 10.1007/s42543-021-00037-7, https://doi.org/10.1007/s42543-021-00037-7 (arXiv:1909.12872, https://arxiv.org/abs/1909.12872)
15. Omprokash Das, Joe Waldron, 'On the log minimal model program for threefolds over imperfect fields of characteristic $p>5$', Journal of the London Mathematical Society 106 (2022), no. 4, 3895-3937. DOI: 10.1112/jlms.12677, https://doi.org/10.1112/jlms.12677 (arXiv:1911.04394, https://arxiv.org/abs/1911.04394)
16. Sean Keel, 'Basepoint freeness for nef and big line bundles in positive characteristic', Annals of Mathematics (2) 149 (1999), no. 1, 253-286. DOI: 10.2307/121025, https://doi.org/10.2307/121025
17. Marta Benozzo, 'On the Iitaka conjecture for anticanonical divisors in positive characteristic', Michigan Mathematical Journal 76 (2026), no. 1. DOI: 10.1307/mmj/20226272, https://doi.org/10.1307/mmj/20226272 (arXiv:2204.02045, https://arxiv.org/abs/2204.02045)
18. Tatsuro Kawakami, Hiromu Tanaka, 'Mumford vanishing for threefolds in positive characteristic', preprint (2026). arXiv:2604.13823, https://arxiv.org/abs/2604.13823