The anticanonical Iitaka inequality $C^{-}_{n,m}$ in positive characteristic without $F$-splitting hypotheses

For a fibration $f:X\to Y$ with normal general fibre $X_y$, the anticanonical Iitaka inequality $C^{-}_{n,m}$ predicts $\kappa(X,-K_X)\le\kappa(X_y,-K_{X_y})+\kappa(Y,-K_Y)$. Chang proved it in characteristic $0$ for sources with at worst klt (kawamata log terminal) singularities, with $-K_X$ effective and stable base locus not dominating the base, and Benozzo–Brivio–Chang (Advances in Mathematics, 2026) extended it to fields of any characteristic under a good-singularity condition on the anticanonical $\mathbb Q$-linear series $|-K_X|_{\mathbb Q}$ along the fibre — in positive characteristic, a global $F$-splitting condition $Q^0_y(X,0)=k(y)$ defined via Frobenius trace maps, plus a base-locus condition with multiplicators prime to $p$. The open problem is whether this positive-characteristic good-singularity hypothesis is removable: over an algebraically closed field of characteristic $p>0$, does the inequality hold for every fibration with normal general fibre and base-point-free restricted anticanonical series (for some multiple prime to $p$), assuming no $F$-splitting or strong $F$-regularity hypothesis on the fibre? The suggested unrestricted version (arbitrary fibrations, no hypotheses) is refuted by the primary sources themselves — ruled-surface counterexamples exist in every characteristic, and Benozzo's Tango–Raynaud examples refute it in characteristics $2$ and $3$ — so the problem below is the surviving open core, which is explicitly posed in the source paper and is open already for smooth fourfold sources over surfaces.

Background

Let $X$ be a normal projective variety over an algebraically closed field. The Iitaka dimension $\kappa(X,L)$ of a $\mathbb Q$-Cartier divisor $L$ (a divisor some positive multiple of which is Cartier, that is, the divisor of a line bundle) measures the rate of growth of the plurisection spaces $H^0(X,mL)$ as a function of $m$: it is the dimension of the image of the rational map defined by $|mL|$ for $m$ sufficiently large and divisible, with the convention $\kappa(X,L)=-\infty$ when no positive multiple of $L$ is effective. The Kodaira dimension of $X$ is $\kappa(X)=\kappa(X,K_X)$, where $K_X$ is the canonical divisor (the divisor of a top-degree rational differential form); it is the basic measure of positivity in the Minimal Model Program (the classification programme for projective varieties up to birational equivalence). A fibration is a surjective projective morphism $f:X\to Y$ between normal varieties with $f_*\mathcal O_X=\mathcal O_Y$ (equivalently, connected fibres), and $X_y=f^{-1}(y)$ denotes the fibre over a general point $y\in Y$. The classical Iitaka conjecture $C_{n,m}$ predicts that a fibration of smooth projective varieties in characteristic $0$ satisfies $\kappa(X)\ge\kappa(X_y)+\kappa(Y)$; it is proved in many central cases and would follow from the Minimal Model Program together with its companion abundance conjecture. In positive characteristic the conjecture is false in general: Cascini, Ejiri, Kollár and Zhang gave counterexamples built from Tango–Raynaud surfaces, which are smooth surfaces in small positive characteristic on which the classical Kodaira vanishing theorem fails.

This problem concerns the anticanonical analogue, in which $K$ is replaced by $-K$ throughout and the inequality sign is reversed. Denote by $C^{-}_{n,m}$ the statement that a fibration $f:X\to Y$ satisfies

$$\kappa(X,-K_X)\le\kappa(X_y,-K_{X_y})+\kappa(Y,-K_Y).$$

Chang proved $C^{-}_{n,m}$ in characteristic $0$ under mild hypotheses: $X$ and $Y$ normal projective and $\mathbb Q$-Gorenstein (that is, some positive multiple of each of $K_X$ and $K_Y$ is Cartier), $X$ with at worst klt singularities (kawamata log terminal, the mild singularity class central to the Minimal Model Program), $-K_X$ effective, and the stable base locus $\mathbf B(-K_X)=\bigcap_{m\ge0}\mathrm{Bs}(-mK_X)$ — where $\mathrm{Bs}(|L|)$ denotes the base locus of a linear series, the closed subset on which all its sections vanish — not dominating $Y$. The base-locus hypothesis cannot be dropped in any characteristic: if $Y$ is a curve of genus $g\ge2$ and $D$ a divisor with $\deg D>2g-2$, the ruled surface (a $\mathbb P^1$-bundle over a curve) $X=\mathbb P(\mathcal O_Y\oplus\mathcal O_Y(-K_Y-D))\to Y$ has $-K_X$ big, so $\kappa(X,-K_X)=2$, while $\kappa(Y,-K_Y)=-\infty$, and the inequality fails even though $f$ is a smooth morphism.

Benozzo, Brivio and Chang extended Chang's theorem to fields of any characteristic, at the cost of a good-singularity condition on the anticanonical $\mathbb Q$-linear series $|-K_X|_{\mathbb Q}$ (the set of effective $\mathbb Q$-divisors $\mathbb Q$-linearly equivalent to $-K_X$) along the general fibre. In characteristic $0$ the condition is the triviality of the asymptotic multiplier ideal $\mathcal I(X_y,0;\|-K_X\|_y)$: the multiplier ideal, which measures the singularities of a general member of the series, must be trivial on the fibre, roughly meaning that some member of $|-K_X|_{\mathbb Q}$ restricts to a divisor with at worst log canonical singularities (the weakest of the standard singularity classes) on $X_y$. In characteristic $p>0$ the same role is played by singularities detected through the Frobenius morphism (the morphism that raises regular functions to their $p$-th powers): a variety is (globally) $F$-split if the Frobenius pullback of its structure sheaf splits back onto it, and strongly $F$-regular if it admits no nonzero Frobenius-pure ideals, the Frobenius analogue of klt singularities. The hypothesis of Benozzo–Brivio–Chang in characteristic $p$ is the condition $Q^0_y(X,0)=k(y)$, a global $F$-splitting-type condition defined through images of Frobenius trace maps on the fibre, together with a base-locus condition: some multiple $-mK_X$ with $p\nmid m$ is Cartier and the restricted linear series $|-mK_X|_y$ (the restriction of the linear series $|-mK_X|$ to the general fibre) is base-point-free on the fibre. The characteristic-$0$ proof runs through the canonical bundle formula for lc-trivial fibrations (fibrations whose canonical divisor plus a suitable boundary is $\mathbb Q$-linearly trivial over the base), which expresses $K_X$ plus the boundary as a pullback from the base; in positive characteristic that formula is false in general, and the known replacement, due to Das and Schwede, is available only under $F$-splitting hypotheses — exactly the hypothesis this problem asks to remove. Without any good-singularity hypothesis the inequality fails in every characteristic (the ruled surfaces above), and in characteristics $2$ and $3$ Benozzo constructed counterexamples to $C^{-}_{7,6}$, again from Tango–Raynaud surfaces, with non-normal general fibres. What is known in positive characteristic without global $F$-splitting: Benozzo proved $C^{-}_{n,1}$, the case of a smooth curve base, for all dimensions $n$ and all $p$, and $C^{-}_{3,m}$, the case of a threefold source, for $p\ge5$ (for smaller $p$ the anticanonical Iitaka fibration of a surface base — the fibration defined by sufficiently large multiples of $-K_Y$ — can be quasi-elliptic, that is, have general fibre a singular genus-one curve), in each case assuming the general fibre regular (all of its local rings regular) and the pair it inherits from $X$ strongly $F$-regular. Benozzo, Brivio and Chang themselves open their final section by asking how restrictive their $F$-splitting condition is, and connect it to open conjectures on the density of $F$-split fibres in families spread out from characteristic $0$. The gap between the characteristic-$0$ theorem and these positive-characteristic results is the subject of the problem below.

Problem Statement

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $f:X\to Y$ be a fibration between normal projective $k$-varieties whose general fibre $X_y$ is normal. Write $\mathbb Z_{(p)}$ for the ring of rational numbers whose denominator is prime to $p$, and assume $Y$ is $\mathbb Z_{(p)}$-Gorenstein (some positive multiple of $K_Y$ with multiplier prime to $p$ is Cartier) and $-K_X$ is $\mathbb Z_{(p)}$-Cartier. Assume the base-locus condition of Benozzo–Brivio–Chang's Theorem 7.2: there exists $m\ge1$ with $p\nmid m$ such that $-mK_X$ is Cartier and the restricted linear series $|-mK_X|_y$ (the restriction of the linear series $|-mK_X|$ to the general fibre) is base-point-free. Does the anticanonical Iitaka inequality

$$\kappa(X,-K_X)\;\le\;\kappa(X_y,-K_{X_y})+\kappa(Y,-K_Y)$$

hold with no $F$-singularity hypothesis on the fibre — that is, is the global $F$-splitting condition $Q^0_y(X,0)=k(y)$ (condition (a) of Theorem 7.2) redundant, so that the characteristic-$0$ phenomenon extends to positive characteristic once generic smoothness (the automatic smoothness of the general fibre of a fibration in characteristic $0$) is replaced by the assumption that the general fibre is normal?

The inequality is open in particular for sources of dimension $\ge4$ over bases of dimension $\ge2$ in any characteristic, for threefold sources over surfaces in characteristics $2$ and $3$, and for fibrations whose general fibre is normal but not smooth. A counterexample would be an explicit fibration satisfying all the hypotheses in which the inequality strictly fails; the known counterexamples — ruled surfaces over curves of genus $\ge2$ with $-K_X$ big, and the Tango–Raynaud constructions in characteristics $2$ and $3$ — violate the base-locus condition or the normality of the general fibre and therefore do not settle the question. The same question for pairs $(X,B)$, with $-K_X-B$ and the induced fibre pair in place of $-K_X$, is the general form of Theorem 7.2 and is equally open; an answer for the boundary-free case stated here is the core of the problem.

References

1. Marta Benozzo, Iacopo Brivio, Chi-Kang Chang, On the superadditivity of anticanonical Iitaka dimension, Advances in Mathematics 491 (2026), article 110860. DOI: 10.1016/j.aim.2026.110860; arXiv:2309.16580. https://doi.org/10.1016/j.aim.2026.110860
2. Marta Benozzo, On the Iitaka conjecture for anticanonical divisors in positive characteristic, Michigan Mathematical Journal 76 (2026), no. 1, 3-23. DOI: 10.1307/mmj/20226272; arXiv:2204.02045. https://doi.org/10.1307/mmj/20226272
3. Chi-Kang Chang, Positivity of anticanonical divisors in algebraic fibre spaces, Mathematische Annalen 385 (2023), no. 1-2, 787-809. DOI: 10.1007/s00208-021-02282-8; arXiv:2011.11561. https://doi.org/10.1007/s00208-021-02282-8
4. Paolo Cascini, Sho Ejiri, János Kollár, 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
5. Omprokash Das, Karl Schwede, The F-different and a canonical bundle formula, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5) 17 (2017), no. 3, 1173-1205. DOI: 10.2422/2036-2145.201510_012; arXiv:1508.07295. https://doi.org/10.2422/2036-2145.201510_012
6. Jefferson Baudin, On the Albanese morphism of varieties with Frobenius stable Kodaira dimension zero, preprint (2024). arXiv:2401.06074. https://arxiv.org/abs/2401.06074