Anticanonical Iitaka inequality $C^-_{n,m}$ in positive characteristic without Frobenius-positivity hypotheses

The anticanonical Iitaka inequality $C^-_{n,m}$ predicts that for a fibration $f:X\to Z$ with general fibre $\Phi$ , the anticanonical Iitaka dimensions satisfy $\kappa(X,-(K_X+\Delta))\le\kappa(\Phi,-(K_\Phi+\Delta_\Phi))+\kappa(Z,-K_Z)$ , i.e. that anticanonical Iitaka dimension is superadditive in fibrations. The inequality is known in characteristic zero (Chang), and in positive characteristic in two regimes: for curve bases in all dimensions and threefolds in characteristic $p\ge5$ when the general-fibre pair is strongly F-regular (Benozzo), and in arbitrary dimension and arbitrary characteristic under an asymptotic test-ideal condition $Q^0_y$ on the anticanonical series restricted to a general fibre (Benozzo–Brivio–Chang, 2026). Whether it holds in positive characteristic under only normality of the general fibre and a non-dominating anticanonical stable base locus — that is, whether any Frobenius-positivity hypothesis is genuinely necessary — remains open; every known counterexample exploits either non-normal fibres or a dominating anticanonical base locus.

Background

The Iitaka dimension $\kappa(X,D)$ of a normal projective variety $X$ with respect to a $\mathbb{Q}$ -Cartier divisor $D$ measures the rate of growth of the spaces of plurisections $H^0(X,mD)$ as $m$ grows; for $D=K_X$ , the canonical divisor, it is the Kodaira dimension, the fundamental birational invariant by which the minimal model program classifies algebraic varieties up to birational equivalence. Iitaka, who introduced this invariant in the early 1970s, also formulated the conjecture $C_{n,m}$ (with $n=\dim X$ and $m=\dim Z$ ): for a fibration $f:X\to Z$ between smooth projective varieties over $\mathbb{C}$ (a fibration being a projective surjective morphism with connected fibres, $f_*\mathcal{O}_X=\mathcal{O}_Z$ ) with general fibre $\Phi$ , Kodaira dimension should be subadditive, $\kappa(X,K_X)\ge\kappa(\Phi,K_\Phi)+\kappa(Z,K_Z)$ .

Chang recently established an anticanonical counterpart in characteristic zero: for a fibration $f:X\to Z$ of normal projective $\mathbb{Q}$ -Gorenstein varieties (meaning the canonical divisor is $\mathbb{Q}$ -Cartier) with at worst klt singularities — the standard birational-geometric class of mild singularities — such that $-K_X$ is effective and the stable base locus of $-K_X$ , i.e. the intersection of the base loci of the linear systems $|-mK_X|$ over all sufficiently divisible $m$ , does not dominate $Z$ , one has

$\kappa(X,-K_X)\le\kappa(\Phi,-K_\Phi)+\kappa(Z,-K_Z).$

This statement, and its version for pairs $(X,\Delta)$ , is denoted $C^-_{n,m}$ : the anticanonical Iitaka dimension is superadditive in fibrations.

In positive characteristic the landscape changes. Generic smoothness of fibrations fails, Kodaira-type vanishing fails, and the canonical bundle formula for lc-trivial fibrations — the tool that transports positivity from $-K_X$ to $-K_Z$ in the characteristic-zero proof — is not known in full generality, although Witaszek established it in important special cases. Moreover, the classical conjecture $C_{n,m}$ itself is false in positive characteristic: Cascini–Ejiri–Kollár–Zhang constructed counterexamples from Tango–Raynaud surfaces, which are smooth projective surfaces in characteristic $p$ built from a curve carrying a non-split rank-two vector-bundle extension, exploiting the failure of Kodaira vanishing.

Benozzo proved the first positive-characteristic instances of $C^-_{n,m}$ . It holds for fibrations onto smooth curves in any dimension, and for threefolds in characteristic $p\ge5$ , under the hypotheses that the general fibre $\Phi$ is regular, that the pair $(\Phi,\Delta_\Phi)$ induced on the fibre is strongly F-regular (the Frobenius analogue of klt singularities, defined by the splitting of sufficiently high iterates of the Frobenius morphism), that $\Delta$ is an effective $\mathbb{Z}_{(p)}$ -divisor (a $\mathbb{Q}$ -divisor whose coefficients have denominators prime to $p$ ) with $K_X+\Delta$ $\mathbb{Q}$ -Cartier, and that $\mathrm{Bs}(-m(K_X+\Delta))$ does not dominate $Z$ for some $m$ prime to $p$ . She also gave counterexamples in characteristics 2 and 3 to $C^-_{7,6}$ , all with non-normal fibres, and recorded that the base-locus hypothesis is necessary in every characteristic (ruled-surface counterexamples already in dimension 2).

The most recent advance, by Benozzo–Brivio–Chang, proves $C^-_{n,m}$ in arbitrary characteristic and arbitrary dimension for fibrations of normal varieties with normal general fibre satisfying the condition $Q^0_y(X,B+f^*D)=k(y)$ — a Frobenius-splitting-type positivity condition on the restriction of the anticanonical $\mathbb{Q}$ -linear series to a general fibre, the positive-characteristic analogue of the triviality of the asymptotic multiplier ideal (the multiplier ideal being the characteristic-zero measure of singularities of a pair, its test ideal being the Frobenius counterpart) — together with a restricted base-point-freeness condition. Their proof replaces the canonical bundle formula with a version available for globally F-split generic fibres and controls the fibres of the anticanonical Iitaka fibration of the base (the rational map induced by the plurisections of $-K_Z$ ) via Frobenius base change and positive-characteristic foliation theory. A related open arithmetic statement, their Conjecture 8.2, predicts that complex fibrations whose anticanonical stable base locus does not dominate the base have reductions modulo a dense set of primes with $Q^0_{y_p}(X_p,\Delta_p)=k(y_p)$ .

Every known positive-characteristic theorem for $C^-_{n,m}$ therefore imposes a Frobenius-positivity hypothesis of one kind or another, while every known counterexample exploits either non-normal fibres or a dominating anticanonical stable base locus. What happens between these two regimes — normal general fibres and a well-behaved anticanonical base locus, but no Frobenius-positivity assumption — is not known.

Problem Statement

Let $k$ be an algebraically closed field of characteristic $p>0$ . Let $f:X\to Z$ be a fibration (a projective surjective morphism with connected fibres, $f_*\mathcal{O}_X=\mathcal{O}_Z$ ) between normal projective varieties whose general fibre $\Phi$ (the fibre over a general closed point of $Z$ ) is normal; let $\Delta$ be an effective $\mathbb{Z}_{(p)}$ -divisor on $X$ (a $\mathbb{Q}$ -divisor whose coefficients have denominators prime to $p$ ) such that $K_X+\Delta$ is $\mathbb{Q}$ -Cartier, with $Z$ $\mathbb{Q}$ -Gorenstein (i.e. $K_Z$ is $\mathbb{Q}$ -Cartier) and $\Delta_\Phi$ defined by $(K_X+\Delta)|_\Phi=K_\Phi+\Delta_\Phi$ ; and suppose that for some integer $m_0>0$ not divisible by $p$ the base locus $\mathrm{Bs}(-m_0(K_X+\Delta))$ does not dominate $Z$ , i.e. its image is contained in a proper closed subset of $Z$ . Decide whether

$\kappa(X,-(K_X+\Delta))\le\kappa(\Phi,-(K_\Phi+\Delta_\Phi))+\kappa(Z,-K_Z)$

holds for every fibration satisfying these hypotheses, in every dimension. An affirmative answer with no hypothesis of Frobenius-splitting type — neither strong F-regularity of the general-fibre pair, as in Benozzo's theorems, nor the asymptotic test-ideal condition $Q^0_y(X,B+f^*D)=k(y)$ , as in Benozzo–Brivio–Chang's Theorem 7.2 — would strictly strengthen every known positive-characteristic result and show that the Frobenius-positivity hypotheses in the literature are unnecessary. A negative answer must exhibit a counterexample with normal general fibre and non-dominating anticanonical stable base locus, a regime reached by no known construction, all of which exploit non-normal fibres or a dominating base locus. The problem is exactly the determination of whether the Frobenius-positivity hypotheses are necessary.

References

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