The Iitaka conjecture $C_{n,m}$ for algebraic fibre spaces beyond dimension six

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 $\kappa(X)\ge\kappa(F)+\kappa(Y)$. The source paper of this candidate (Birkar, published 2009) proved the conjecture for total dimension $n\le 6$; the hard new case was a fourfold fibre with $\kappa(F)=0$ over a surface base, where abundance for the fibre is unknown, and Birkar circumvented the gap by a direct argument combining the canonical bundle formula, the nef reduction map, and a semipositivity theorem. The problem is the open extension: prove $C_{n,m}$ for $n\ge 7$, toward the conjecture in arbitrary dimension. The standard route, Kawamata's reduction to good minimal models of the fibre, is blocked by the two open problems the source itself acknowledges: minimal model existence for varieties of nonnegative Kodaira dimension in dimension $\ge 6$ (known up to dimension 5) and abundance in dimension $\ge 4$ (known up to dimension 3). Later work settles large special classes — bases of dimension at most 2 (Kawamata; Cao), bases of maximal Albanese dimension (Cao–Păun; Hacon–Popa–Schnell, which also resolves part (1) of Ueno's conjecture K in all dimensions) — and Chang (2025) reaches $n=7$ except one configuration with $\kappa(F)=\kappa(Y)=0$ and vanishing irregularities. The general conjecture, the two minimal-model-program ingredients, and part (2) of Ueno's conjecture K all remain open.

Background

The canonical divisor $K_X$ of a smooth complex projective variety $X$ is the divisor class of a holomorphic volume form. For each positive integer $\ell$, the pluricanonical system $|\ell K_X|$ is the linear system of holomorphic $\ell$-tuple canonical forms, and the Kodaira dimension $\kappa(X)$ is the maximum, over all sufficiently divisible $\ell$, of the dimension of the image of the rational map defined by $|\ell K_X|$; by convention $\kappa(X)=-\infty$ when all pluricanonical systems are empty. Thus $\kappa(X)\in\{-\infty,0,1,\dots,\dim X\}$ is a birational invariant, and the classes $\kappa=-\infty$, $\kappa=0$, and $\kappa=\dim X$ (general type) are the coarse landmarks of the birational classification of algebraic varieties; a modern introduction to the circle of ideas below is Fujino's SpringerBriefs volume 'Iitaka Conjecture: An Introduction'.

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 $F$ denote a general fibre. The Iitaka conjecture $C_{n,m}$, formulated in the 1970s (it appears in Ueno's classification-theory notes) and open ever since, asserts the subadditivity of Kodaira dimension under fibrations:

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

where the right-hand side is read as $-\infty$ if either summand is $-\infty$. Viehweg strengthened it to $C^{+}_{n,m}$, replacing $\kappa(Y)$ by $\max\{\kappa(Y),\operatorname{var}(f)\}$ with $\operatorname{var}(f)$ the variation of the isomorphism types of the fibres, and proved the strengthened version when the base is of general type via his weak positivity theorem for direct images of pluricanonical sheaves.

The structural key to $C_{n,m}$ is a theorem of Kawamata (1985): the conjecture holds whenever the general fibre $F$ has a good minimal model, meaning a birational model $F'$ with terminal singularities (the mildest singularity class on which the minimal model program operates) whose canonical divisor $K_{F'}$ is nef (non-negative on every curve) and semiample (some positive multiple is base-point-free). This reduces the conjecture to the two central conjectures of the minimal model program (MMP): the minimal model conjecture, that every smooth projective variety of nonnegative Kodaira dimension has a minimal model (terminal singularities, nef canonical divisor), and the abundance conjecture, that every minimal model is good. Lazić and Peternell later reformulated the abundance input as the generalized nonvanishing conjecture — for a projective pair with a nef canonical class (allowing mild singularities), the Kodaira dimension of the canonical class is at least its numerical dimension, the largest power of that class which is not numerically trivial — together with a companion generalized abundance conjecture; both are verified so far only in dimension at most 2, in dimension 3 when the numerical dimension or the irregularity is positive, and in special nef dimensions. At the time of the source paper, minimal models for varieties of nonnegative Kodaira dimension were known up to dimension 5 (Birkar, by an induction showing that the log minimal model program in dimension $d-1$ yields minimal models for effective pairs in dimension $d$), and abundance up to dimension 3 (Kawamata 1992 and related threefold work); minimal models for varieties of general type exist in all dimensions (Birkar–Cascini–Hacon–McKernan). Consequently $C_{n,m}$ was known when the base is a curve (Kawamata 1982), when the fibre is of general type (Kollár 1987), and whenever $\dim F\le 3$; bases of dimension $m\ge 2$ with small Kodaira dimension were out of reach.

The source paper of this candidate, Birkar's six-page note published as Compositio Mathematica 145 (2009), proved that $C_{n,m}$ holds for $n\le 6$ (Theorem 1.3), and that $C_{n,m}$ holds when $m=2$ and $\kappa(F)=0$ with no bound on $n$ (Theorem 1.4). If $m=1$, the result follows from Kawamata's theorem for curve bases; if $n-m\le 3$, the fibre has dimension at most 3 and hence a good minimal model, so Kawamata's theorem applies; together these cover every case with $n\le 5$, as well as $n=6$ with $m\ne 2$. The genuinely new case was $n=6$, $m=2$, $\dim F=4$, $\kappa(F)=0$: minimal models of fourfolds exist, but abundance for a $\kappa=0$ fourfold is unknown, so Kawamata's reduction does not apply. Birkar circumvented the missing abundance by combining the Fujino–Mori canonical bundle formula (which, for a fibration with $\kappa(F)=0$ fibres, expresses after birational modification a multiple of $K_{X'}$ as the pullback of $K_{Z'}+B+L$ with $B$ effective and $L$ nef downstairs), the nef reduction map associated with a nef divisor (a fibration, up to birational modification, whose fibres span exactly the curve directions on which the divisor has degree zero; the dimension of its base is called the nef dimension of the divisor), and a semipositivity theorem of Campana–Peternell.

The paper also relates $C_{n,m}$ to Ueno's conjecture K on varieties of Kodaira dimension zero. The Albanese map $\alpha\colon X\to A$ of a smooth projective variety is the universal morphism to an abelian variety (a projective algebraic group) $A=\operatorname{Alb}(X)$; the dimension of $A$ equals the irregularity $q(X)=h^1(X,\mathcal O_X)$ of $X$, and a variety is said to have maximal Albanese dimension when its Albanese image has dimension equal to its own dimension, that is, when the Albanese map is generically finite. For $\kappa(X)=0$, Kawamata proved that $\alpha$ is surjective with connected fibres (as recorded in the source paper and in Chang's 2025 formulation), and the conjecture predicts: (1) the general fibre $F$ of $\alpha$ satisfies $\kappa(F)=0$; and (2) after a finite étale cover $A'\to A$ (an unramified covering of abelian varieties), the fibre product $X\times_A A'$ is birational to the product $F\times A'$ over $A$. Birkar's Theorem 1.3 yields part (1) in dimension at most 6; Chen and Hacon proved the partial bound $\kappa(F)\le q(X)$ in general.

The problem considered here is the direct open extension of this paper: establish $C_{n,m}$ in total dimension $n\ge 7$, toward the conjecture in arbitrary dimension. The open problems the source acknowledges — minimal models for varieties of nonnegative Kodaira dimension in dimension $\ge 6$, abundance in dimension $\ge 4$ (in particular for $\kappa=0$ fibres of dimension at least 4), and part (2) of Ueno's conjecture K — are precisely the obstructions on the standard route. Two methodological directions are recorded in the literature: the fibre-side direction, through Kawamata's reduction, which yields $C_{n,m}$ for $n-m\le d$ once minimal model existence and abundance are available up to fibre dimension $d$, and the direct direction, which avoids good minimal models of the fibre and argues from the fibration itself, as do the partial results over curve and surface bases, over bases of maximal Albanese dimension, and up to total dimension seven. Characteristic zero is essential to the statement: over algebraically closed fields of positive characteristic, subadditivity of Kodaira dimension is false in general (Cascini–Ejiri–Kollár–Zhang constructed violating fibrations), so any proof must use tools without a positive-characteristic analogue, such as generic vanishing (Hodge-theoretic vanishing statements for the cohomology of sheaves twisted by a generic topologically trivial line bundle, the engine behind the results over abelian-variety bases) or the characteristic-zero MMP.

Problem Statement

Let $k$ be an algebraically closed field of characteristic zero (for instance $k=\mathbb{C}$). Let $f\colon X\to Y$ be an algebraic fibre space between smooth projective varieties over $k$, with $\dim X=n\ge 7$, $\dim Y=m$, and general fibre $F$. Prove the Iitaka conjecture in this range:

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

where the right-hand side is read as $-\infty$ when either summand is $-\infty$. Equivalently: extend Birkar's theorem that $C_{n,m}$ holds for $n\le 6$ to all higher total dimensions, with the eventual target being $C_{n,m}$ in full generality (all $n$ and $m$).

The problem covers in particular the sharpest currently open configuration, Chang's exceptional case: $n=7$, $m=3$ with $\kappa(F)=\kappa(Y)=0$ and $q(X)=q(F)=q(Y)=\tilde q(Y)=0$, where $\tilde q(Y)$ denotes the augmented irregularity of $Y$, that is, the maximum of the irregularities $q(Y')=h^1(Y',\mathcal O_{Y'})$ over all quasi-étale covers $Y'\to Y$ (finite covers unramified in codimension one).

The statement must hold for all algebraic fibre spaces in the stated dimensions: restrictions on the base, the fibre, the Kodaira dimensions, or the irregularities would answer a narrower derived problem, not this one. The problem is one of characteristic zero only. Progress on the attached but logically independent part (2) of Ueno's conjecture K — the birational product structure after an étale base change of the Albanese variety of a $\kappa=0$ variety — may accompany an answer but does not substitute for the inequality above.

References

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