For an algebraic fiber space f:X→Yf:X\to Y of smooth complex projective varieties with general fiber FF, Iitaka's Cn,mC_{n,m} conjecture predicts the subadditivity of the Kodaira dimension: κ(X)≥κ(Y)+κ(F)\kappa(X)\ge\kappa(Y)+\kappa(F). The settled regimes — base or fiber of general type; fiber admitting a good minimal model; one-dimensional base; surface base; base of maximal Albanese dimension; total dimension at most 6; base Albanese dimension at least dim⁡Y−2\dim Y-2; and dimension 7 under nonvanishing hypotheses — leave a precise open core in characteristic zero: fibrations over bases of dimension at least 3 with Albanese dimension at most dim⁡Y−3\dim Y-3 that are not of general type, with total space of dimension at least 7 and general fibers not established to admit a good minimal model. The conjecture is known to fail in positive characteristic, which shows that no proof valid simultaneously in all characteristics can succeed in characteristic zero. The problem is open.

Background

The Kodaira dimension κ(X)\kappa(X) of a smooth complex projective variety XX is the fundamental birational invariant measuring the growth of pluricanonical forms: κ(X)\kappa(X) is the maximum dimension of the image of XX under the maps given by the pluricanonical linear systems ∣mKX∣|mK_X| for m=1,2,…m=1,2,\dots, with κ(X)=−∞\kappa(X)=-\infty if no pluricanonical form exists; a variety with κ(X)=dim⁡X\kappa(X)=\dim X is said to be of general type. An algebraic fiber space is a surjective morphism f:X→Yf:X\to Y of smooth complex projective varieties with connected fibers, and the fiber over a general point of YY is called the general fiber FF. Since the classification program of the 1970s, a central problem of higher-dimensional birational geometry has been how κ(X)\kappa(X) relates to κ(Y)\kappa(Y) and κ(F)\kappa(F): the elementary easy-addition inequality κ(X)≤κ(F)+dim⁡Y\kappa(X)\le\kappa(F)+\dim Y always holds, while the reverse subadditivity

κ(X)≥κ(Y)+κ(F)\kappa(X)\ge\kappa(Y)+\kappa(F)

(with the convention that any sum involving −∞-\infty equals −∞-\infty, so that the statement is vacuous unless κ(Y)≥0\kappa(Y)\ge 0 and κ(F)≥0\kappa(F)\ge 0) is Iitaka's conjecture Cn,mC_{n,m}, where n=dim⁡Xn=\dim X and m=dim⁡Ym=\dim Y. The inequality is a pillar of classification theory: it is the engine that assembles the birational geometry of XX from that of its base and fiber, and it underpins the inductive structure of the minimal model program — the study of birational models on which the canonical class behaves well; a good minimal model of FF is a birational model whose canonical class is semi-ample, meaning some positive multiple of it is base-point-free.

Several regimes were settled classically. Viehweg's weak positivity theorem (1983), a positivity property of direct image sheaves of pluricanonical bundles, gives the case where the base is of general type; Kollár (1987) settled fibers of general type; Kawamata (1985) proved the conjecture whenever the general fiber admits a good minimal model, a class that by later existence results includes for example fibers of maximal Albanese dimension; and Kawamata (1982) settled one-dimensional bases. Here the Albanese morphism aY:Y→Alb(Y)a_Y:Y\to\mathrm{Alb}(Y) is the universal morphism from YY to an abelian variety (a projective complex torus), where Alb(Y)\mathrm{Alb}(Y) is dual to H0(Y,ΩY1)H^0(Y,\Omega_Y^1) modulo torsion; the Albanese dimension α(Y)\alpha(Y) is the dimension of the image aY(Y)a_Y(Y), and YY has maximal Albanese dimension when α(Y)=dim⁡Y\alpha(Y)=\dim Y. By Kawamata's structure theorem (1981), a finite étale cover (a finite unramified cover) of a maximal-Albanese base of Kodaira dimension 00 is an abelian variety, and more generally such bases decompose into an abelian part and a general-type part; this reduces the maximal-Albanese case of Cn,mC_{n,m} to fibrations over abelian varieties together with the general-type base case. Chen–Hacon (2011) announced the maximal-Albanese case in this setting, but the proof is incomplete — Hacon–Popa–Schnell (2018) identify a serious mistake in §4 of that paper — and the case was established by Cao–Păun (2017), who proved the log version of the inequality over abelian bases through a study of the positivity of direct images of pluricanonical bundles carrying singular Hermitian metrics (Hermitian metrics on line bundles whose local potential functions are allowed to be non-smooth). The techniques effective in this regime are, however, bound to the Albanese geometry of the base and do not currently extend to bases with small Albanese dimension: Cao–Păun's positivity of direct images exploits the group structure of abelian bases, while the older generic-vanishing methods — the description, via the Fourier–Mukai transform (an equivalence between coherent sheaves on an abelian variety and on its dual), of the loci in the Picard variety (the moduli space of line bundles on YY) where twisted pluricanonical cohomology jumps — used in the Chen–Hacon approach, likewise require the base to have maximal Albanese dimension.

Beyond the maximal-Albanese regime: Birkar (2009) proved Cn,mC_{n,m} for total spaces of dimension at most 66; Junyan Cao (2018) settled arbitrary surface bases; Chang (2025) proved it in dimension 77 under nonvanishing hypotheses on the fibration, reducing to his threefold generalized nonvanishing theorem, and proved the weaker version Cn,m−C^-_{n,m} for all n≤7n\le 7; Benammar Ammar (2025) covered bases with α(Y)≥dim⁡Y−2\alpha(Y)\ge\dim Y-2, and (2026, announced as Part 1) certain cases over Calabi–Yau threefold bases. In positive characteristic the conjecture is false in general (Cascini–Ejiri–Kollár–Zhang 2021), so a proof in the remaining characteristic-zero cases cannot be characteristic-free. The open core includes fibrations over simply-connected bases with trivial canonical class — Calabi–Yau bases, or hyperkähler bases (irreducible holomorphic symplectic manifolds, i.e. simply-connected manifolds whose space of holomorphic 22-forms is generated by an everywhere-nondegenerate symplectic form) — with general fibers of intermediate Kodaira dimension (0≤κ(F)<dim⁡F0\le\kappa(F)<\dim F).

Problem Statement

Let f:X→Yf:X\to Y be an algebraic fiber space (a surjective morphism with connected fibers) between smooth complex projective varieties, with general fiber FF, and let n=dim⁡Xn=\dim X, m=dim⁡Ym=\dim Y. Prove, or refute by an explicit example, Iitaka's Cn,mC_{n,m} conjecture in characteristic zero:

κ(X)≥κ(Y)+κ(F),\kappa(X)\ge\kappa(Y)+\kappa(F),

where κ\kappa denotes the Kodaira dimension and the sum uses the convention that any sum involving −∞-\infty equals −∞-\infty. The problem is to settle this inequality for arbitrary ff. The regimes already established — base or fiber of general type; fiber admitting a good minimal model; one-dimensional base; surface base; base of maximal Albanese dimension; total dimension at most 6; base Albanese dimension at least dim⁡Y−2\dim Y-2; and the dimension-7 cases covered by nonvanishing hypotheses — leave as the unresolved core the fibrations whose base has dimension at least 3 and Albanese dimension at most dim⁡Y−3\dim Y-3 and is not of general type, whose total space has dimension at least 7, and whose general fiber is not established to admit a good minimal model. A solution that restricts the base, fiber, or dimension class beyond the source problem does not answer the question; this restriction concerns proofs of the inequality, since an explicit counterexample in any class refutes the general claim.

References

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