The Iitaka conjecture Cn,mC_{n,m} predicts that for an algebraic fiber space f:X→Yf:X\to Y between smooth projective complex varieties with general fiber FF, the Kodaira dimension is subadditive: κ(X)≥κ(F)+κ(Y)\kappa(X)\ge\kappa(F)+\kappa(Y). Junyan Cao proved the log (klt-pair) version κ(KX+Δ)≥κ(KF+ΔF)+κ(Y)\kappa(K_X+\Delta)\ge\kappa(K_F+\Delta_F)+\kappa(Y) when the base has dimension at most two, using surface-specific tools: Zariski decomposition, abundance in dimension two, positivity of direct images, and the uniformization theorem for compact Kähler orbifolds with trivial first Chern class. This record concerns the open extension of that theorem to arbitrary projective bases of dimension at least three, where none of these tools is available in comparable generality. The problem remains open despite substantial partial progress (bases of maximal Albanese dimension, threefold bases with positive irregularity (q(Y)=dim⁡H1(Y,OY)>0q(Y)=\dim H^1(Y,\mathcal{O}_Y)>0) or Kodaira dimension at least two, and some Calabi–Yau threefold base cases).

Background

For a smooth projective complex variety VV, the Kodaira dimension κ(V)=κ(KV)\kappa(V)=\kappa(K_V) measures the rate of growth of the spaces of pluricanonical sections H0(V,mKV)H^0(V,mK_V) as mm grows (their dimensions are the plurigenera); it takes the values −∞,0,1,…,dim⁡V-\infty,0,1,\dots,\dim V and is the fundamental birational invariant around which the classification of algebraic varieties is organized. For κ(V)≥0\kappa(V)\ge0 the pluricanonical maps for sufficiently divisible mm define the Iitaka fibration of VV, the rational map onto a variety of dimension κ(V)\kappa(V) that accounts for the pluricanonical sections. An algebraic fiber space is a surjective holomorphic map f:X→Yf:X\to Y with connected fibers between smooth projective varieties; its general fiber FF is the fiber over a general point of YY. A pair (X,Δ)(X,\Delta), where Δ\Delta is a Q\mathbb{Q}-effective divisor with rational coefficients, is called klt (Kawamata–Viehweg log terminal) if the discrepancy of every divisor over XX exceeds −1-1 (equivalently, all its log discrepancies are positive); this is the standard mild-singularity class used in the minimal model program (MMP), and one writes ΔF=Δ∣F\Delta_F=\Delta|_F for the restriction to the general fiber. Iitaka's Cn,mC_{n,m} conjecture (1970s) predicts subadditivity of the Kodaira dimension under fibrations: κ(X)≥κ(F)+κ(Y)\kappa(X)\ge\kappa(F)+\kappa(Y), where n=dim⁡Xn=\dim X and m=dim⁡Ym=\dim Y; the log version replaces XX and FF by the pairs (X,Δ)(X,\Delta) and (F,ΔF)(F,\Delta_F). The inequality is the easy direction reversed: the opposite bound κ(X)≤κ(F)+dim⁡Y\kappa(X)\le\kappa(F)+\dim Y ('easy addition') is elementary, and only the lower bound is deep.

The classical cases settled before 2010 are: dim⁡Y=1\dim Y=1 (Kawamata, 1982), bases and fibers of general type (via Viehweg's weak positivity in the early 1980s), and total spaces of dimension at most six (Birkar, 2009). A major newer case is the abelian-variety base with klt boundary (Cao–Păun, 2017), extended by Hacon–Popa–Schnell to bases of maximal Albanese dimension, i.e. bases whose Albanese map — the natural morphism to an abelian variety generated by the holomorphic one-forms — is generically finite onto its image; Hashizume proved the log version for fibrations whose general fiber has abundant log canonical divisor (abundance means the divisor's Iitaka dimension equals its numerical dimension, i.e. it behaves semi-ample in the relevant sense). Cao (2018) settled all bases of dimension at most two, with boundary: κ(KX+Δ)≥κ(KF+ΔF)+κ(Y)\kappa(K_X+\Delta)\ge\kappa(K_F+\Delta_F)+\kappa(Y) for dim⁡Y≤2\dim Y\le 2.

Cao's surface argument explains exactly what fails in higher dimensions. After running the minimal model program on the base surface, the case κ(Y)≥1\kappa(Y)\ge1 reduces to the curve case, and κ(Y)=0\kappa(Y)=0 reduces, up to a finite étale cover, to a torus base (Cao–Păun) or a K3 surface base. On a K3 base, positivity results for direct images (due to Berndtsson and Păun–Takayama) equip det⁡f∗(m(KX/Y+Δ))\det f_*(m(K_{X/Y}+\Delta)) with a singular hermitian metric of semipositive curvature, so this determinant line bundle is pseudo-effective (a limit of classes of effective Q\mathbb{Q}-divisors). For surfaces, Zariski decomposition (every pseudo-effective divisor on a surface splits as an effective 'negative' part plus a nef part intersecting it trivially) together with abundance in dimension two (on a K3 surface every nef line bundle — one of non-negative degree on every curve — is effective and semiample, meaning some positive multiple is generated by its global sections) turns pseudo-effectivity into actual sections. If the numerical dimension of the determinant (the dimension of the largest subvariety on which the divisor has positive degree) is zero, the metric is hermitian flat outside finitely many exceptional curves, and one invokes the uniformization theorem for compact Kähler orbifolds with trivial first Chern class — the orbifold analogue of the Beauville–Bogomolov decomposition into Calabi–Yau factors (simply connected with holonomy SUSU), hyperkähler factors (simply connected with holonomy SpSp), and a torus — which forces the fundamental group of the complement of the exceptional curves to be almost abelian, so that parallel transport produces pluricanonical sections on XX.

For bases of dimension three or more every one of these steps loses its engine: abundance is unproven in dimension at least three (a pseudo-effective divisor on the base need not be effective, so the determinant gives no sections), there is no Zariski decomposition of comparable strength, the flatness/monodromy analysis of direct images must be controlled in codimension at least three, and no substitute for the surface uniformization and fundamental-group argument is known beyond the torus and product situations. The problem is known to be characteristic-zero-specific: Cascini, Ejiri, Kollár, and Zhang (2021) constructed fibrations over algebraically closed fields of every positive characteristic that violate subadditivity of Kodaira dimension. Popa's 2022 survey of Kodaira-dimension conjectures notes that Iitaka's conjecture, like the stronger additivity conjectures proposed there, is implied by the main conjectures of the minimal model program (existence of good minimal models); an unconditional proof must therefore either go through major MMP ingredients or circumvent them.

Problem Statement

Let f:X→Yf:X\to Y be an algebraic fiber space (a surjective morphism with connected fibers) between smooth projective complex varieties, where dim⁡Y≥3\dim Y\ge3, let (X,Δ)(X,\Delta) be a klt pair (Kawamata–Viehweg log terminal, as defined in the Background) with Δ\Delta a Q\mathbb{Q}-effective divisor, let FF be a general fiber of ff, and set ΔF=Δ∣F\Delta_F=\Delta|_F. Prove, or disprove by an explicit counterexample over the complex numbers, the log Iitaka inequality

κ(KX+Δ) ≥ κ(KF+ΔF)+κ(Y).\kappa(K_X+\Delta)\ \ge\ \kappa(K_F+\Delta_F)+\kappa(Y).

The inequality is already known for dim⁡Y≤2\dim Y\le2 (Cao, 2018), so the problem is precisely the case of bases of dimension at least three, in all regimes of κ(Y)\kappa(Y) and of the geometry of the base; in particular it includes threefold bases with q(Y)=κ(Y)=0q(Y)=\kappa(Y)=0, where q(Y)=dim⁡H1(Y,OY)q(Y)=\dim H^1(Y,\mathcal{O}_Y) is the irregularity, the dimension of the Albanese variety (e.g. Calabi–Yau threefold bases) and all higher-dimensional simply-connected bases with κ(Y)=0\kappa(Y)=0, for which no general result is known. The statement must be proved over the field of complex numbers: in positive characteristic subadditivity of Kodaira dimension fails in general.

References

  1. Junyan Cao, Kodaira dimension of algebraic fiber spaces over surfaces, Algebraic Geometry (2018), 728–741. DOI: 10.14231/ag-2018-020, https://doi.org/10.14231/ag-2018-020 (preprint: arXiv:1511.07048, https://arxiv.org/abs/1511.07048)
  2. Yujiro Kawamata, Kodaira dimension of algebraic fiber spaces over curves, Inventiones Mathematicae 66 (1982), 57–71. DOI: 10.1007/BF01404756, https://doi.org/10.1007/BF01404756
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  5. Christopher Hacon, Mihnea Popa, Christian Schnell, Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Păun, Contemporary Mathematics 712 (2018), 143–195. DOI: 10.1090/conm/712/14346, https://doi.org/10.1090/conm/712/14346 (preprint: arXiv:1611.08768, https://arxiv.org/abs/1611.08768)
  6. Mihnea Popa, Conjectures on the Kodaira dimension, preprint (2021, revised 2022). arXiv:2111.10900, https://arxiv.org/abs/2111.10900
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  8. 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
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