For a fibration f:X→Yf:X\to Y between smooth complex projective varieties with general fiber FF and κ(Y)≥0\kappa(Y)\ge 0, Fanjun Meng (2023, published 2024) conjectured that for every sufficiently divisible positive integer mm with f∗ωX⊗m≠0f_*\omega_X^{\otimes m}\ne 0 one has κ(X)=κ(F)+κ(Y,det⁡^(f∗ωX⊗m)⊗ωY)\kappa(X)=\kappa(F)+\kappa(Y,\widehat{\det}(f_*\omega_X^{\otimes m})\otimes\omega_Y), where det⁡^\widehat{\det} is the reflexive determinant of the pluricanonical pushforward. Meng proved the equality when the base is of maximal Albanese dimension, of general type, or a curve, and showed that the conjecture implies the Viehweg (and hence Iitaka) subadditivity conjecture assuming the good minimal model conjecture for the general fiber. No resolution or refutation beyond these known cases is known as of August 2026: the conjecture remains open for arbitrary smooth projective bases of non-negative Kodaira dimension, for instance bases of Kodaira dimension 00 and irregularity 00 (that is, whose Picard variety of topologically trivial line bundles is a point).

Background

The Kodaira dimension κ(X)\kappa(X) of a smooth complex projective variety measures the rate of polynomial growth of the spaces of pluricanonical forms: it is the Iitaka dimension of the canonical bundle KXK_X, i.e. the integer kk such that h0(X,ωX⊗n)h^0(X,\omega_X^{\otimes n}) grows like nkn^k; one writes κ(Y,L)\kappa(Y,L) for the analogous Iitaka dimension of a line bundle LL on a projective variety YY, and κ=−∞\kappa=-\infty when all plurigenera vanish. A fibration is a surjective morphism f:X→Yf:X\to Y between smooth projective varieties with connected fibers; FF denotes a general fiber. A guiding question of birational classification is how κ(X)\kappa(X) relates to κ(F)\kappa(F) and κ(Y)\kappa(Y): the Iitaka conjecture Cn,mC_{n,m} (subadditivity) predicts κ(X)≥κ(F)+κ(Y)\kappa(X)\ge\kappa(F)+\kappa(Y). It is known in several cases, including when the base YY is a curve (Kawamata 1982), when the general fiber has a good minimal model (Kawamata 1985, where a good minimal model is a mild birational model whose canonical bundle is semiample, meaning some positive power of it is basepoint-free and defines a morphism), when YY is an abelian variety, that is, a projective complex torus (Cao–Păun 2017), and when YY is of maximal Albanese dimension, meaning the Albanese morphism of YY (the universal morphism to an abelian variety) is generically finite onto its image. For bases with κ(Y)≥0\kappa(Y)\ge 0, Viehweg strengthened the conjecture to κ(X)≥κ(F)+max⁡{κ(Y),Var(f)}\kappa(X)\ge\kappa(F)+\max\{\kappa(Y),\mathrm{Var}(f)\}, where Var(f)\mathrm{Var}(f) is the variation of the fibration, the birational invariant measuring how the fibers move in moduli (equivalently, the smallest dimension of a base through which the family factors birationally non-trivially); see Kawamata 1985 for the standard formulation. Popa later proposed a hierarchy of superadditivity conjectures refining these statements for morphisms of smooth quasi-projective varieties.

For fibrations over an abelian variety AA a finer invariant than κ(A)\kappa(A) enters: the determinant of the pushforward of pluricanonical bundles. For a coherent sheaf EE on a smooth projective variety, det⁡^E\widehat{\det} E denotes the reflexive hull of the determinant det⁡E\det E; on the smooth base this reflexive hull is a line bundle, to which κ(Y,−)\kappa(Y,-) applies. Meng–Popa (2021, published in the International Mathematics Research Notices in 2024) and Meng (2022) proved lower bounds of the form κ(X)≥κ(F)+dim⁡V0(A,f∗ωX⊗m)\kappa(X)\ge\kappa(F)+\dim V^0(A,f_*\omega_X^{\otimes m}), where V0(A,F)={α∈Pic0(A):h0(A,F⊗α)>0}V^0(A,\mathcal F)=\{\alpha\in\mathrm{Pic}^0(A):h^0(A,\mathcal F\otimes\alpha)>0\} is a cohomological support locus, a closed subset of the Picard variety of topologically trivial line bundles. Meng's 2023 paper upgraded these to exact equalities: over an abelian base, κ(X)=κ(F)+κ(A,det⁡^f∗ωX⊗m)=κ(F)+dim⁡V0(A,f∗ωX⊗m)\kappa(X)=\kappa(F)+\kappa(A,\widehat{\det} f_*\omega_X^{\otimes m})=\kappa(F)+\dim V^0(A,f_*\omega_X^{\otimes m}) for all m>1m>1 (Theorem 1.1 and Corollary 1.2 there, with a twisted version, Theorem 1.4), and over a base of maximal Albanese dimension, the inequality κ(V)≥κ(Y,det⁡^f∗ωX⊗m⊗ωY)\kappa(V)\ge\kappa(Y,\widehat{\det} f_*\omega_X^{\otimes m}\otimes\omega_Y) where VV is the smooth locus of ff (Theorem 1.6 there). Building on these results, Meng posed Conjecture 1.8: a single equality predicting that over any smooth projective base of non-negative Kodaira dimension, κ(X)\kappa(X) is exactly κ(F)\kappa(F) plus the Iitaka dimension of the line bundle det⁡^(f∗ωX⊗m)⊗ωY\widehat{\det}(f_*\omega_X^{\otimes m})\otimes\omega_Y for sufficiently divisible mm. The conjecture refines the classical picture: assuming the good minimal model conjecture for the general fiber, it implies Viehweg's conjecture, because Kawamata's theorem gives κ(Y,det⁡^f∗ωX/Y⊗m)≥Var(f)\kappa(Y,\widehat{\det} f_*\omega_{X/Y}^{\otimes m})\ge\mathrm{Var}(f) for suitable mm, whence κ(Y,det⁡^f∗ωX⊗m⊗ωY)≥max⁡{κ(Y),Var(f)}\kappa(Y,\widehat{\det} f_*\omega_X^{\otimes m}\otimes\omega_Y)\ge\max\{\kappa(Y),\mathrm{Var}(f)\}. Meng proved the conjecture when YY is of maximal Albanese dimension (in a stronger form for pairs (X,Δ)(X,\Delta) with Kawamata log terminal, klt, singularities — the standard mild singularity class in the minimal model program, the birational classification framework for algebraic varieties), of general type, or a curve, and these are all the cases known.

Problem Statement

Let f:X→Yf:X\to Y be a fibration between smooth complex projective varieties with general fiber FF and κ(Y)≥0\kappa(Y)\ge 0. For every sufficiently divisible positive integer mm such that f∗ωX⊗m≠0f_*\omega_X^{\otimes m}\ne 0, is it true that

κ(X)=κ(F)+κ ⁣(Y, det⁡^ ⁣(f∗ωX⊗m)⊗ωY),\kappa(X)=\kappa(F)+\kappa\!\left(Y,\,\widehat{\det}\!\left(f_*\omega_X^{\otimes m}\right)\otimes\omega_Y\right),

with the convention (as in the source) that the case κ(F)=−∞\kappa(F)=-\infty is trivial because then κ(X)=−∞\kappa(X)=-\infty? Prove or refute the conjecture in the generality stated - that is, for smooth projective bases of non-negative Kodaira dimension beyond the cases already known: bases of maximal Albanese dimension, bases of general type, and curves.

References

  1. Fanjun Meng, "On surjective morphisms to abelian varieties and a generalization of the Iitaka conjecture", Mathematische Zeitschrift 307 (2024), no. 2, Paper No. 28, DOI 10.1007/s00209-024-03505-9, arXiv:2306.01326, https://arxiv.org/abs/2306.01326
  2. Junyan Cao and M. Păun, "Kodaira dimension of algebraic fiber spaces over abelian varieties", Inventiones mathematicae 207 (2017), 345-387, DOI 10.1007/s00222-016-0672-6, https://doi.org/10.1007/s00222-016-0672-6
  3. 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
  4. Yujiro Kawamata, "Minimal models and the Kodaira dimension of algebraic fiber spaces", Journal für die reine und angewandte Mathematik 363 (1985), 1-46, DOI 10.1515/crll.1985.363.1, https://doi.org/10.1515/crll.1985.363.1
  5. Fanjun Meng and Mihnea Popa, "Kodaira dimension of fibrations over abelian varieties", International Mathematics Research Notices (2024), no. 9, 7466-7487, DOI 10.1093/imrn/rnad209, arXiv:2111.14165, https://arxiv.org/abs/2111.14165
  6. Fanjun Meng, "Estimates on the Kodaira dimension for fibrations over abelian varieties", Journal of Algebra 695 (2026), 237-254, DOI 10.1016/j.jalgebra.2026.01.042, arXiv:2207.08359, https://arxiv.org/abs/2207.08359
  7. Mihnea Popa, "Conjectures on the Kodaira dimension", arXiv:2111.10900 (2021), https://arxiv.org/abs/2111.10900
  8. Chi-Kang Chang, "Generalized nonvanishing conjecture and Iitaka conjecture", arXiv:2312.04015 (2023), https://arxiv.org/abs/2312.04015
  9. Houari Benammar Ammar, "Note on Iitaka Conjecture Cn,mC_{n,m}", arXiv:2510.06412 (2025), https://arxiv.org/abs/2510.06412