Iitaka's subadditivity conjecture for the logarithmic Kodaira dimension

Iitaka's subadditivity conjecture for the logarithmic Kodaira dimension $\bar\kappa$ is the log version of the classical Iitaka conjecture $C_{n,m}$, asserting that for a dominant morphism $g: V \to W$ of complex algebraic varieties the logarithmic Kodaira dimension of the source is at least the sum of those of a general fiber and the base. Fujino (J. Math. Soc. Japan 69, 2017) reduced this conjecture, via a subadditivity inequality for Nakayama's numerical Kodaira dimension $\kappa_\sigma$, to the generalized abundance conjecture $\kappa_\sigma = \kappa_\iota$ for $\mathbb{Q}$-factorial projective dlt pairs, dimension-wise equivalent to the good minimal model conjecture; the reduction survives a proof gap found via Lesieutre's example and repaired in Fujino's 2020 corrigendum. The conjecture remains open: it is known unconditionally only for total dimension at most 5 (Hashizume) and in special boundary/positivity cases, and both the generalized abundance conjecture and the good minimal model conjecture are open in dimension at least 4.

Background

For a smooth projective complex variety $X$ with canonical divisor $K_X$, the Kodaira dimension $\kappa(X)$ measures the rate of growth of the spaces of pluricanonical sections $H^0(X, mK_X)$ as $m$ grows; it is the fundamental birational invariant by which the classification of algebraic varieties is organized. A foundational open problem of the subject, Iitaka's subadditivity conjecture (usually called Conjecture $C_{n,m}$), predicts that birational complexity can only grow under fibration: for a surjective morphism $f: X \to Y$ of projective varieties with general fiber $F$, one should have $\kappa(X) \ge \kappa(F) + \kappa(Y)$. It is known when the base is a curve (Kawamata, 1982), when the total dimension is at most 6 (Birkar, 2009), and when the base has maximal Albanese dimension — that is, its Albanese morphism (the universal map to an abelian variety) is generically finite onto that abelian variety (Cao–Păun, 2016) — but it remains open in general.

Iitaka extended the same question from projective varieties to open (non-complete) varieties. For a smooth variety $U$, choose a smooth projective compactification $X$ such that the boundary $D = X \setminus U$ is a simple normal crossing divisor (one whose components are smooth and meet like coordinate hyperplanes); the logarithmic Kodaira dimension is $\bar\kappa(U) := \kappa(X, K_X + D)$, which is independent of the chosen compactification. The log version of Iitaka's conjecture asserts that for every dominant morphism $g: V \to W$ of complex algebraic varieties, $\bar\kappa(V) \ge \bar\kappa(F') + \bar\kappa(W)$, where $F'$ is an irreducible component of a sufficiently general fiber. This log conjecture contains the closed conjecture $C_{n,m}$ as the special case of complete varieties.

Fujino's 2017 paper is the central modern attack on the log conjecture. It passes through two refinements of the Kodaira dimension introduced to handle boundary divisors and $\mathbb{R}$-divisors. Nakayama's numerical Kodaira dimension $\kappa_\sigma(X, D)$ (from his monograph Zariski-decomposition and abundance, 2004) is a birational invariant of a pseudoeffective divisor $D$ defined through the asymptotic contributions of valuations of prime divisors on all birational models; it depends only on numerical positivity data and always satisfies $\kappa_\sigma \ge \kappa$. The invariant $\kappa_\iota(X, D)$ is defined for $\mathbb{R}$-Cartier $\mathbb{R}$-divisors by the growth of $H^0(X, \mathcal{O}_X(\lfloor mD \rfloor))$ and coincides with $\kappa$ for $\mathbb{Q}$-divisors. Fujino established an Iitaka-type inequality for $\kappa_\sigma$ along a fibration of pairs $(X, D_X) \to (Y, D_Y)$ with simple normal crossing boundaries, and observed that the log conjecture would follow from the equality $\kappa_\sigma(X, K_X + D) = \kappa(X, K_X + D)$ for the relevant pairs — a special case of the generalized abundance conjecture: for every $\mathbb{Q}$-factorial projective dlt pair $(X, \Delta)$ (divisorial log terminal: the standard mild-singularity class of the minimal model program), $\kappa_\sigma(X, K_X + \Delta) = \kappa_\iota(X, K_X + \Delta)$. This conjecture is equivalent, dimension by dimension, to the good minimal model conjecture — the assertion that a dlt pair with pseudo-effective log canonical divisor admits a minimal model on which the log canonical divisor is semiample (some positive multiple is basepoint-free) — one of the central open problems of the minimal model program.

Two post-publication developments shape the current status. First, Lesieutre constructed a pseudo-effective $\mathbb{R}$-divisor on a smooth projective threefold for which the different numerical notions of Iitaka dimension disagree ($\kappa_\sigma = 1$ while $\kappa_\nu = 2$); this disproved a lemma used in the original proof of Fujino's main theorem. Fujino's corrigendum (2020) established slightly weaker mixed inequalities (with one ordinary $\kappa$ and one $\kappa_\sigma$ on the right-hand side) and showed that these still suffice both for the reduction of the log conjecture to generalized abundance and for the paper's unconditional corollaries, so the reduction stands. Second, Hashizume (2020) proved that the log conjecture holds whenever the log canonical divisor of a sufficiently general fiber is abundant (numerically equivalent to an effective $\mathbb{Q}$-divisor with semiample positive part), in particular whenever the general fiber pair has a good minimal model; since good minimal models exist unconditionally in dimension at most 3, this makes the log conjecture unconditional for total dimension at most 5 and for relative dimension at most 3. His introduction also records the cases settled before it: the conjecture holds when the log canonical divisor of the general fiber is big or when that of the base is big, and Fujino's original paper had already settled the affine-source case, that is, dominant morphisms $g: V \to W$ with $V$ affine. Analytic and convex-geometric approaches — multiplier ideals (sheaves measuring singularities through integrability conditions) in the work of He–Zhou, and Okounkov bodies (convex bodies attached to a divisor that encode the asymptotic shape of its spaces of sections) in the work of Choi–Park — reproduce and extend the numerical half of the statement, but the passage from $\kappa_\sigma$ to $\kappa$ is precisely the abundance input, and Lesieutre's example shows that input cannot be supplied by numerics alone. In positive characteristic subadditivity of Kodaira dimension is false (Cascini–Ejiri–Kollár–Zhang), so any proof over $\mathbb{C}$ must use genuinely characteristic-zero tools.

Problem Statement

Prove or refute Iitaka's subadditivity conjecture for the logarithmic Kodaira dimension over the complex numbers, in full generality: for every dominant morphism $g: V \to W$ of algebraic varieties, with $F'$ an irreducible component of a sufficiently general fiber, $\bar\kappa(V) \;\ge\; \bar\kappa(F') + \bar\kappa(W).$ Equivalently, in compactified form: for every surjective morphism $f: X \to Y$ of smooth projective varieties with connected fibers and simple normal crossing divisors $D_X$, $D_Y$ satisfying $\mathrm{Supp}\, f^* D_Y \subseteq \mathrm{Supp}\, D_X$, with sufficiently general fiber $F$, $\kappa(X, K_X + D_X) \;\ge\; \kappa(F, K_F + D_X|_F) + \kappa(Y, K_Y + D_Y).$ The conjecture is known unconditionally only in the cases listed in the background (total dimension $\le 5$, relative dimension $\le 3$, affine source, and the big/abundant/maximal-Albanese boundary cases); the required advance is a proof (or refutation) in the remaining generality. No dimensional, characteristic, or boundary restrictions may be added to the claim: the statement above is the target.

References

1. Osamu Fujino, On subadditivity of the logarithmic Kodaira dimension, Journal of the Mathematical Society of Japan 69 (2017), no. 4, 1565–1581, DOI 10.2969/jmsj/06941565 (preprint arXiv:1406.2759), https://doi.org/10.2969/jmsj/06941565
2. Osamu Fujino, Corrigendum to “On subadditivity of the logarithmic Kodaira dimension”, Journal of the Mathematical Society of Japan 72 (2020), no. 4, DOI 10.2969/jmsj/82568256 (preprint arXiv:1904.11639), https://arxiv.org/abs/1904.11639
3. Kenta Hashizume, Log Iitaka conjecture for abundant log canonical fibrations, Proceedings of the Japan Academy, Series A, Mathematical Sciences 96 (2020), no. 10, DOI 10.3792/pjaa.96.017 (preprint arXiv:1902.10923), https://doi.org/10.3792/pjaa.96.017
4. John Lesieutre, Notions of numerical Iitaka dimension do not coincide, Journal of Algebraic Geometry 31 (2021), no. 1, 113–126, DOI 10.1090/jag/763, https://doi.org/10.1090/jag/763
5. Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs 14, Mathematical Society of Japan, Tokyo, 2004, DOI 10.2969/msjmemoirs/014010000, https://doi.org/10.2969/msjmemoirs/014010000
6. Yujiro Kawamata, Kodaira dimension of algebraic fiber spaces over curves, Inventiones Mathematicae 66 (1982), no. 1, 57–71, DOI 10.1007/BF01404756, https://doi.org/10.1007/BF01404756
7. Caucher Birkar, The Iitaka conjecture $C_{n,m}$ in dimension six, Compositio Mathematica 145 (2009), no. 6, 1442–1446, DOI 10.1112/s0010437x09004187, https://doi.org/10.1112/s0010437x09004187
8. Junyan Cao and Mihai Păun, Kodaira dimension of algebraic fiber spaces over abelian varieties, Inventiones Mathematicae 205 (2016), no. 1, 345–387, DOI 10.1007/s00222-016-0672-6, https://doi.org/10.1007/s00222-016-0672-6
9. Frédéric Campana, Kodaira additivity, birational isotriviality and specialness, Moscow Mathematical Journal 23 (2023), no. 3, 319–330, DOI 10.17323/1609-4514-2023-23-3-319-330 (preprint arXiv:2207.05412), https://doi.org/10.17323/1609-4514-2023-23-3-319-330
10. Sung Rak Choi and Jinhyung Park, On subadditivity of Okounkov bodies for algebraic fiber spaces, International Mathematics Research Notices 2023 (2023), no. 24, 21781–21802, DOI 10.1093/imrn/rnac336, https://doi.org/10.1093/imrn/rnac336
11. Bojie He and Xiangyu Zhou, On the subadditivity of generalized Kodaira dimensions, arXiv:2405.08265 (2024), https://arxiv.org/abs/2405.08265
12. Chi-Kang Chang, Generalized nonvanishing conjecture and Iitaka conjecture, Mathematische Zeitschrift 310 (2025), no. 2, DOI 10.1007/s00209-025-03727-5 (preprint arXiv:2312.04015), https://doi.org/10.1007/s00209-025-03727-5
13. Houari Benammar Ammar, Note on Iitaka Conjecture $C_{n,m}$, arXiv:2510.06412 (2025), https://arxiv.org/abs/2510.06412
14. Houari Benammar Ammar, Kodaira dimension of algebraic fiber spaces over threefolds: Part 1, arXiv:2605.08598 (2026), https://arxiv.org/abs/2605.08598
15. Zhiyuan Jiang, An approach to the abundance conjecture for Kähler varieties via algebraic reduction, arXiv:2604.03879 (2026), https://arxiv.org/abs/2604.03879
16. Sung Gi Park, Logarithmic base change theorem and smooth descent of positivity of log canonical divisor, Algebra & Number Theory 20 (2026), no. 3, 445–476, DOI 10.2140/ant.2026.20.445 (preprint arXiv:2210.02825), https://doi.org/10.2140/ant.2026.20.445
17. Paolo Cascini, Sho Ejiri, János Kollár and Lei Zhang, Subadditivity of Kodaira dimension does not hold in positive characteristic, arXiv:2003.13206 (2020), https://arxiv.org/abs/2003.13206