The Iitaka conjecture predicts that for a fibration f ⁣:X→Zf\colon X\to Z of smooth projective varieties the Kodaira dimension is subadditive: κ(X)≥κ(Z)+κ(F)\kappa(X)\ge\kappa(Z)+\kappa(F). Birkar, Chen and Zhang proved the three-dimensional case over the algebraic closure of a finite field of characteristic p>5p>5, and Ejiri and Zhang later removed the finite-field restriction for fibrations over curves whose geometric generic fiber is smooth. This record formulates the surviving open core of that line of work as a single decision problem: for a fibration from a smooth projective threefold to a smooth projective curve over an algebraically closed field of characteristic p>0p>0, with WW a smooth birational model of the geometric generic fiber, decide whether κ(X)≥κ(Z)+κ(W)\kappa(X)\ge\kappa(Z)+\kappa(W) always holds. The question is open in characteristics 22, 33 and 55 (already for smooth geometric generic fibers) and in characteristic p>5p>5 for singular geometric generic fibers over algebraically closed fields other than F‾p\overline{\mathbb{F}}_p.

Background

For a smooth projective variety XX over an algebraically closed field, the Kodaira dimension κ(X)\kappa(X) measures the rate of growth of the spaces of pluricanonical sections H0(X,mKX)H^0(X,mK_X) as mm grows (KXK_X the canonical divisor); it is −∞-\infty if all these spaces vanish, and otherwise the maximal dimension of the image of the pluricanonical maps. The Kodaira dimension is the basic coarse invariant in the birational classification of algebraic varieties. Iitaka's subadditivity conjecture Cn,mC_{n,m}, proposed over the complex numbers, concerns a fibration, i.e. a surjective projective morphism f ⁣:X→Zf\colon X\to Z between smooth projective varieties with f∗OX=OZf_*\mathcal{O}_X=\mathcal{O}_Z (so fibers are connected): writing FF for the generic fiber, it predicts κ(X)≥κ(Z)+κ(F)\kappa(X)\ge\kappa(Z)+\kappa(F). Over the complex numbers many cases are known (relative dimension at most two, bases of general type, fibers of general type, bases of maximal Albanese dimension, total dimension at most six), but the conjecture is open in general.

In positive characteristic the statement needs care. Even when XX and ZZ are smooth, the generic fiber XηX_\eta, a variety over the function field k(Z)k(Z), can fail to be smooth over k(Z)k(Z) because of inseparable phenomena, and then the geometric generic fiber XηˉX_{\bar\eta} (the fiber after extending scalars to an algebraic closure of k(Z)k(Z)) can be singular or even non-reduced. The conjecture is therefore usually posed for fibrations whose geometric generic fiber is integral and smooth; the more general "weak form" replaces κ(F)\kappa(F) by κ(W)\kappa(W), where WW is a smooth projective birational model of the underlying reduced geometric generic fiber (which exists in the threefold-over-curve setting because resolution of singularities for surfaces holds in every characteristic). The corresponding "strong form", using κ(Xη)\kappa(X_\eta) with XηX_\eta taken over k(Z)k(Z), is now known to fail in general in positive characteristic, so the weak form is the version expected to survive.

Progress in positive characteristic has been built on two pillars. One is the three-dimensional log minimal model program in characteristic p>5p>5 (the machinery of birational contractions and flips that produces models on which the canonical class is nef, together with abundance-type semi-ampleness statements; a divisor is semi-ample when some positive multiple is base-point-free, and nef when it has non-negative degree on every curve). The other is the study of direct images of pluricanonical sheaves; here the characteristic-zero hope fails outright, since for fibrations built from Tango curves over suitable bases these direct images can be anti-ample.

In this landscape, Birkar, Chen and Zhang proved C3,mC_{3,m} for threefolds over k=F‾pk=\overline{\mathbb{F}}_p with p>5p>5; the new content was C3,1C_{3,1} (fibrations over curves), while C3,2C_{3,2} had been settled earlier for separable fibrations of relative dimension one in arbitrary positive characteristic by Chen and Zhang. The restriction to F‾p\overline{\mathbb{F}}_p was used because over such fields a numerically trivial Q\mathbb{Q}-divisor is automatically torsion, and nef and big divisors on surfaces are semi-ample; over general fields the same conclusions require substantial extra work. Patakfalvi had earlier proved the conjecture over bases of general type when the geometric generic fiber has non-nilpotent Hasse-Witt matrix (the matrix of Frobenius, the pp-th power map on coordinates, acting on H1(O)H^1(\mathcal{O}) of the fiber, whose nilpotence measures pathological pp-behavior), in arbitrary characteristic. Ejiri and Zhang then proved C3,1C_{3,1} over every algebraically closed field of characteristic p>5p>5, handling exactly the small Kodaira-dimension cases of the fiber that the finite-field arguments had left exceptional.

Two regimes nevertheless remain untouched. First, characteristics 22, 33 and 55: the three-dimensional minimal model program and the accompanying vanishing and semi-positivity inputs are established only for p>5p>5, and no proof of C3,1C_{3,1} in small characteristic is known, even for smooth geometric generic fibers (the case κ(Z)=−∞\kappa(Z)=-\infty, i.e. rational bases, is trivial, so the substance concerns elliptic and higher-genus bases). Second, singular geometric generic fibers: over F‾p\overline{\mathbb{F}}_p with p>5p>5 the source paper settled the weak form, but over general algebraically closed fields only special cases are known (bases of general type under nefness and relative semi-ampleness conditions, fibers with sufficiently large Frobenius-stable canonical rings, fibrations over abelian varieties with Frobenius-stable Kodaira-dimension-zero generic fibers). Counterexamples of Cascini, Ejiri, Kollár and Zhang show that the strong form fails in general (in total dimension up to 2p−12p-1, dimension 44 for p=2p=2, always with uniruled base and singular geometric generic fibers), but no threefold counterexample is known, and those authors explicitly state that the weak form is still expected to hold. This leaves a precise question: does the weak form of Iitaka subadditivity hold for every fibration from a smooth projective threefold to a smooth projective curve over every algebraically closed field of positive characteristic?

Problem Statement

Let kk be an algebraically closed field of characteristic p>0p>0, and let f ⁣:X→Zf\colon X\to Z be a fibration (a surjective projective morphism with f∗OX=OZf_*\mathcal{O}_X=\mathcal{O}_Z) from a smooth projective threefold XX onto a smooth projective curve ZZ. Let XηˉX_{\bar\eta} be the geometric generic fiber, and let WW be a smooth projective birational model of the underlying reduced scheme of XηˉX_{\bar\eta}, which exists in every characteristic by resolution of singularities for surfaces. Does the weak form of Iitaka's subadditivity

κ(X) ≥ κ(Z)+κ(W)\kappa(X)\ \ge\ \kappa(Z)+\kappa(W)

hold for every such fibration over every such field?

When XηˉX_{\bar\eta} is smooth, W=XηˉW=X_{\bar\eta} and this is the classical conjecture C3,1C_{3,1}. The published results leave exactly two open regimes: (i) characteristics p∈{2,3,5}p\in\{2,3,5\}, where even C3,1C_{3,1} with smooth geometric generic fiber is open in general (only special cases are known, such as suitable general-type bases; the case κ(Z)=−∞\kappa(Z)=-\infty of rational bases is trivial, so the substance lies in elliptic and higher-genus bases); and (ii) p>5p>5 with singular geometric generic fiber over algebraically closed fields other than F‾p\overline{\mathbb{F}}_p (over F‾p\overline{\mathbb{F}}_p itself the weak form is already settled).

References

  1. Caucher Birkar, Yifei Chen, Lei Zhang, Iitaka conjecture for 3-folds over finite fields, Nagoya Mathematical Journal 229 (2018), 21-51, DOI: 10.1017/nmj.2016.61, https://doi.org/10.1017/nmj.2016.61 (arXiv:1507.08760, https://arxiv.org/abs/1507.08760)
  2. Sho Ejiri, Lei Zhang, Iitaka's Cn,mC_{n,m} conjecture for 3-folds in positive characteristic, Mathematical Research Letters 25 (2018), no. 3, 783-802, DOI: 10.4310/mrl.2018.v25.n3.a3, https://doi.org/10.4310/mrl.2018.v25.n3.a3 (arXiv:1604.01856, https://arxiv.org/abs/1604.01856)
  3. Yifei Chen, Lei Zhang, The subadditivity of the Kodaira dimension for fibrations of relative dimension one in positive characteristics, Mathematical Research Letters 22 (2015), no. 3, 675-696, DOI: 10.4310/mrl.2015.v22.n3.a3, https://doi.org/10.4310/mrl.2015.v22.n3.a3 (arXiv:1305.6024, https://arxiv.org/abs/1305.6024)
  4. Paolo Cascini, Sho Ejiri, János Kollár, Lei Zhang, Subadditivity of Kodaira dimension does not hold in positive characteristic, Commentarii Mathematici Helvetici 96 (2021), 465-481, DOI: 10.4171/cmh/517, https://doi.org/10.4171/cmh/517 (arXiv:2003.13206, https://arxiv.org/abs/2003.13206)
  5. Zsolt Patakfalvi, On subadditivity of Kodaira dimension in positive characteristic over a general type base, Journal of Algebraic Geometry 27 (2018), 21-53, DOI: 10.1090/jag/688, https://doi.org/10.1090/jag/688
  6. Lei Zhang, Subadditivity of Kodaira dimensions for fibrations of three-folds in positive characteristics, Advances in Mathematics 354 (2019), article 106741, DOI: 10.1016/j.aim.2019.106741, https://doi.org/10.1016/j.aim.2019.106741 (arXiv:1601.06907, https://arxiv.org/abs/1601.06907)
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  8. Sho Ejiri, Direct images of pluricanonical bundles and Frobenius stable canonical rings of fibers, Algebraic Geometry 11 (2024), no. 1, 71-110, DOI: 10.14231/ag-2024-003, https://doi.org/10.14231/ag-2024-003 (arXiv:1909.07000, https://arxiv.org/abs/1909.07000)
  9. Sho Ejiri, Numerical Kodaira dimension of algebraic fiber spaces in positive characteristic, arXiv:2212.09245, https://arxiv.org/abs/2212.09245
  10. Jefferson Baudin, On the Albanese morphism of varieties with Frobenius stable Kodaira dimension zero, arXiv:2401.06074, https://arxiv.org/abs/2401.06074
  11. Omprokash Das, Joe Waldron, On the log minimal model program for threefolds over imperfect fields of characteristic p>5p>5, Journal of the London Mathematical Society 106 (2022), no. 4, 3895-3937, DOI: 10.1112/jlms.12677, https://doi.org/10.1112/jlms.12677 (arXiv:1911.04394, https://arxiv.org/abs/1911.04394)
  12. Marta Benozzo, On the Iitaka conjecture for anticanonical divisors in positive characteristic, Michigan Mathematical Journal 76 (2026), no. 1, 3-23, DOI: 10.1307/mmj/20226272, https://doi.org/10.1307/mmj/20226272 (arXiv:2204.02045, https://arxiv.org/abs/2204.02045)