Iitaka Subadditivity Conjecture for Kodaira Dimension

Iitaka Subadditivity Conjecture for Kodaira Dimension

OpenAI's claim

Manuscript 1 (family 033, 2026-09-26): The logarithmic Iitaka--Viehweg lower bound is claimed for projective fibrations over a smooth quasi-projective base of nonnegative logarithmic Kodaira dimension, using variation of the entire generic fiber; Manuscript 2 (family 033, 2026-09-26): For compact manifolds in Fujiki class $\mathcal C$ carrying rational normal-crossing boundaries, the orbifold Iitaka lower bound is claimed, implying its ordinary and logarithmic specializations. Manuscript 3 (family 033, 2026-09-27): The ordinary Kodaira dimension inequality is claimed for projective connected-fiber morphisms in characteristic zero, without restricting total dimension or assuming nonnegative Kodaira dimension upstairs; Manuscript 4 (family 033, 2026-09-26): For reduced normal-crossing projective pairs that are smooth on every boundary stratum over the open base, OpenAI claims logarithmic Kodaira additivity, including the cases with negative-infinite dimension. Manuscript 5 (family 034, 2026-09-24): The Campana--Peternell Kodaira inequality and the specified Schnell fiber-space conclusion are claimed from a pseudo-effective difference between a canonical multiple and the chosen effective or pulled-back divisor.

Mathematical statement of the claimed result

Manuscript 1.

For a projective connected-fiber surjective morphism $f:U\to V$ with $U$ and $V$ smooth, connected and complex quasi-projective. Assume $\overline\kappa(V)\ge0$. With geometric generic fiber $F$, OpenAI claims \[ \overline\kappa(U)\ge\kappa(F)+\max\{\overline\kappa(V),\operatorname{Var}(f)\}. \] Logarithmic Kodaira dimension is $\kappa(\overline U,K_{\overline U}+D)$ on a smooth reduced-normal-crossing compactification. Birational variation minimizes $\operatorname{trdeg}_{\mathbb C}k$ over algebraically closed fields $k$ permitting a birational definition of the entire generic fiber. The projective specialization asserts $\kappa(X)\ge\kappa(F)+\max\{\kappa(Y),\operatorname{Var}(f)\}$ for a connected-fiber morphism $f:X\to Y$ between smooth connected projective varieties with $\kappa(Y)\ge0$.

Manuscript 2.

Let $X$ be smooth compact connected in Fujiki class $\mathcal C$ (bimeromorphic to a compact Kähler space), with rational effective simple-normal-crossing boundary $\Delta$ of coefficients in $[0,1]$. For a connected-fiber surjective holomorphic $f:X\to Y$ onto normal compact irreducible $Y$, and a very general smooth fiber $F$, OpenAI claims \[ \kappa(X,K_X+\Delta)\ge\kappa(F,K_F+\Delta|_F)+\kappa(f\mid\Delta). \] The last term is the birationally invariant neat orbifold-base Kodaira dimension: for each base prime, the infimum of the multiplicities of its dominating pullback components weighted by reciprocal $1-\Delta$ coefficients determines boundary coefficient $1-1/m$. It uses $\kappa(\mathrm{point})=0$ and $(-\infty)+a=-\infty$, including $a=-\infty$.

Manuscript 3.

Over any algebraically closed characteristic-zero field, for a surjective projective morphism $f:X\to Z$ whose fibers are connected and whose source and target are smooth connected projective varieties and geometric generic fiber $F$, OpenAI claims \[ \kappa(X)\ge\kappa(F)+\kappa(Z). \] Kodaira dimension is the maximal image dimension of positive complete pluricanonical systems, with $-\infty$ for empty systems; $(-\infty)+a=-\infty$ and a point has dimension zero. This is the manuscript's ordinary Iitaka theorem. Its referenced auxiliary Hodge-line input assumes a rationally polarized pure variation whose highest nonzero filtration is a line and unipotent boundary monodromy: its Deligne--Schmid extension is nef and is big precisely when the highest-line period map is generically immersive. That technical input is not the main subadditivity statement.

Manuscript 4.

Take a connected-fiber surjective morphism $f:X\to Y$ of smooth connected projective complex varieties, reduced simple-normal-crossing divisors $E,D$ with $\operatorname{Supp}(f^*D)\subseteq\operatorname{Supp}E$, and $V=Y\setminus D$. Require $X$ and each nonempty boundary intersection stratum of $E$, restricted over $V$, to be smooth over $V$. For a very general fiber $F$ and $E_F=E|_F$, OpenAI claims \[ \kappa(X,K_X+E)=\kappa(Y,K_Y+D)+\kappa(F,K_F+E_F). \] The independent main theorem proves the upper inequality; equality uses the companion logarithmic subadditivity result. The zero divisors and $-\infty$ Kodaira dimensions are included. Smoothness is required over the original open base, with no assertion of smoothness above $D$.

Manuscript 5.

Let $f:X\to Y$ be a connected-fiber surjective morphism of smooth connected projective complex varieties, with smooth geometric generic fiber $F$ satisfying $\kappa(F)=0$. If some ample Cartier divisor $H$ on $Y$ and integer $m_0>0$ have $m_0K_X-f^*H$ pseudo-effective, OpenAI claims \[ \kappa(X)=\dim Y. \] The manuscript also invokes its companion's canonical good-model theorem: every smooth connected projective complex $T$ with pseudo-effective $K_T$ has a $K_T$-negative birational contraction to a normal projective rational-factorial klt $V$ with $K_V$ semiample, and on a common smooth resolution $p:W\to T$, $q:W\to V$, \[ p^*K_T=q^*K_V+E,\qquad E\ge0,\quad q_*E=0. \] The good-model statement is explicitly credited to the companion theorem, not presented as an independent proof here.

Outline of the argument

Manuscript 1 (family 033, 2026-09-26): Canonical root covers isolate discriminant and moduli terms without losing pluriform orders. Flatness of the relevant Hodge lines is converted into birational constancy by singular-weight extension estimates and a big adjoint model; marked-family positivity and finite-cover descent then account for whole-fiber variation. Manuscript 2 (family 033, 2026-09-26): A relative Iitaka reduction first replaces the original fibers by the part for which general-type addition applies. Root-cover comparisons retain the boundary multiplicities, and positivity of the resulting period line supplies the missing contribution from the base.

Manuscript 3 (family 033, 2026-09-27): A canonical root cover of a Kodaira-zero fiber gives a one-dimensional top Hodge space. The argument distinguishes the full period image from the rank seen by that line, establishes the required positivity on the compactification, and applies the canonical bundle formula to recover subadditivity. Manuscript 4 (family 033, 2026-09-26): Iterated logarithmic cotangent and Higgs maps turn a selected pluriform into base sections with controlled coefficients. Period-image positivity and a canonical bundle formula supply the reverse inequality; combining it with subadditivity yields equality under the stated stratum-smoothness assumptions.

Manuscript 5 (family 034, 2026-09-24): The total-space canonical model, with its good-model property, supplies the geometric reduction. Cartier intersection theory and proper pushforward compare the base with the canonical image without requiring either map to factor through the other; transfer of sections then yields the Kodaira and fiber formulas.

Formal verification

For family 033, the scope page documents the following selections. Only the case of a negative-infinite logarithmic fiber dimension is selected; it forces negative-infinite dimension upstairs and vanishing of positive pluriforms. Finite-dimensional fibers and the negative-base case remain outside the linked statement. Manuscript 1 is not among the papers selected by that page, so the family link does not formalize its main conclusion. It does not establish formal coverage of the complete problem clause. Manuscript 2 is not among the papers selected by that page, so the family link does not formalize its main conclusion. It does not establish formal coverage of the complete problem clause. Manuscript 3 is not among the papers selected by that page, so the family link does not formalize its main conclusion. It does not establish formal coverage of the complete problem clause. This entry does not assert that the selected declarations formalize every part of manuscript 4's main conclusion. Comparator files: LogKodairaFiberNegative.lean. Declarations: OAI.ReverseLogKodaira.SmoothProjectiveVariety.veryGenerally_fiber_negative_branch.

Family 034 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

• OpenAI, Logarithmic Kodaira dimension and whole-fiber variation, 2026-09-26; citation key OAI:Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:projective-variation; Statement source: thm:main.
• OpenAI, Orbifold and logarithmic Iitaka subadditivity, 2026-09-26; citation key OAI:Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: main:orbifold.
• OpenAI, Projective Hodge lines and ordinary Iitaka subadditivity, 2026-09-27; citation key OAI:Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:ordinary.
• OpenAI, The reverse logarithmic Kodaira inequality and additivity, 2026-09-26; citation key OAI:The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: cor:additivity.
• Family 033: Lean scope.
• Comparator LogKodairaFiberNegative.json.
• Comparator LogKodairaFiberNegative.lean.
• OpenAI, Schnell fiber spaces and good canonical models, 2026-09-24; citation key OAI:Schnell-fiber-spaces-and-good-canonical-models-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:canonical-good-models; Statement source: thm:schnell.
• OpenAI catalogue entry.

Iitaka Subadditivity Conjecture for Kodaira Dimension

Definitions and mathematical statement

For a smooth projective variety $Z$, its Kodaira dimension $\kappa(Z)$ is the growth exponent of the spaces $H^0(Z,mK_Z)$, with value $-\infty$ if all positive pluricanonical spaces vanish. For the surjective morphism and general fiber in the record, the target is \[ \kappa(X)\ge\kappa(Y)+\kappa(F). \] The usual fibration convention uses connected fibers and the characteristic-zero setting of the cited source; the catalogue's abbreviated sentence does not separately spell out those conventions. When one term on the right is $-\infty$, the inequality is interpreted in the extended ordered sense.

Formulation and concept references: [S1].

Short English statement

Is the Kodaira dimension of a projective fibration at least the sum of those of its base and general fiber?

Research attempt

Current frontier and source audit.

Birkar's introduction explicitly works over an algebraically closed field of characteristic zero and defines an algebraic fibre space to have connected fibres. Those are the conventions incorporated by the present entry. Theorem 1.3 proves the conjecture for $\dim X\leq6$; the introduction also records the good-minimal-model fibre case. Neither unrestricted existence of good minimal models nor abundance is assumed here. [S1]

Established methods include positivity of pluricanonical direct images; Cao--Păun prove the abelian-base case, with a logarithmic extension. A December 2025 manuscript by Ammar states the case $\dim\operatorname{alb}_Y(Y)\geq\dim Y-2$. That recent result is reported as a manuscript theorem, not used as an independently audited ingredient. The complete arbitrary-base, arbitrary-fibre claim is not supplied by these hypotheses. [E1][E2]

Attack route.

A basis of pluricanonical forms on the generic fibre extends meromorphically over the base. The central question is the cost of removing its poles. The route below turns this into a precise section-counting lemma, then asks whether positivity can bound that cost in directions supported by the base's canonical system. A second route tests descent through a finite étale product cover. A third, numerical-positivity-only route is rejected by a degree-zero line-bundle example.

Attempt: a pole-cost criterion.

Write $K_{X/Y}=K_X-f^*K_Y$ and \[ E_m=f_*\mathcal O_X(mK_{X/Y}),\qquad r_m=\operatorname{rank}E_m=h^0(F,mK_F). \] The last equality is generic base change; the general fibre is understood in that sense for each $m$. The sheaf is torsion-free, and all generic-rank arguments below take place on a dense open subset before being interpreted as morphisms of coherent sheaves on $Y$. Projection formula gives \begin{equation}\label{wsref-0-65-71-3a-33-30-30-30-35-31-36-38-3a-70-72-6f-6a-65-63-74-69-6f-6e} h^0(X,mK_X)=h^0(Y,E_m\otimes\mathcal O_Y(mK_Y)). \end{equation} If either Kodaira dimension on the right of the target is $-\infty$, the inequality is automatic. Assume from now on that both are nonnegative.

Lemma 1 (canonical-direction pole cost). Choose an integer $q>0$ and an effective divisor $D\sim qK_Y$. Suppose along an unbounded sequence of multiples $m$ of $q$ there are integers $b_m\geq0$ with $b_m=o(m)$ and morphisms \begin{equation}\label{wsref-0-65-71-3a-33-30-30-30-35-31-36-38-3a-69-6e-6a-65-63-74-69-6f-6e} \mathcal O_Y(-b_mD)^{\oplus r_m}\hookrightarrow E_m \end{equation} that have rank $r_m$ at the generic point. Choose $q$ sufficiently divisible that the standard lower plurigenus-growth bounds for $F$ and $Y$ hold on its multiples. Then $\kappa(X)\geq\kappa(F)+\kappa(Y)$.

Proof. A generically injective map from the displayed locally free sheaf is injective everywhere as a sheaf map: its kernel is torsion-free and zero at the generic point. Tensoring by $\mathcal O_Y(mK_Y)$ and taking sections gives \[ h^0(X,mK_X)\geq r_m\, h^0\left(Y,(m-qb_m)K_Y\right). \] Eventually $m-qb_m\geq m/2$ and it is still a multiple of $q$. Thus the right side is at least $c m^{\kappa(F)+\kappa(Y)}$ along the sequence. This yields the asserted Kodaira dimension. The lower growth estimates need no finite generation theorem: choose pluricanonical sections whose ratios realize the Iitaka dimension in a fixed degree; independent monomials in those ratios give the required polynomial lower bound in multiples of that degree. $\square$

The hypothesis is concretely testable: $r_m$ global sections of $E_m(b_mD)$ forming a generic basis yield (eq:30005168:injection). It is stronger than what the conjecture requests and has not been shown to hold in general. In particular, when $K_Y$ is torsion one may have $D=0$, leaving no ample twisting room at all.

Attempt: what sublinear ample twisting actually proves.

A more plausible output of generic generation is an injection \[ \mathcal O_Y(-b_mH)^{\oplus r_m}\hookrightarrow E_m, \qquad b_m=o(m), \] for a fixed ample effective divisor $H$. This suffices when $K_Y$ is big. Indeed, choose $q$ and $G\geq0$ with $qK_Y\sim H+G$. For $m=q\ell$, \[ mK_Y-b_mH\sim(\ell-b_m)H+\ell G. \] For sufficiently large $m$, $\ell-b_m\geq\ell/2$, so multiplication by the section of $\ell G$ and ample section growth give \[ h^0(Y,mK_Y-b_mH)\geq c m^{\dim Y}. \] The previous proof then gives $\kappa(X)\geq\kappa(F)+\dim Y$. This isolates, rather than re-proves from scratch, the geometric room available in the established general-type-base case. [S1]

That reasoning fails sharply for a non-big base. On an elliptic curve, $K_Y\sim0$ and an effective divisor $H$ of positive degree satisfies \[ h^0(Y,mK_Y-H)=h^0(Y,-H)=0 \] for every $m$. Even constant pole cost $b_m=1=o(m)$ cannot be absorbed. Thus "the poles grow sublinearly" is insufficient unless their direction is also controlled. Lemma 1 is one possible replacement, but its canonical-direction hypothesis is exactly an unresolved part of this route.

Attempt: finite étale descent without a false invariants count.

The following restricted case can be handled completely. Suppose a finite étale cover $Y'\to Y$ trivializes the fibration: \[ X'=X\times_Y Y'\simeq F\times Y'. \] Then $\kappa(X)=\kappa(F)+\kappa(Y)$.

Here are the descent details. For a finite Galois étale cover $\pi:Z\to W$ with group $G$, étaleness identifies $K_Z=\pi^*K_W$ and \[ R(W,K_W)=R(Z,K_Z)^G. \] For any element $s$ of the latter graded canonical ring, the monic polynomial $\prod_{g\in G}(T-gs)$ has invariant coefficients. Hence $R(Z,K_Z)$ is integral over its invariant subring. Their fraction fields have equal transcendence degree, so their Kodaira dimensions agree. If there are no positive-degree sections downstairs, the norm of a nonzero homogeneous section upstairs would produce one, so this also handles $\kappa=-\infty$. Passing through a Galois closure handles a non-Galois finite étale cover. None of this asserts that invariant sections occupy a fixed fraction of every individual graded piece.

On a product, the complete pluricanonical section space is the tensor product of the two factor spaces. Its rational map is the Segre product of the two pluricanonical maps wherever defined. Choosing a common sufficiently divisible degree gives $\kappa(F\times Y')=\kappa(F)+\kappa(Y')$, with the usual $-\infty$ convention. Applying étale invariance to $X'\to X$ and $Y'\to Y$ proves the restricted statement. This is a self-contained recovery of a familiar isotrivial situation, not a proof for arbitrary fibrations.

Stress test: positivity is not a supply of untwisted sections.

Let $L$ be a non-torsion degree-zero line bundle on an elliptic curve. It is nef and numerically trivial, yet $H^0(Y,L^m)=0$ for every $m>0$. A nonzero section would have an effective zero divisor of degree zero, hence no zeros, and would trivialize $L^m$, a contradiction. Consequently one cannot replace the generic-generation hypothesis in Lemma 1 by numerical semipositivity, even for line bundles. This example is not asserted to be a pluricanonical direct image of a counterexample fibration: the geometric origin of $E_m$ is exactly the additional structure a successful proof must exploit. The direct-image methods in Cao--Păun use much more than arbitrary numerical positivity. [E1]

Likewise, a generically chosen fibre basis can always be cleared of poles with some ample twist depending on $m$, but that observation supplies neither a sublinear bound on the twist nor support in a canonical divisor. Also, replacing a fibration by a finite ramified cover changes the canonical divisor by ramification; the étale descent proof does not apply verbatim to such a cover.

Outcome.

Outcome: Conditional reduction.

The attempt proves a canonical-direction pole-cost criterion and explains exactly why the apparently weaker sublinear ample-pole criterion only works directly over a big-canonical base. It also gives a complete finite-étale-product special case. What is missing is geometric control of the pluricanonical direct images strong enough to remove poles on a general non-big base. Assuming all good minimal models or abundance would use another unresolved conjectural package, not complete this argument. No novelty claim is made for the special cases or standard descent facts.

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Sources

• S1 C. Birkar, Iitaka conjecture C_{n,m} in dimension six, arXiv:0806.4413 [math.AG], 2008. https://arxiv.org/abs/0806.4413. Catalogue formulation source.
• E1 Junyan Cao and Mihai Păun, Kodaira dimension of algebraic fiber spaces over abelian varieties, Invent. Math. 207 $2017$, 345--387; arXiv:1504.01095v3 $2016$, Theorem 1.1 and Sections 2--4. https://arxiv.org/html/1504.01095v3.
• E2 Houari Benammar Ammar, Note on Iitaka Conjecture $C_{n,m}$, arXiv:2510.06412v2 (12 December 2025), Theorem 1.2. Recent manuscript result, not used in any proof in this notebook. https://arxiv.org/html/2510.06412v2.