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Math-2481 · mathematics · Algebraic geometry

Known Cases of Iitaka Subadditivity

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For a surjective morphism $f:X\to Y$ of smooth projective varieties over a field of characteristic zero, with connected fibers and general fiber $F$, must the Kodaira dimensions satisfy \[\kappa(X)\geq\kappa(Y)+\kappa(F)?\] The Kodaira dimension measures the asymptotic growth of pluricanonical sections; the question is ordinary Iitaka subadditivity.

Formulation references: [S1], [S2].

Short English statement

For a surjective morphism $f:X\to Y$ of smooth projective varieties over a field of characteristic zero, with connected fibers and general fiber $F$, must the Kodaira dimensions satisfy \[\kappa(X)\geq\kappa(Y)+\kappa(F)?\] The Kodaira dimension measures the asymptotic growth of pluricanonical sections; the question is ordinary Iitaka subadditivity.

Sources

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Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2481-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2481

The ordinary Iitaka theorem claims the displayed inequality for projective fibrations with connected fibers in characteristic zero, without a dimension bound. Its algebraically closed base convention extends to the usual characteristic-zero formulation by base change of pluricanonical section spaces and the geometric-fiber convention. This is the full mathematical assertion currently recorded on this page. The variation and orbifold papers give further extensions with their stated conventions.

Known Cases of Iitaka Subadditivity

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.

Mathematical statement of the claimed result

Mathematical claim 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 $\kappa(Y)\ge0$.
Mathematical claim 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 $a=-\infty$.
Mathematical claim 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.

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.

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. Manuscript 2 is not among the papers selected by that page, so the family link does not formalize its main conclusion. Manuscript 3 is not among the papers selected by that page, so the family link does not formalize its main conclusion.

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.

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