Iitaka subadditivity, variation, and logarithmic additivity

1 · The essentials

These manuscripts ask how the complexity of a geometric space relates to the complexity of its parts. They study maps that organize one space into a family of smaller spaces, called fibers, indexed by another space, called the base. The central claim is that the complexity of the total space is at least the combined complexity of the base and a typical fiber. The main manuscript treats compact complex spaces with certain permitted boundaries, and the main result has a Lean formalization. A companion manuscript strengthens the lower bound by also taking account of how much the whole fiber changes across the family. Another obtains an exact addition rule under extra smoothness and boundary conditions. These conclusions apply to the particular classes of spaces and maps specified in the manuscripts, not to every geometric family.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, think of a patterned quilt: one can examine the pattern within each patch, the arrangement of the patches, and how the patch designs change across the quilt. The manuscripts relate corresponding kinds of geometric complexity, but their mathematical measure is not a literal count of patterns.

Why it matters

The family connects three questions: a lower bound for complexity, a stronger bound that detects changing fibers, and an exact addition rule. The manuscripts describe their conclusions as resolving Campana’s orbifold subadditivity conjecture and Popa’s logarithmic variation conjecture, with Popa’s logarithmic additivity conjecture resolved in the stated projective, reduced-boundary, stratum-smooth setting. A further companion treats b-semiampleness for compact log-smooth Kähler fibrations without assuming projectivity or Campana’s orbifold Iitaka conjecture.

2 · Going further

The basic ordinary inequality is $κ (X) \geq κ (F) + κ (Z)$ , where $F$ is the geometric generic fiber of a map $f : X \to Z$ . Manuscript 4 shows this in the manuscript for surjective projective morphisms with connected fibers between smooth connected projective varieties over algebraically closed fields of characteristic zero. The main manuscript instead treats Campana’s orbifold Iitaka subadditivity conjecture on compact manifolds in Fujiki class $C$ , with rational simple-normal-crossing boundaries, including coefficient one. Its abstract states that ordinary and logarithmic subadditivity follow.

Manuscript 2 strengthens logarithmic subadditivity. For a projective surjective morphism $f : U \to V$ with connected fibers between smooth complex quasi-projective varieties, assuming $κ ˉ (V) \geq 0$ , it shows in the manuscript that $κ ˉ (U) \geq κ (F) + max {κ ˉ (V), Var (f)}$ . Here $F$ is the general fiber, and variation measures the birational field of definition of the whole geometric generic fiber. This whole-fiber specification is part of the stated result, rather than an unspecified measure of change.

Manuscript 3 supplies the complementary direction: a reverse logarithmic Kodaira inequality. It assumes a surjective connected-fiber morphism $f : (X, E) \to (Y, D)$ of smooth projective reduced simple-normal-crossing pairs, with $Supp (f^{*} D) \subseteq Supp E$ , and requires $X$ and every boundary stratum to be smooth over $Y ∖ Supp D$ . The argument structure stated in the abstract is to combine this reverse inequality with logarithmic subadditivity, obtaining additivity, including both negative-infinity cases. The supplied abstracts do not describe the internal arguments for the other inequalities, so no further proof mechanism can be extracted from them.

Manuscript 5 concerns b-semiampleness. It assumes a surjective holomorphic map $f : Y \to X$ with connected fibers between smooth compact connected Kähler manifolds, an effective rational simple-normal-crossing divisor $Δ$ with coefficients in $[0, 1]$ , and $K_{Y} + Δ \sim_{Q} f^{*} L$ for $L \in Pic (X)_{Q}$ . It shows in the manuscript that there is a smooth compact Kähler modification $S \to X$ on which the threshold-moduli line satisfies $M_{S_{1}} = ν^{*} M_{S}$ under every further smooth compact Kähler modification $ν : S_{1} \to S$ , and some positive multiple of $M_{S}$ is represented by a holomorphic line bundle generated by global sections. Horizontal coefficient-one components are allowed. The abstract does not state that this result is a step in the main manuscript’s argument.

Useful notions

Maps, bases, and fibers
Smooth complex and projective varieties
Kodaira and logarithmic Kodaira dimension
Divisors and simple-normal-crossing boundaries
Line bundles and global sections

What was known before / what is new

Before

The source identifies ordinary Iitaka subadditivity and Campana’s orbifold Iitaka subadditivity as conjectures addressed by these manuscripts. It also names Popa’s logarithmic variation and logarithmic additivity conjectures. It provides no account of earlier partial results or their chronology.

New

The main manuscript treats rational simple-normal-crossing boundaries on compact Fujiki-class- $C$ manifolds, explicitly allowing coefficient one. The companions give a logarithmic bound involving whole-fiber variation and an additivity result under the stated boundary and smoothness hypotheses. They also address ordinary subadditivity over algebraically closed characteristic-zero fields and b-semiampleness in the compact log-smooth Kähler setting.

Limits and assumptions

The hypotheses differ across the manuscripts and cannot be interchanged. In particular, the variation inequality assumes a projective morphism and a base of nonnegative logarithmic Kodaira dimension; the additivity result assumes projective reduced simple-normal-crossing pairs and smoothness of the total space and every boundary stratum away from the base boundary. The b-semiampleness statement requires the specified rational pullback relation for $K_{Y} + Δ$ . The main result has a Lean formalization, but the supplied material does not identify its exact scope or establish formal verification of every companion claim. Those companion results are therefore reported as shown in their manuscripts. The source does not specify what remains open outside these settings.

Mini glossary

Fiber: A fiber is the part of the source space lying over a point of the base under the given map.
Kodaira dimension: Kodaira dimension, written $κ$ , is the geometric complexity measure compared in the ordinary subadditivity inequality.
Logarithmic Kodaira dimension: Logarithmic Kodaira dimension, written $κ ˉ$ , is the corresponding measure used in the logarithmic statements.
Whole-fiber variation: In Manuscript 2, variation measures the birational field of definition of the whole geometric generic fiber.
Subadditivity: Here subadditivity means that the dimension measure of the total space is at least the sum of the corresponding measures of the fiber and base.
Additivity: Here additivity is the exact addition rule obtained by combining logarithmic subadditivity with the reverse inequality under Manuscript 3’s hypotheses.

Keywords : Iitaka subadditivity · Whole-fiber variation · Logarithmic additivity

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves Campana's orbifold Iitaka subadditivity conjecture for smooth Fujiki-class- $C$ manifolds with rational simple-normal-crossing boundaries. For projective fibrations $f : U \to V$ of smooth complex quasi-projective varieties with connected fibers, general fiber F, and $κ ˉ (V) \geq 0$ , proves Popa's inequality $κ ˉ (U) \geq κ (F) + max {κ ˉ (V), Var (f)}$ , where variation measures the whole geometric generic fiber.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

10/09/2026

To be verified

We prove the compact log-smooth Kähler case of b-semiampleness. Let $f : Y \to X$ be a surjective holomorphic map with connected fibers between smooth compact connected Kähler manifolds, and let Δ be an effective rational divisor with simple normal crossing support and coefficients in $[0, 1]$ , with $K_{Y} + Δ \sim_{Q} f^{*} L$ for $L \in Pic (X)_{Q}$ . There is a smooth compact Kähler modification $S \to X$ for which the threshold-moduli line satisfies $M_{S_{1}} = ν^{*} M_{S}$ in $Pic (S_{1})_{Q}$ for every smooth compact Kähler modification $ν : S_{1} \to S$ , and some positive multiple of M_S is represented by a holomorphic line bundle generated by global sections. Horizontal components of coefficient one are allowed; neither projectivity nor a Campana orbifold Iitaka hypothesis is assumed.

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