CATALOGUE 033
Campana's orbifold Iitaka conjecture and logarithmic subadditivity
Five papers on subadditivity and variation of Kodaira dimension, logarithmic additivity, and moduli-line semiampleness. Start with the official-order inventory, then follow the result-level connections into the relevant proof sections.
Papers in official catalogue order
These summaries give the principal scope; consult the cited original results for the complete hypotheses and statements.
Page counts include references. On narrow screens, scroll the table sideways.
| # | Manuscript and overview | Principal claim | Role and available overview |
|---|---|---|---|
| 01 | Orbifold and logarithmic Iitaka subadditivity →Short name: Orbifold Iitaka2026-09-26 · 55 pagesOriginal paper (GitHub) ↗ | Orbifold subadditivity for class-C SNC pairs, including coefficient one; logarithmic and ordinary characteristic-zero consequences. Theorem 1.1; Corollaries 6.2–6.3 · pp. 2–3, 40–41 ↗ | Section comparison on the relative Iitaka base, Hodge adjoint positivity, and twist cancellation; distinguishes the results supplied to later papers. Read the proof overview → |
| 02 | Logarithmic Kodaira dimension and whole-fiber variation →Short name: Whole-fiber variation2026-09-26 · 75 pagesOriginal paper (GitHub) ↗ | A lower bound involving variation of the whole geometric generic fiber over a base of nonnegative logarithmic Kodaira dimension. Smooth-family consequences use additional inputs. Theorem 1.1; Corollaries 1.2–1.3 · pp. 2, 5 ↗ | Builds a parameter field from sections and descends the whole fiber using root-cover constancy, markings, and finite-cover descent. Read the proof overview → |
| 03 | The reverse logarithmic Kodaira inequality and additivity →Short name: Reverse log additivity2026-09-26 · 51 pagesOriginal paper (GitHub) ↗ | An upper bound for projective pairs whose boundary strata are smooth over the open base; the logarithmic lower bound then gives additivity. Theorem 1.1; Corollary 1.2; Proposition 1.3 · pp. 2–3 ↗ | Constructs a base section retaining a prescribed zero and removes a numerical twist for a positive-dimensional period quotient. Read the proof overview → |
| 04 | Projective Hodge lines and ordinary Iitaka subadditivity →Short name: Projective Hodge lines2026-09-27 · 40 pagesOriginal paper (GitHub) ↗ | Ordinary characteristic-zero subadditivity and adjoint positivity for variations whose entire highest Hodge piece is a line. Theorems 1.1–1.2 · p. 2 ↗ | An alternative proof through boundary and finite-ramification rank loss, canonical bundle data, two-base comparison, and interpolation. Read the proof overview → |
| 05 | B-semiampleness for compact log-smooth Kähler fibrations →Short name: Kähler B-semiampleness2026-09-10 · 38 pagesOriginal paper (GitHub) ↗ | Stabilization and semiampleness of the moduli b-line for compact Kähler fibrations with relative log-canonical class pulled back from the base. Theorem 1.1(a)–(b) · p. 3 ↗ | Identifies thresholds with Hodge extensions and descends global generation of the specified line from semiample product factors. Read the proof overview → |
Connections within 033
Orbifold Iitaka supplies two different inputs to the main proof of Whole-fiber variation. Reverse log additivity first proves its upper bound independently; the lower bound enters only its additivity corollary. Each paper name below links to the relevant article section; result numbers link to the pinned original.
| Source result | Recipient and use | Content supplied |
|---|---|---|
| Orbifold Iitaka · Corollary 6.2 · pp. 40–41 | Whole-fiber variation · Theorem 2.1; Lemmas 2.8–2.9 · pp. 6, 10–12 | Direct input. Supplies the logarithmic lower bound and forces two nonnegative terms on an Iitaka fiber to vanish, enabling the parameter-field construction. |
| Orbifold Iitaka · Theorem 3.1 · pp. 17–18 | Whole-fiber variation · Theorem 3.8 / Proposition 3.9; Proposition 3.12 · pp. 21, 23 | Direct input. Gives adjoint positivity for the restricted Hodge line. The section bound forces its adjoint base to be a point and makes M rationally trivial. |
| Orbifold Iitaka · Corollary 6.2 · pp. 40–41 | Reverse log additivity · Corollary 1.2; Theorem 7.8 / §7.6 · pp. 2, 50 | Input to a consequence. Supplies the lower bound for the same pair and very general fiber; combining it with Reverse log additivity’s upper bound gives additivity. |
The first two inputs enter the parameter-field and restricted-line stages; neither by itself gives constancy of the whole fiber. The overview in Whole-fiber variation retains the subsequent constancy, marking, and descent steps. The upper-bound argument in Reverse log additivity treats all signs of the Kodaira dimensions before introducing the lower bound.
The additional relative-Iitaka construction in Whole-fiber variation §9 uses three further inputs. It is separate from the main proof, which ends in §7.4.
| Source result | Recipient and use | Content supplied |
|---|---|---|
| Orbifold Iitaka · Proposition 2.7 · pp. 10, 16–17 | Whole-fiber variation · Proposition 9.1(i), (iii) · pp. 69–70 | Input within an additional result. Preserves relative and absolute section systems and their ratios on the fixed original X, giving relative bigness in the additional ordinary construction. |
| Orbifold Iitaka · Lemma 7.1 / Corollary 7.2 · pp. 43–46 | Whole-fiber variation · Proposition 9.1(iv), (v) · pp. 69–70 | Input within an additional result. For a reduced boundary, the least-index root cover makes the entire top piece rank one and realizes the normalized M as a line in a pure integral variation. |
| Orbifold Iitaka · Lemma 7.4 · pp. 46–47 | Whole-fiber variation · Proposition 9.1: initial normalized boundary · pp. 69–70 | Input within an additional result. Normalizes B by comparing the minimum over actual source components with the lct over all valuations, on the same model as the section and Hodge-line comparison. |
These inputs preserve sections, identify the entire highest line for a reduced boundary, and normalize the boundary on a common model. Their stated uses were compared; this does not verify the full valuation arguments or all of §§8–9.
Connections with catalogue 034
Different results travel in the two directions. The return from Log Abundance supplies good models to a smooth-family consequence in Whole-fiber variation, not to its main variation theorem.
| Source result | Recipient and use | Content supplied |
|---|---|---|
| Orbifold Iitaka · Corollary 6.2 · pp. 40–41 | Log Abundance · Theorem 1.2 → Lemma 6.1 · pp. 6, 30–31 | Direct input. Applied with zero boundaries to the Albanese fibration, it forces irregularity zero for a nonvanishing counterexample; this is Log Abundance’s sole use of subadditivity. |
| Log Abundance · Corollary 11.2 · pp. 73–74 | Whole-fiber variation · Corollary 1.3 proof · p. 5 | Input to a consequence. Provides a semiample good model and effective exceptional comparison for each closed fiber whose canonical class is pseudo-effective by BDPP; canonicality then permits Taji’s theorem. |
| Orbifold Iitaka · Lemma 2.6 · p. 9 | Kähler abundance · §5.1; reused in Lemma 5.7 · pp. 91, 98 | Direct input. Passes to the relative Iitaka base with Kodaira-zero general fiber so that lower-dimensional induction can continue. |
| Orbifold Iitaka · Lemma 5.2 · p. 32 | Kähler abundance · §5.1 · p. 91 | Direct input. Compares the big fibers, after lowering the horizontal boundary, with a projective stable family, excluding the big case at algebraic dimension zero. |
| Orbifold Iitaka · Proposition 2.7 / Theorem 3.1 · pp. 10–17, 17–18 | Kähler abundance · Lemma 5.2 · pp. 91–94 | Direct input. Compares section spaces and the actual rational Hodge line, giving M=p*P and adjoint bigness on the projective base; signs above codimension two require separate work. |
| Orbifold Iitaka · Lemma 3.3 · pp. 18–19 | Kähler abundance · Lemma 5.7, interpolation · pp. 98–99 | Direct input. Produces a section after a big base twist; interpolation with another effective divisor forces positive Kodaira dimension. |
The Albanese application in Log Abundance uses zero boundaries; its manuscript also allows Hacon–Popa–Schnell for that application. The Kähler rows concern specified auxiliary results, with the receiving manuscript’s own assumptions retained. No transitive path through Log Abundance is recorded as a direct input to Schnell.
Explicit premise. Orbifold Iitaka, Theorem 1.1 · pp. 2–3 matches Conditional Kähler models, Assumption 2.2 · pp. 7–8: arbitrary-dimensional class-C orbifold subadditivity, including coefficient one and the invariant base convention. The other Kähler MMP and abundance premises remain. See the premise statements.
Alternative proofs, inclusion, and background
Projective Hodge lines, Theorem 1.1 · p. 2 and Orbifold Iitaka, Corollary 6.3 · p. 41 prove the same ordinary characteristic-zero subadditivity statement by different routes. The latter also includes the logarithmic statement. The ordinary claim is a zero-boundary case of that logarithmic conclusion; this inclusion of results does not create a proof dependency. Compare the two-base interpolation with the characteristic-zero transfer.
Projective Hodge lines, Lemma 8.4 / Theorem 8.1 · pp. 28–29 is an alternative adjoint comparison in Whole-fiber variation, Remark 3.17 · p. 25. Tensor replacement must be performed again on the restriction; the receiving proof uses the direct comparison. Separately, additivity and the variation lower bound recover the nonnegative-base branch for smooth families with finite nonnegative fiber Kodaira dimension (comparison · p. 5); they do not cover negative bases or every special base.
Weak positivity and interpolation in Orbifold Iitaka and Projective Hodge lines are similar methods with different Hodge-line hypotheses. The root-line calculations in Orbifold Iitaka and Kähler B-semiampleness also use different boundaries: orbifold inf-multiplicity versus the lct discriminant. Kähler B-semiampleness · p. 3 explicitly avoids the orbifold Iitaka theorem; this is not a claim that all possible inter-paper dependencies have been investigated. Its whole proof overview follows its own stabilization and product-factor route.
The connection-form Ax–Schanuel argument in Reverse log additivity is recorded as a method background: the manuscript gives the semisimple case it needs. The external-input discussion distinguishes this from importing an external theorem.
Sources and checking scope
Official catalogue / OpenAI Research Catalog · 033 · p. 5. The field is Algebraic and complex geometry; the inventory retains the official order. The 033 manuscripts are fixed at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Connections to 034 retain its published snapshot adc7f1241b42e322a6451854ab7e4b4c146bf78a (Log Abundance: 2026-09-24; Kähler abundance: 2026-10-04; conditional Kähler models: 2026-10-05).
The revision of 11 October 2026 compared missing proof-target statements and their conclusions against the five pinned PDFs, rechecked their hashes, and matched the recorded connections to the articles. The twelve inputs and one premise match inherit the statement/use checks of 8 October; no new complete verification of external proofs is claimed. Full connection records, including conditions and limits, remain in the source record.
- Orbifold Iitaka: rank loss, the full stable-family construction, valuation/BFMT comparisons, and core consequences.
- Whole-fiber variation: all regularization and L² estimates, the original Hanamura and SGA1 statements, and §§8–9 in full.
- Reverse log additivity: homogeneous replacement, compact flags, full boundary/representation calculations.
- Projective Hodge lines: multivariable extensions, all auxiliary proofs, and the alternatives in §§8–9.
- Kähler B-semiampleness: the complete parameter-space construction, polarization changes, and external compactification theory.
Unlisted relations remain uninvestigated. Statement/use comparison, a proof overview, and independent verification of a complete proof are distinct. No expert review or formal verification is asserted.