This guide may contain errors or inaccurate explanations. Check important claims and proofs against the original papers.
CATALOGUE 034
From a first section to a good model, and from a good model to a uniform Iitaka degree. Follow the claims of 14 manuscripts and the results their arguments use.
This is a reading map of the manuscripts’ claims and uses of cited results. The arrows record selected source-to-use connections; they are not a complete dependency graph or a certification of the proofs.
The summaries give the principal scope, not every technical hypothesis. Follow the theorem reference for the full statement. A uniform degree here is an existence statement unless a manuscript explicitly gives a numerical value.
Page counts include references. On narrow screens, scroll the table sideways.
| # | Manuscript and overview | Principal claim | Role and available overview |
|---|---|---|---|
| 01 | Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivityShort name: Kähler abundance2026-10-04 · 148 pagesOriginal paper (GitHub) ↗ | Under logarithmic Iitaka subadditivity (Assumption 1.1), claims semiampleness of analytically nef adjoints of compact Kähler lc pairs with effective rational boundary in every dimension. Theorem 1.2 · p. 3 ↗ | Uses the projective good-model argument in Log Abundance as the anchor for the fibration and simple-space cases. Read the proof overview → |
| 02 | Uniform indices for semi-log-canonical log Calabi–Yau pairsShort name: slc indices2026-10-05 · 42 pagesOriginal paper (GitHub) ↗ | For fixed dimension and a finite rational coefficient set, claims a common trivializing multiple for connected projective slc log Calabi–Yau pairs. The main theorem covers dimension at least four. Theorem 1.1 · p. 2 ↗ | Adds character bounds and residue gluing across the conductor to the normal lc index theorem. Read the proof overview → |
| 03 | Conditional good minimal models for compact Kähler fourfoldsShort name: Conditional Kähler models2026-10-05 · 138 pagesOriginal paper (GitHub) ↗ | For globally strongly Q-factorial compact Kähler klt fourfold pairs with pseudo-effective adjoint, claims good minimal models under three premises: orbifold subadditivity, the specified MMP, and abundance after nonvanishing. Theorems 1.1–1.2 · p. 4; Assumptions 2.2–2.4 · pp. 8–9 ↗ | Supplies nonvanishing at the nef MMP endpoint, then applies Kähler fourfold abundance after nonvanishing. Its stated conditional scope is retained. Read the proof overview → |
| 04 | Log abundance in characteristic zeroShort name: Log Abundance2026-09-24 · 85 pagesOriginal paper (GitHub) ↗ | Claims semiampleness for nef adjoints of projective lc rational pairs over algebraically closed characteristic-zero fields, and good models for pseudo-effective lc real pairs over C. Theorem 1.1 · p. 2; Theorem 11.1 · p. 73 ↗ | A central qualitative input for models and nonvanishing. Some manuscripts instead use only its independent tools in §§7–9. Read the proof overview → |
| 05 | Minimal metrics and interior injectivity for nef adjointsShort name: Minimal metrics2026-09-27 · 21 pagesOriginal paper (GitHub) ↗ | Claims zero Lelong numbers for minimal metrics on nef projective complex klt adjoints, and H¹ injectivity for SNC interior boundaries with zero-Lelong semipositive endpoint metrics. Theorems 1.1–1.2 · pp. 1–2 ↗ | Provides the analytic input for lifting sections from the whole reduced boundary in fourfold nonvanishing. Read the proof overview → |
| 06 | Fourfold nonvanishing by minimal metrics and moving jetsShort name: Fourfold nonvanishing2026-09-27 · 38 pagesOriginal paper (GitHub) ↗ | Claims canonical nonvanishing for smooth projective complex fourfolds with pseudo-effective canonical divisor; Hashizume’s reduction gives sections for nef lc rational adjoints in dimension at most four. Theorem 1.1 / Corollary 1.2 · p. 2 ↗ | The first-section step, using metrics, interior injectivity, and Log Abundance’s finite-data Frobenius theorem. Read the proof overview → |
| 07 | Lifting sections from the reduced support of an adjointShort name: Supported lifting2026-09-27 · 40 pagesOriginal paper (GitHub) ↗ | For projective Q-factorial dlt pairs, claims positive Iitaka dimension from a nonzero effective adjoint multiple supported in the floor and semiample on its entire reduced support. Deduces abundance after nonvanishing through dimension four. Theorems 1.1–1.3 / Corollary 1.4 · p. 2 ↗ | The step from an existing section to semiampleness. Fourfold nonvanishing enters Corollary 1.4, separately from the proof of Theorem 1.2. Read the proof overview → |
| 08 | Schnell fiber spaces and good canonical modelsShort name: Schnell2026-09-24 · 11 pagesOriginal paper (GitHub) ↗ | Claims a twisted nonvanishing inequality for smooth projective fibre spaces with general fibre of Kodaira dimension zero, with Campana–Peternell consequences. See the overview for the exact hypotheses. Theorem 1.1 / Corollaries 1.2–1.3 · pp. 1–2 ↗ | Uses the good canonical model supplied by Log Abundance to obtain a contradiction between mixed-intersection signs. Read the proof overview → |
| 09 | Uniform log Iitaka fibrations and bounded moduli denominatorsShort name: Log Iitaka2026-10-04 · 32 pagesOriginal paper (GitHub) ↗ | For fixed dimension at least five and a finite rational coefficient set, claims that one rounded complete system generates the full Iitaka field of every projective lc pair with nonnegative Iitaka dimension. Theorem 1.1 · p. 2; Theorem 4.2 · pp. 7–8 ↗ | Uses Log Abundance, normal lc indices, and arithmetic Stein degrees to control moduli denominators in the presence of horizontal boundary. Read the proof overview → |
| 10 | Uniform Pluricanonical Iitaka FibrationsShort name: Uniform Iitaka2026-10-03 · 44 pagesOriginal paper (GitHub) ↗ | Claims a dimension-only pluricanonical Iitaka degree for smooth projective varieties with nonnegative Kodaira dimension, and uniform normal lc log Calabi–Yau indices for rational DCC coefficient sets. Theorems 1.1–1.2 · p. 2 ↗ | Inducts simultaneously on Iitaka degrees and normal lc indices. The index theorem feeds the Log Iitaka and slc-index manuscripts. Read the proof overview → |
| 11 | Relative denominators and effective systems for log Calabi-Yau fibrationsShort name: Relative denominators2026-09-27 · 23 pagesOriginal paper (GitHub) ↗ | For projective lc log Calabi–Yau fibrations of total dimension at most four with finite rational coefficients, claims an exact pullback presentation and a uniform b-Cartier multiple of the moduli b-divisor. Theorem 1.1 · p. 2; Proposition 6.1 · p. 15 ↗ | For a big base divisor, supplies a uniform complete system generating the base field; a relative input to effective fourfold Iitaka fibrations. Read the proof overview → |
| 12 | Arithmetic Stein-degree bounds for log Calabi–Yau pairsShort name: Stein degree2026-09-25 · 15 pagesOriginal paper (GitHub) ↗ | For projective lc log Calabi–Yau pairs over a characteristic-zero field k with H⁰(O_X)=k, claims a bound depending on dimension and t for the constant-field degree of boundary components of coefficient at least t. Theorem 1.1 · p. 1 ↗ | Controls finite field extensions in Stein factorization when adjunction or residue passes to a boundary component. Read the proof overview → |
| 13 | Uniform effective log Iitaka fibrations for fourfoldsShort name: Fourfold Iitaka2026-09-26 · 52 pagesOriginal paper (GitHub) ↗ | Claims a uniform canonical index for projective klt Calabi–Yau fourfolds and a uniform effective Iitaka degree for pseudo-effective lc fourfold adjoints with finite rational coefficients. Theorems 1.1–1.2 · pp. 1–2 ↗ | Combines Log Abundance nonvanishing, semiampleness from Supported lifting, and relative denominator bounds. Read the proof overview → |
| 14 | Abundance after nonvanishing for compact Kähler fourfoldsShort name: Kähler after nonvanishing2026-09-27 · 92 pagesOriginal paper (GitHub) ↗ | Claims semiampleness for analytically nef adjoints of compact Kähler klt fourfolds with effective rational boundary and nonvanishing, without projectivity or Q-factoriality. Theorem 1.1 · p. 3 ↗ | Handles the Kähler fourfold step after nonvanishing, proving its analytic boundary arguments within the manuscript. Read the proof overview → |
These are reading groups for this guide. The numbered manuscript inventory above retains the official catalogue order. Short names used in diagrams are paired with the full titles there.
01
Obtain a first section and a good model.
Log Abundance supplies an all-dimensional qualitative input. In dimension four, metrics and nonvanishing meet lifting and abundance after nonvanishing. Schnell applies good models.
02
Control the degree by dimension and coefficients.
Combines arithmetic Stein degrees, normal lc indices, and moduli denominators. The slc index problem adds descent of trivializations across the conductor.
03
Keep the nonprojective hypotheses visible.
Separates conditional all-dimensional lc abundance from fourfold klt models and abundance after nonvanishing. Similar titles alone do not create dependency edges.
Log Abundance supplies all-dimensional good models. Alongside it, the fourfold papers separate nonvanishing from lifting. Schnell is an application of the geometry of the resulting good model.
Fourfold nonvanishing uses Minimal metrics, Theorems 1.1–1.2, for zero-Lelong metrics and interior injectivity on the endpoint residual bundles. These supply surjectivity of restriction. Three-dimensional semi-dlt abundance and descent separately produce a nonzero section on the whole boundary; combining the two steps gives nonvanishing upstairs.
The moving-jet argument imports Log Abundance, Lemmas 7.1–7.5, to control centres, degrees and covers. Theorem 9.1 then rules out coexistence of the constructed jets, flag and movable curve by Frobenius comparison. These are auxiliary inputs, rather than an application of Log Abundance’s abundance main theorem.
Fourfold nonvanishing · Proof overview →Log Abundance · Proof overview →
Supported lifting, Theorem 1.2, proves fourfold semiampleness assuming nonnegative Kodaira dimension. Fourfold nonvanishing, Corollary 1.2, supplies the first section, giving Supported lifting, Corollary 1.4. A separate input to its conditional Theorem 1.3 is FN Lemma 8.1: rational linear algebra converting an effective real presentation to a rational one. The two uses must be distinguished.
Log Abundance, Corollary 11.2, supplies a model with semiample canonical divisor and an effective exceptional comparison. Schnell transfers sections to bound the Kodaira dimension, then uses the mixed intersections of Lemma 2.3 for the remaining inequality. Model existence is the imported input; the subsequent intersection argument is carried out in Schnell.
Good models alone do not bound the degree uniformly. The arguments separately control generic-fibre indices, moduli denominators on the base and degrees of finite covers. The diagram shows the six papers addressing these tasks and an input from Log Abundance.
In Fourfold Iitaka, Log Abundance supplies nonvanishing from pseudo-effectivity and Supported lifting makes the nef adjoint semiample after nonvanishing. Relative denominators, Theorem 1.1 and Proposition 6.1, control relative denominators and complete systems on a big base uniformly. The qualitative step supplies the fibration; the effective step captures its function field in a uniform degree.
Fourfold Iitaka · Proof overview →Relative denominators · Proof overview →
Uniform Iitaka combines Log Abundance good models with Stein degree, Theorem 1.1, bounding constant-field extensions of boundary components and hence norm descent. Arbitrary-dimensional chain lemmas from Fourfold Iitaka lead from covering families to estimates on leaves. The output is a uniform complete system generating the full Iitaka field, not just an image of the right dimension.
Uniform Iitaka · Proof overview →Stein degree · Proof overview →
Log Iitaka applies Uniform Iitaka, Theorem 1.2, to the generic fibre, fixing a uniform trivializing multiple and an exact pullback presentation. Log Abundance supplies the semiample model, while Stein degree controls finite extensions in the residue induction. The finite rational coefficient set of the log Iitaka main theorem should be distinguished from the rational DCC coefficient scope of the normal lc index theorem.
slc indices applies the normal lc index theorem on normalization components, but descent to the original slc pair needs more. Product covers and stability under further covers from Log Iitaka, the rational-image lemma from Uniform Iitaka, and Log Abundance good models enter the character bounds. Residues and cycles must then be made compatible. The slc paper itself supplies the final conductor gluing.
Conditional all-dimensional lc abundance, conditional klt fourfold models, and klt fourfold abundance after nonvanishing have different scopes. Dimension-free auxiliary lemmas are drawn separately from the main theorems of the papers containing them.
Conditional Kähler models proves nonvanishing at the specified nef MMP endpoint, then invokes Kähler after nonvanishing, Theorem 1.1, for semiampleness. This is Assumption 2.4. Orbifold subadditivity (Assumption 2.2) and the specified MMP (Assumption 2.3) remain necessary; the after-nonvanishing theorem alone does not supply all premises.
Conditional Kähler models · Proof overview →Kähler after nonvanishing · Proof overview →
Kähler abundance, Proposition 2.10, uses the full Log Abundance induction for projective good models and effective exceptional comparison. The argument then splits into a positive-dimensional fibration case and a simple case of algebraic dimension zero. Theorem 6.1 in the latter gives meromorphic nonvanishing, not directly a holomorphic section. Signed-boundary rigidity and lower-dimensional induction return to generation of the original nef adjoint. Assumption 1.1 remains in the main theorem.
Kähler abundance · Proof overview →Kähler abundance · Proof overview →
Conditional Kähler models, Lemmas 7.16–7.17, give dimension-free strata adjunction and strict comparison, used in Kähler abundance, Theorem 3.13. Torsion-freeness in Kähler after nonvanishing, Lemma 7.1, is another input: Kähler abundance, Lemma 4.3, uses it to pass from residue matching to section extension. Neither connection applies a fourfold main theorem in arbitrary dimension.
Kähler abundance · Proof overview →Kähler after nonvanishing · Proof overview →
The diagrams select principal routes. This explorer includes the other recorded connections. Choose a manuscript and direction, or follow an arrow above to open its result-level record. The initial view shows the three inputs recorded for Log Iitaka.
The following records include all 23 selected connections. Each gives the supplied result, its use, and the scope of the editorial check.
23 selected connections
No connection in this direction has been recorded in this survey. This does not mean there are no dependencies. See the manuscript summary and reading scope below.
Converts a real effective presentation to a rational one using finite rational linear constraints. This linear-algebra lemma uses no nonvanishing theorem and is distinct from the input to Corollary 1.4.
Supplies actual adjunction on normalized strata and strict discrepancy comparison over the exceptional locus. Both lemmas are dimension-free; the fourfold main theorem is not applied in all dimensions.
Uses torsion-freeness of the first higher direct image when the pair is klt off its floor and an actual positive adjoint multiple is pulled back from the base, to turn generic-fibre residue matching into extension of restricted sections.
Independent tools controlling centre degrees, moving multiplicities, slices, and finite covers of a fixed branch complement.
Supplies the rational-image property of boundary-free nonabelian factors for the Hodge-rank theorem. The source lemma itself uses Log Abundance nonvanishing.
Uses a quasi-étale product cover and stability under further covers to control the character on the volume form factor by factor.
Supplies a good minimal model, semiampleness, and effective exceptional comparison for a pseudo-effective canonical divisor.
Identifies endpoint residual bundles with positive rational multiples of nef klt adjoints, supplying zero-Lelong metrics and interior injectivity. A nonzero section on the whole boundary additionally uses three-dimensional semi-dlt abundance and descent.
Rules out coexistence of the fixed jets, flag, and movable curve constructed in dimension four. This input is the independent Frobenius theorem, not Log Abundance’s abundance theorem.
Provides the first section on the original variety; combined with the independent Theorem 1.2 to obtain fourfold log abundance.
Makes a nef adjoint with known nonvanishing semiample, in the model comparison preserving rounded section spaces in every degree.
Controls moduli denominators and supplies a uniform degree generating the full field of a positive-dimensional big base.
Provides the qualitative first section from pseudo-effectivity. Uniform degree comes later from index and relative denominator arguments.
Supplies all-dimensional projective lc good models, including nonvanishing and semiampleness on the model.
Bounds the Stein degree of a coefficient-one component, controlling the index when a principal divisor descends by a field norm.
Constructs a rational fibration from chains of covering families and bounds section orders and degrees on its leaves. Uses the arbitrary-dimensional chain lemmas, rather than the fourfold main theorem.
Supplies a uniform trivializing degree on the generic fibre, fixing an exact pullback presentation.
Bounds the field extension from the Stein curve of a boundary component in the residue induction.
Passes from nonnegative Iitaka dimension to a good model, where the effective-degree argument uses the semiample fibration.
Trivializes all lc normalization components in one even degree. The receiving paper adds conductor gluing.
Supplies good models of perturbed projective klt pairs in the Hodge-rank argument.
Uses Log Abundance’s full induction under the stated logarithmic subadditivity assumption to supply projective good models and effective exceptional comparison.
Gives semiampleness once nonvanishing is obtained for the nef Kähler klt fourfold adjoint. The other two premises remain separate.
Arrows run from the supplied result to its use. Numbers 01–14 are positions within catalogue 034. Citations open the manuscript on GitHub.
Read dependencies at the level of results: Fourfold nonvanishing uses Log Abundance’s independent geometric tools and Frobenius theorem. Its abundance main theorem is not that input. In Supported lifting, fourfold nonvanishing enters Corollary 1.4; Theorem 1.2 already assumes a first section. These distinctions matter when tracing apparent back-and-forth citations.
| Source result | Used in | Input and role | Editorial check |
|---|---|---|---|
| Fourfold nonvanishing by minimal metrics and moving jetsLemma 8.1 · p. 35 | Supported lifting: Theorem 1.3, proof · p. 27 | Converts a real effective presentation to a rational one using finite rational linear constraints. This linear-algebra lemma uses no nonvanishing theorem and is distinct from the input to Corollary 1.4. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Conditional good minimal models for compact Kähler fourfoldsLemmas 7.16–7.17 · pp. 58–60 | Kähler abundance: Theorem 3.13, proof · pp. 29–30 | Supplies actual adjunction on normalized strata and strict discrepancy comparison over the exceptional locus. Both lemmas are dimension-free; the fourfold main theorem is not applied in all dimensions. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Abundance after nonvanishing for compact Kähler fourfoldsLemma 7.1 · pp. 51–52 | Kähler abundance: Lemma 4.3, proof · p. 74 | Uses torsion-freeness of the first higher direct image when the pair is klt off its floor and an actual positive adjoint multiple is pulled back from the base, to turn generic-fibre residue matching into extension of restricted sections. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Log abundance in characteristic zeroLemmas 7.1–7.5 · pp. 45–52 | Fourfold nonvanishing: §3 / Proposition 6.4 · pp. 8–12, 24 | Independent tools controlling centre degrees, moving multiplicities, slices, and finite covers of a fixed branch complement. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Uniform Pluricanonical Iitaka FibrationsLemma 5.2 · pp. 21–22 | slc indices: Lemma 6.6 · pp. 32–34 | Supplies the rational-image property of boundary-free nonabelian factors for the Hodge-rank theorem. The source lemma itself uses Log Abundance nonvanishing. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Uniform log Iitaka fibrations and bounded moduli denominatorsProposition 5.1 / Lemmas 5.2–5.3 · pp. 12–13 | slc indices: Input 6.3 / Lemma 6.4 · pp. 30–32 | Uses a quasi-étale product cover and stability under further covers to control the character on the volume form factor by factor. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Log abundance in characteristic zeroCorollary 11.2 · pp. 73–74 | Schnell: Theorem 3.1 / §4.1 · pp. 7–8 | Supplies a good minimal model, semiampleness, and effective exceptional comparison for a pseudo-effective canonical divisor. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Minimal metrics and interior injectivity for nef adjointsTheorems 1.1–1.2 · pp. 1–2 | Fourfold nonvanishing: Theorems 4.1–4.2 / §5 · pp. 13–20 | Identifies endpoint residual bundles with positive rational multiples of nef klt adjoints, supplying zero-Lelong metrics and interior injectivity. A nonzero section on the whole boundary additionally uses three-dimensional semi-dlt abundance and descent. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Log abundance in characteristic zeroTheorem 9.1 · pp. 62–69 | Fourfold nonvanishing: Theorem 7.1 / §7 · pp. 30–34 | Rules out coexistence of the fixed jets, flag, and movable curve constructed in dimension four. This input is the independent Frobenius theorem, not Log Abundance’s abundance theorem. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Fourfold nonvanishing by minimal metrics and moving jetsCorollary 1.2 · p. 2 | Supported lifting: Corollary 1.4, proof · p. 29 | Provides the first section on the original variety; combined with the independent Theorem 1.2 to obtain fourfold log abundance. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Lifting sections from the reduced support of an adjointTheorem 1.2 · p. 2 | Fourfold Iitaka: Theorem 2.1 / Proposition 2.2 · pp. 6–7 | Makes a nef adjoint with known nonvanishing semiample, in the model comparison preserving rounded section spaces in every degree. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Relative denominators and effective systems for log Calabi-Yau fibrationsTheorem 1.1 / Proposition 6.1 · pp. 2, 15 | Fourfold Iitaka: §3 / §10 · pp. 8, 48–49 | Controls moduli denominators and supplies a uniform degree generating the full field of a positive-dimensional big base. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Log abundance in characteristic zeroTheorems 1.1, 11.1 / Lemma 2.2 · pp. 2, 73, 8 | Fourfold Iitaka: §10 · p. 48 | Provides the qualitative first section from pseudo-effectivity. Uniform degree comes later from index and relative denominator arguments. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Log abundance in characteristic zeroTheorem 11.1 · p. 73 | Uniform Iitaka: Theorem 2.1 · p. 4 | Supplies all-dimensional projective lc good models, including nonvanishing and semiampleness on the model. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Arithmetic Stein-degree bounds for log Calabi–Yau pairsTheorem 1.1 · p. 1 | Uniform Iitaka: Proposition 3.2, proof · pp. 7–8 | Bounds the Stein degree of a coefficient-one component, controlling the index when a principal divisor descends by a field norm. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Uniform effective log Iitaka fibrations for fourfoldsLemma 5.1 / Corollary 5.2 / Lemma 5.4 · pp. 18, 20–21 | Uniform Iitaka: Lemma 6.2 · pp. 24–25 | Constructs a rational fibration from chains of covering families and bounds section orders and degrees on its leaves. Uses the arbitrary-dimensional chain lemmas, rather than the fourfold main theorem. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Uniform Pluricanonical Iitaka FibrationsTheorem 1.2 · p. 2 | Log Iitaka: Theorem 2.1 / Proposition 7.1 · pp. 5, 23–24 | Supplies a uniform trivializing degree on the generic fibre, fixing an exact pullback presentation. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Arithmetic Stein-degree bounds for log Calabi–Yau pairsTheorem 1.1 · p. 1 | Log Iitaka: Theorem 2.3 / Lemma 4.6 · pp. 5, 10–11 | Bounds the field extension from the Stein curve of a boundary component in the residue induction. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Log abundance in characteristic zeroTheorem 11.1 · p. 73 | Log Iitaka: Theorem 2.2 / §8 · pp. 5, 29 | Passes from nonnegative Iitaka dimension to a good model, where the effective-degree argument uses the semiample fibration. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Uniform Pluricanonical Iitaka FibrationsTheorem 1.2 · p. 2 | slc indices: Theorem 2.2 / §7.3 · pp. 5, 37 | Trivializes all lc normalization components in one even degree. The receiving paper adds conductor gluing. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Log abundance in characteristic zeroTheorem 11.1 · p. 73 | slc indices: Theorem 2.3 / Lemma 5.6 · pp. 5, 20–21 | Supplies good models of perturbed projective klt pairs in the Hodge-rank argument. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Log abundance in characteristic zeroProposition 2.5 / Theorem 9.6 / §10 · pp. 9–12, 69–73 | Kähler abundance: Proposition 2.10 · p. 12 | Uses Log Abundance’s full induction under the stated logarithmic subadditivity assumption to supply projective good models and effective exceptional comparison. | Input statement and the receiving connection compared; the complete input proof is not verified. |
| Abundance after nonvanishing for compact Kähler fourfoldsTheorem 1.1 · p. 3 | Conditional Kähler models: Assumption 2.4 · p. 9 | Gives semiampleness once nonvanishing is obtained for the nef Kähler klt fourfold adjoint. The other two premises remain separate. | Input statement and the receiving connection compared; the complete input proof is not verified. |
Log Abundance’s Lemma 6.1 uses Corollary 6.2 of the orbifold Iitaka manuscript in catalogue 033, in its zero-boundary projective case. The Log Abundance overview also explains how the existing Hacon–Popa–Schnell result supplies this particular application.
See the statements and application in Log Abundance →The receiving manuscript identifies orbifold subadditivity and the specified Kähler MMP as Assumptions 2.2–2.3. Together with abundance after nonvanishing from 034, they form its three premises. The companion statements and selected application conditions are compared in the conditional Kähler overview. The complete proofs of the premises have not been independently verified.
Assumptions 2.2–2.4 · pp. 7–9 ↗ Read the conditional overview →Established inputs such as Hashizume’s reduction, gluing, lower-dimensional abundance, and effective birationality are not exhaustively mapped here. The individual articles identify their selected external inputs separately. The all-dimensional Kähler article also records Hodge-line, relative Iitaka and weak-effectivity inputs from OILS, outside the 14-paper inventory.
OpenAI Research Catalog · 034 · p. 5 ↗
Official catalogue at the reference commit ↗
CONTENTS.md: Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity
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