CATALOGUE 036
Numerical semiampleness and generalized minimal models
Four manuscripts address numerical semiampleness of nef adjoints and model existence for generalized lc pairs with fixed nef b-data. A semiample numerical representative, equality of real Bott–Chern classes, and semiampleness of the original adjoint are distinct conclusions. The numerical-dimension-one paper supplies a separate construction of minimal models using surface intersections.
1. 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 | Numerical semiampleness of nef adjoint classes on compact Kähler manifolds →Short name: Numerical semiampleness of nef adjoint classes on compact Kähler manifoldsOctober 6, 2026 · 52 pagesOriginal paper (GitHub) ↗ | For a smooth compact Kähler rational SNC klt pair, with K+B pseudo-effective and M and K+B+M nef, a semiample rational line represents the same real Bott–Chern class. Theorems 1.1–1.2 · pp. 2–3 ↗ | Combines vanishing in algebraic dimension zero, descent of nef lines and positive parts on a projective base. Read the proof overview → |
| 02 | Numerical Semiampleness of Nef Adjoint Divisors →Short name: Numerical Semiampleness of Nef Adjoint DivisorsOctober 3, 2026 · 37 pagesOriginal paper (GitHub) ↗ | For a projective klt rational pair in characteristic zero with K+B pseudo-effective, M nef on X and K+B+M nef, obtains numerical equivalence to a semiample rational divisor on X. Theorem 1.1; Corollary 1.2 · pp. 1–2 ↗ | Bigness from full nef dimension and induction for positive parts via nef reduction. Read the proof overview → |
| 03 | Minimal models and Mori fibre spaces for generalized log canonical Q-pairs →Short name: Minimal models and Mori fibre spaces for generalized log canonical Q-pairsSeptember 24, 2026 · 79 pagesOriginal paper (GitHub) ↗ | For a projective generalized lc rational pair with fixed nef rational b-data, gives a minimal model for a pseudo-effective adjoint and a Mori fibre space otherwise. Theorem 1.1; Corollary 1.2 · p. 2 ↗ | Transfers a local phase argument and smooth relative weak Zariski decompositions to the NQC generalized MMP. Read the proof overview → |
| 04 | Minimal models in numerical dimension one →Short name: Minimal models in numerical dimension oneSeptember 24, 2026 · 14 pagesOriginal paper (GitHub) ↗ | A smooth complex projective X of dimension at least three with pseudo-effective KX and section-growth numerical dimension one has a Q-factorial terminal minimal model; a separate input makes the same endpoint good. Theorem 1.1; Corollary 1.2 · p. 2 ↗ | A separate route to nefness using a model fixed in codimension two and two surfaces. Read the proof overview → |
2. From model existence to numerical semiampleness
The diagram shows a chain of reductions linking three manuscripts. Downward edges are direct uses of specific results, not containment of every conclusion or verification of complete proofs. Box numbers follow the official inventory; their placement follows the dependency order.
References in the diagram
- OpenAI26a OpenAI — Numerical semiampleness of nef adjoint classes on compact Kähler manifolds. October 6, 2026.
- OpenAI26b OpenAI — Numerical Semiampleness of Nef Adjoint Divisors. October 3, 2026.
- OpenAI26c OpenAI — Minimal models and Mori fibre spaces for generalized log canonical Q-pairs. September 24, 2026.
- OpenAI26d OpenAI — Minimal models in numerical dimension one. September 24, 2026.
- OpenAI26LA OpenAI — Log abundance in characteristic zero. September 24, 2026.
OpenAI26c supplies model existence, OpenAI26b the projective positive part, and OpenAI26a the Kähler real-Chern-class conclusion. OpenAI26LA is an additional input for ordinary adjoints. OpenAI26d denotes the separate surface argument discussed below.
The projective paper uses the finite MMP in the Q-factorial generalized klt setting to control two adjoints. It combines ordinary log abundance with its new bigness argument to construct the positive part. The Kähler paper applies this positive-part theorem to an ordinary klt pair on a projective base obtained after algebraic reduction. This is not an arrow running an arbitrary generalized MMP on a Kähler space.
[OpenAI26c, Theorem 1.1; Section 3.3, pp. 2, 14–16] [OpenAI26b, Theorem 2.3, p. 4] [OpenAI26b, Proposition 8.1, p. 30] [OpenAI26a, Proposition 2.5; Section 7.3, pp. 7–8, 48–49]
Follow the projective article and the Kähler article for hypotheses and proof steps. The Kähler paper also uses ordinary Kähler log abundance (the October 6, 2026 version in 034) and Orbifold Iitaka (033); their versions and distinct roles are recorded in its reference guide.
The numerical-dimension-one article fixes positive perturbations in codimension two and proves nefness using growth with a fixed twist and a surface through a negative curve. Its Theorem 1.1 does not use the generalized existence theorem above. Corollary 1.2 separately adds ordinary log abundance to make the same terminal endpoint a good minimal model.
[OpenAI26d, Theorem 1.1; Corollary 1.2, p. 2] [OpenAI26LA, Theorem 1.1, p. 2]
Corollary 1.2 of the generalized lc article likewise treats ordinary pairs with M=0, to which ordinary abundance can subsequently be added. No semiampleness conclusion is extrapolated to nonzero nef b-data. Unlisted citation relations remain uninvestigated rather than being classified as absent.
3. Sources and checking scope
The official overview and CONTENTS are fixed at commit 1581d4cbdd69a0faf34b2089614fe2a814316a6a. The subject is the second in official display order, “Algebraic and complex geometry”; entry 036 appears on PDF p. 5. The four titles, order, versions and page counts were checked, followed by the article statements, principal reductions, and statements and applications of direct inputs. Remaining textbook checks and unverified external proofs are described in each article. These accounts explain the manuscripts' claims and do not constitute specialist review or formal verification of the entire arguments.