Numerical semiampleness and generalized minimal models

1 · The essentials

These manuscripts study when geometric objects can be replaced by others that are easier to work with while preserving specified information. The main manuscript starts with a particular combination of line bundles on a smooth, compact complex space. Under stated positivity conditions, it claims that another line bundle has the same measured class and has enough sections to define a map after taking a suitable multiple. This is a statement about the measured class, not necessarily about the original line bundle itself. A companion manuscript claims a corresponding result for projective spaces with certain allowed singularities. Two further manuscripts concern the existence of simpler geometric models under related positivity assumptions. The main result is not formalized in Lean and has not been formally verified.

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

As an imperfect image, imagine replacing a folded paper pattern with another pattern that has the same recorded measurements but is easier to assemble. Matching the measurements does not mean that the original pattern itself becomes easier to assemble; likewise, the main claim concerns a replacement with the same class, not the original line bundle.

Why it matters

The semiampleness claims connect positivity assumptions with the existence of a representative whose multiples define maps. Their numerical formulation is important: it separates what can be achieved within a class from what is asserted about a particular line bundle. The companion model-existence claims address when the geometry admits a minimal model or, in the stated alternative, a Mori fibre space.

2 · Going further

The main manuscript considers a smooth connected compact Kähler manifold $X$ , an effective rational simple normal crossing divisor $B$ with coefficients less than one, and a nef rational holomorphic line bundle $M$ . Assuming that $K_{X} + B$ is pseudo-effective and $K_{X} + B + M$ is nef, it claims that the first Chern class of $K_{X} + B + M$ in real Bott–Chern cohomology is represented by a semiample rational line bundle. The conclusion allows a flat change of line bundle and explicitly does not assert semiampleness of the original adjoint.

The second manuscript gives the projective counterpart, described in its abstract as the Generalised Abundance Conjecture. Here $(X, B)$ is a projective klt rational pair over an algebraically closed field of characteristic zero, and $M$ is a nef rational Cartier divisor. Under the same two positivity assumptions, the claimed conclusion is numerical equivalence of $K_{X} + B + M$ to a semiample rational Cartier divisor. The abstracts provide no argument or intermediate steps; the central distinction they describe is between finding a semiample representative and establishing semiampleness of the given adjoint.

The third manuscript separately claims the existence form of the minimal-model conjecture for projective generalized log canonical rational pairs over algebraically closed fields of characteristic zero. It claims a minimal model when the adjoint is pseudo-effective and a Mori fibre space otherwise, keeping the nef rational Cartier b-divisor fixed. Setting the nef part to zero gives its stated result for ordinary log canonical rational pairs.

The fourth manuscript focuses on smooth connected complex projective varieties of dimension at least three. If $K_{X}$ is pseudo-effective and $κ_{σ} (X, K_{X}) = 1$ , with $κ_{σ}$ defined through section growth with a fixed ample twist, it claims the existence of a projective rationally factorial terminal minimal model. The supplied abstracts do not specify logical dependencies among these manuscripts.

Useful notions

Divisors and line bundles
Canonical divisors and adjoint expressions
Nefness and pseudo-effectivity
Sections and semiampleness
Numerical equivalence and first Chern classes
Singularities of pairs and minimal models

What was known before / what is new

Before

The source identifies the projective numerical semiampleness statement with the Generalised Abundance Conjecture. It also names the existence form and the numerical-dimension-one case of the minimal-model conjecture. It gives no account of earlier partial results, historical development, or methods.

New

The main manuscript claims a semiample representative for the real Bott–Chern first Chern class of the stated nef Kähler adjoint. The second claims numerical semiampleness under the projective klt hypotheses. The third claims the minimal-model/Mori-fibre-space alternative for generalized log canonical rational pairs with fixed nef b-data. The fourth claims terminal minimal models in the specified smooth projective numerical-dimension-one setting.

Limits and assumptions

The Kähler claim requires smoothness, connectedness, compactness, an effective rational simple normal crossing boundary with coefficients less than one, and a nef rational holomorphic line bundle; both the pseudo-effectivity of $K_{X} + B$ and the nefness of $K_{X} + B + M$ are essential stated hypotheses. Its conclusion is equality of a Bott–Chern class with that of a semiample representative, not semiampleness of the original adjoint. The projective semiampleness claim requires a klt rational pair over an algebraically closed characteristic-zero field. The generalized model-existence claim has its own projective, generalized log canonical, rational, and characteristic-zero hypotheses and keeps the nef b-divisor fixed. The numerical-dimension-one claim is restricted to smooth connected complex projective varieties of dimension at least three. The main result is not formalized in Lean and has not been formally verified. The abstracts supply no proof details and do not identify further open cases; in particular, they leave semiampleness of the original Kähler adjoint unasserted.

Mini glossary

Adjoint: The expression combining the canonical divisor with the boundary and nef part, written here as $K_{X} + B + M$ .
Semiample: A line bundle or rational divisor is semiample if a suitable positive multiple has global sections generating it everywhere.
Numerical semiampleness: In the projective statement, this means numerical equivalence to a semiample rational Cartier divisor, rather than semiampleness of the given divisor.
Bott–Chern class: The cohomological class used in the Kähler manuscript to express equality between the first Chern class of the adjoint and that of a semiample representative.
Numerical dimension one: The condition $κ_{σ} (X, K_{X}) = 1$ , using the section-growth definition with a fixed ample twist specified in the fourth manuscript.

Keywords : Numerical semiampleness · Kähler adjoints · Generalized minimal models

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 numerical semiampleness for nef adjoints $K_{X} + B + M$ with $K_{X} + B$ pseudo-effective and M nef rational, for projective klt rational pairs over algebraically closed characteristic-zero fields and smooth compact Kähler rational klt simple-normal-crossing pairs, using Bott–Chern cohomology in the latter case. Separately, projective generalized log canonical rational pairs over such fields admit minimal models for pseudo-effective adjoints and Mori fiber spaces otherwise, with nef b-data fixed.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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Numerical semiampleness of nef adjoint classes on compact Kähler manifolds

06/10/2026

To be verified

Let X be a smooth connected compact Kähler manifold, let B be an effective rational simple normal crossing divisor with coefficients less than one, and let M be a nef rational holomorphic line bundle on X. If $K_{X} + B$ is pseudo-effective and $K_{X} + B + M$ is nef, we prove that its first Chern class in real Bott–Chern cohomology is represented by a semiample rational line bundle. This numerical statement allows a flat change of line bundle; it does not assert semiampleness of the original adjoint.

PDF GitHub folder

Numerical Semiampleness of Nef Adjoint Divisors

03/10/2026

To be verified

We prove the Generalised Abundance Conjecture: if $(X, B)$ is a projective klt ℚ-pair over an algebraically closed field of characteristic zero, $K_{X} + B$ is pseudo-effective, M is a nef ℚ-Cartier divisor on X, and $K_{X} + B + M$ is nef, then $K_{X} + B + M$ is numerically equivalent to a semiample ℚ-Cartier divisor on X.

PDF GitHub folder

Minimal models and Mori fibre spaces for generalized log canonical Q-pairs

24/09/2026

To be verified

We resolve the existence form of the minimal-model conjecture for projective generalized log canonical ℚ-pairs over algebraically closed fields of characteristic zero. Such a pair admits a minimal model when its adjoint divisor is pseudo-effective, and a Mori fibre space otherwise, while keeping the nef ℚ-Cartier b-divisor fixed. Taking the nef part to be zero gives the corresponding result for ordinary log canonical ℚ-pairs.

PDF GitHub folder

Minimal models in numerical dimension one

24/09/2026

To be verified

We resolve the numerical-dimension-one case of the minimal-model conjecture for smooth connected complex projective varieties of dimension at least three. If K_X is pseudo-effective and $κ_{σ} (X, K_{X}) = 1$ , with κ_σ defined by section growth with a fixed ample twist, then X admits a projective ℚ-factorial terminal minimal model.

PDF GitHub folder