Numerical semiampleness and the existence of minimal models

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Family 036 · Algebraic and complex geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-12

1. The question

Can the numerical positivity of a line bundle determine a map, even when the line itself has no sections? This question sounds contradictory until one distinguishes a bundle from its numerical class. Different bundles can have identical degrees on every curve but very different spaces of sections.

Family 036 studies canonical divisors with an additional nef contribution. Two manuscripts claim that the resulting class has a semiample representative: a different bundle with the same numerical information and enough sections to define a map. Another claims existence of minimal models or Mori fiber spaces for generalized log canonical pairs. A fourth gives a separate minimal-model construction in numerical dimension one.

These conclusions concern different stages of birational classification. Existence of a nef model is not itself semiampleness, and numerical semiampleness does not generally make the originally specified bundle semiample.

2. Background and essential definitions

Divisors and their numerical classes

A line bundle assigns a one-dimensional vector space to each point, varying algebraically or holomorphically. Global sections can define a map to projective space if they do not all vanish at any point. A bundle is semiample when some positive tensor power has this property.

For a projective variety, a rational Cartier divisor represents a rational line bundle: an integral multiple is an actual line bundle. It is nef if its degree on every curve is nonnegative. It is pseudoeffective if its numerical class is a limit of classes of effective divisors. Two divisors are numerically equivalent, written $D\equiv L$, if they have the same degree on every curve.

The canonical divisor $K_X$ records top-degree differential forms. A rational boundary $B\geq0$ records weighted hypersurfaces, and $K_X+B$ is an ordinary adjoint. A pair is klt, short for Kawamata log terminal, when its log discrepancies are positive. Concretely, on a resolution with

$$K_Y+B_Y=\pi^*(K_X+B),$$

all coefficients of $B_Y$ are less than one. Log canonical, or lc, allows equality to one. These conditions control singularities so that birational comparison and extension methods can be used.

The simplest reason for a numerical conclusion

Take a complex elliptic curve $E$, whose canonical bundle is trivial. Let $M$ be a non-torsion line bundle of degree zero, meaning that no positive power is trivial. Every $M^{\otimes m}$ has degree zero and has no nonzero section: a section would have an effective zero divisor of degree zero, so would vanish nowhere and trivialize the bundle.

Nevertheless, $M$ is nef and numerically equivalent to $\mathcal O_E$, the trivial bundle, which is generated by a constant section. Therefore numerical semiampleness is possible even though semiampleness of $M$ is false. This is why adding an arbitrary nef line to a canonical divisor changes the appropriate target.

Kähler classes and generalized pairs

A compact Kähler manifold has a positive closed real differential form compatible with its complex structure. It need not embed in projective space. The first Chern class of a holomorphic line bundle lies in real Bott–Chern cohomology $H^{1,1}_{\mathrm{BC}}(X,\mathbb R)$, which records closed real $(1,1)$-forms modulo the appropriate second derivatives of functions. Equality there captures curvature information but can forget a flat line-bundle factor.

A generalized pair $(X,B+M)$ permits the nef contribution to live on a higher birational model. One fixes a nef rational Cartier divisor on that model and uses its compatible pullbacks and pushforwards elsewhere. This compatible collection is a nef b-divisor. Its trace on $X$ appears in $K_X+B+M$, but the higher-model data must remain fixed when discrepancies are compared.

A minimal model is a birational model with nef adjoint and improved discrepancies, extracting no new divisors. A Mori fiber space has a map to a lower-dimensional base, with negative adjoint along the fibers and relative Picard number one. The last condition says that only one independent numerical divisor direction remains relative to that base.

Numerical dimension before nefness

The fourth manuscript uses Nakayama’s section-growth invariant $\kappa_\sigma(X,D)$. For a smooth projective $n$-fold, it is the largest integer $j$ for which some fixed ample Cartier divisor $A$ satisfies

$$\limsup_{m\to\infty} \frac{h^0(X,\mathcal O_X(mD+A))}{m^j}>0.$$

The ample twist is chosen once before taking the limit. Numerical dimension one in this convention rules out positive quadratic growth for every fixed twist. It does not assert a uniform linear upper bound, and it does not manufacture a section of the untwisted $mD$.

3. Development history and previous partial results

Mori’s contraction and flip methods made birational simplification into a systematic program. The subsequent theory of singular pairs is developed in Kollár–Mori’s monograph. The work of Birkar–Cascini–Hacon–McKernan supplied minimal models and finite-generation results in crucial projective settings, particularly the big-boundary and general-type ranges.

Approaching a merely pseudoeffective divisor is harder than treating each positive perturbation. A finite program for every positive perturbation parameter does not imply that the combined sequence stops when the parameter tends to zero. Weak Zariski decompositions, which separate positive and effective parts on birational models, became an important route to this limit problem.

Lazić–Peternell formulated generalized abundance as numerical semiampleness. Their On Generalised Abundance, I, cited in M1 and M2, proves the surface case and a threefold case where the ordinary adjoint has positive numerical dimension. Their broader reductions identify an additional obstacle on varieties with numerically trivial canonical class. Subsequent work studies numerical nonvanishing and restricted nef-reduction dimensions.

For generalized minimal models, Birkar, Han–Li, Lazić–Tsakanikas, and Tsakanikas–Xie developed the decomposition and transfer machinery described in M3 §1. M3 credits these earlier transfer theorems and aims to supply their missing smooth input. It does not claim to originate the generalized-pair framework.

At numerical dimension zero, Nakayama’s methods and results of Druel and Gongyo established important existence and termination statements. M4 addresses the next numerical dimension using a separate surface argument. In the analytic setting, the compact Kähler threefold work of Höring–Peternell and the subsequent abundance results provide precedents, while the present all-dimensional numerical statement uses substantial companion inputs from family 034.

4. What the new manuscripts claim

M1: a semiample representative of a Kähler class

Let $X$ be a smooth connected compact Kähler manifold. Let $B\geq0$ be a rational simple-normal-crossing boundary with coefficients strictly less than one, and let $M$ be a nef rational holomorphic line bundle on $X$. Assume

$$K_X+B\text{ is pseudoeffective},\qquad D=K_X+B+M\text{ is nef}.$$

M1 claims a semiample rational line bundle $L$ on the same manifold with

$$c_1(L)=c_1(D)\quad\text{in }H^{1,1}_{\mathrm{BC}}(X,\mathbb R).$$

The conclusion allows changing the flat part. It does not state that $D$ itself is semiample. Nor does it allow replacing the nef line on $X$ by arbitrary higher-model nef data in the starting hypothesis.

M1 uses the projective positive-part theorem of M2 and the ordinary Kähler decomposition and model constructions of family 034, together with the stated logarithmic Iitaka input from family 033. These are explicit inputs to its argument.

M2: projective generalized abundance

For a projective klt rational pair over an algebraically closed characteristic-zero field, M2 assumes pseudoeffective $K_X+B$, a nef $\mathbb Q$-Cartier divisor $M$ on $X$, and nef $D=K_X+B+M$. It claims a semiample $\mathbb Q$-Cartier divisor $L$ with $D\equiv L$.

In particular, if $K_X+B\equiv0$, every nef rational Cartier divisor is numerically semiample. This still permits the elliptic-curve phenomenon above. The proof uses the companion ordinary log-abundance theorem and specified terminating programs. Ordinary abundance is an input, not a new consequence established independently by setting $M=0$.

M3: existence with fixed generalized nef data

M3 treats projective generalized lc rational pairs in characteristic zero. If $D=K_X+B+M$ is pseudoeffective, it claims a minimal model. If it is not pseudoeffective, it claims a Mori fiber space. The fixed nef b-divisor is retained, and discrepancies improve in the specified sense.

Neither the original variety nor the endpoint is assumed $\mathbb Q$-factorial in the statement. The conclusion supplies a terminating choice of program. It does not assert termination of every arbitrary program, semiampleness of the endpoint generalized adjoint, or a nonzero pluricanonical section.

M4: a separate numerical-dimension-one construction

For a smooth connected complex projective variety of dimension at least three, M4 assumes pseudoeffective $K_X$ and $\kappa_\sigma(X,K_X)=1$ in the fixed-twist convention above. It claims a finite canonical-negative sequence ending at a projective $\mathbb Q$-factorial terminal variety $Y$ with nef $K_Y$.

On a common resolution, the canonical pullback from $X$ equals the pullback from $Y$ plus an effective exceptional divisor, with the required strict comparison for contracted divisors. This records that the endpoint is a minimal model, not just some birational variety with a nef class.

M4’s construction does not use a general minimal-model existence theorem. Its further good-model corollary invokes the separate log-abundance theorem from family 034.

5. Why the advance matters

Importance: 5/5, provisional. The family claims broad progress on two central problems: existence of generalized minimal models and numerical abundance of nef adjoint classes. Together they would provide both birational endpoints and geometric representatives of classes which otherwise offer only intersection data.

The Kähler extension matters because projective constructions cannot be assumed on the original space. The numerical-dimension-one argument is also useful as a distinct route with a carefully specified growth hypothesis.

The score reflects the breadth of the family if its claims are correct. It does not erase the dependence of its abundance statements on the companion theorems.

6. The main idea

Reduce numerical abundance to a core positivity problem

M2 follows generalized-abundance reductions to the case of a nef divisor on a klt Calabi–Yau pair, where the ordinary adjoint is numerically trivial. The central additional claim is that full nef dimension forces bigness. Full nef dimension says every curve through a very general point has positive degree; bigness says the divisor has positive volume. These are different conditions.

Moving-point jet estimates and a Frobenius comparison are used to rule out full nef dimension with zero volume. Once the needed positive-part statement is available on a birational model, the proof descends a semiample numerical representative to the original variety.

M1 extends this strategy to compact Kähler manifolds. Algebraic reduction maps a modification to a projective base carrying its meromorphic functions. The challenge is to descend the nef contribution and control flat twists, which may vary from fiber to fiber. The paper uses families of cycles and countably many global choices to organize that descent, then applies the projective theorem. A separate argument handles algebraic dimension zero.

Control indices in an attempted infinite program

M3’s local input concerns singularities with cyclic symmetry. It selects divisorial valuations, measurements of orders of vanishing, whose discrepancies stay bounded while their values detect the symmetry characters. The manuscript connects this to the canonical indices appearing along a hypothetical infinite smooth program.

Its global contradiction first bounds those indices, then uses generation and vanishing results to force an adjoint divisor to be both globally generated and negative on a contracted curve. A generated line has nonnegative degree there, so the two properties conflict. The established relative and generalized transfer results then produce the stated endpoints.

Use two different surfaces

M4 starts with shrinking positive boundary perturbations. After a finite prefix, their adjoints are generated outside subsets of codimension at least three. A very general surface avoids those subsets; if the limiting canonical square there were positive, extension of sections would force quadratic growth with a fixed ample twist, contradicting $\kappa_\sigma=1$.

That surface might miss a hypothetical negative curve. A second surface is chosen through a point of such a curve. Vanishing-order estimates would force a fixed exceptional contribution with uniformly negative square, making the perturbations have positive square bounded away from zero. This contradicts the zero-square limit. The two surfaces have different jobs, and together they establish nefness of the finite endpoint.

7. Scope, limitations, and open questions

The numerical conclusions do not recover the flat part of the original bundle. The elliptic-curve example shows that this limitation is unavoidable without additional hypotheses. M1’s equality is in real Bott–Chern cohomology; M2’s is projective numerical equivalence.

Both numerical-abundance statements require pseudoeffectivity of the ordinary adjoint $K_X+B$, not merely of the total sum. Their nef contribution lives on the original space. M3 permits higher-model nef data, but its weaker conclusion is existence of an endpoint, without generalized semiampleness.

M4 uses a precise fixed-ample-twist numerical invariant before a nef model exists. Replacing it by an intersection formula for a non-nef canonical divisor would change the hypothesis. The theorems are characteristic-zero statements, despite positive-characteristic techniques inside some arguments.

Uniform effective degrees, recovery of the original flat factor, and stronger termination assertions are additional questions. The current claims and their dependencies are reported without independent proof verification.

8. Sources and further reading

M1 §1 explains the flat ambiguity and its Kähler inputs. M2 §§1–2 state projective numerical abundance and the reduction to full-nef-dimension positivity. M3 §§1 and 3 distinguish existence from arbitrary termination. M4 §1 defines its numerical invariant and outlines the two-surface argument.

For the foundational birational framework, see the linked Kollár–Mori and BCHM sources. Lazić–Peternell’s On Generalised Abundance, I is listed in M1 and M2 with precise theorem references. An optional introduction to numerical equivalence can help separate class-level and line-bundle statements.

Source manuscripts

• M1. OpenAI, Numerical semiampleness of nef adjoint classes on compact Kähler manifolds (October 6, 2026).
• M2. OpenAI, Numerical Semiampleness of Nef Adjoint Divisors (October 3, 2026).
• M3. OpenAI, Minimal models and Mori fibre spaces for generalized log canonical Q-pairs (September 24, 2026).
• M4. OpenAI, Minimal models in numerical dimension one (September 24, 2026).

Sources consulted on 12 October 2026 at repository revision fd4aeeb. Download the manuscript citations.