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Math-25-A2 · revision r1 · on Math-25

OpenAI claim summary for Math-25

Literature report · LLM-generated · unknown ·

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Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 034, 2026-09-27): The main theorem makes a nef klt adjoint semiample only for compact Kähler fourfolds already assumed to have nonnegative Iitaka dimension; the catalogue omits both restrictions.

Manuscript 2 (family 034, 2026-10-05): The fourfold good-model and nef-endpoint nonvanishing theorems explicitly assume Campana, MMP and effective-abundance inputs and strong Q-factoriality; neither settles the unrestricted unconditional catalogue statement.

Manuscript 3 (family 034, 2026-09-27): The abundance-after-nonvanishing theorem covers projective lc pairs in dimension at most four with nonnegative Iitaka dimension; this problem has neither the dimension bound nor an assumed first section.

Manuscript 4 (family 034, 2026-10-04): The main Kähler theorem explicitly assumes logarithmic Iitaka subadditivity; it gives semiampleness and hence nonvanishing only for nef adjoints under that additional assumption, whereas our catalogue assertions are unconditional.

Manuscript 5 (family 034, 2026-09-24): The log-abundance main theorem proves semiampleness of every nef rational lc adjoint over an algebraically closed characteristic-zero field, strictly extending the catalogue projective klt rational-boundary target.

Manuscript 6 (family 034, 2026-09-27): Zero-Lelong minimal metrics together with the stated fourfold semiampleness interface cover nef complex klt fourfold adjoints after nonvanishing; the theorem is not abundance in every dimension or nonvanishing for all pseudo-effective lc pairs.

Manuscript 7 (family 034, 2026-09-26): The effective Iitaka system concerns dimension-four lc adjoints and describes their section field; even with a nef adjoint this does not cover the catalogue semiampleness assertion in every dimension.

Manuscript 8 (family 036, 2026-09-24): The good-minimal-model corollary treats the numerical-dimension-one smooth canonical case, rather than all projective klt rational adjoints.

Manuscript 9 (family 036, 2026-10-03): The theorem yields a semiample divisor numerically equivalent to the nef adjoint; our formulation requires a generated Cartier multiple of the actual adjoint, not a numerical replacement.

Manuscript 10 (family 036, 2026-10-04): A semiample line with the same real Bott–Chern class is supplied for smooth Kähler rational klt data, but numerical equivalence does not make the actual adjoint line semiample as this problem requires.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Abundance Conjecture for the minimal model program
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Manuscript 2.
  6. Manuscript 3.
  7. Manuscript 4.
  8. Manuscript 5.
  9. Manuscript 6.
  10. Manuscript 7.
  11. Manuscript 8.
  12. Manuscript 9.
  13. Manuscript 10.
  14. Outline of the argument
  15. Formal verification
  16. Status caveat
  17. Sources

Abundance Conjecture for the minimal model program

OpenAI's claim

Manuscript 1 (family 034, 2026-09-27): A nef rational adjoint on a compact Kähler klt fourfold is claimed to be semiample once a nonzero section exists, including triviality of a multiple in Iitaka dimension zero; Manuscript 2 (family 034, 2026-10-05): Under explicit orbifold-subadditivity, fourfold-program, and abundance assumptions, good minimal models are claimed for the specified compact Kähler klt fourfold pairs; Manuscript 3 (family 034, 2026-09-27): An effective adjoint supported on the coefficient-one boundary is claimed to acquire positive Iitaka dimension when its restriction is semiample on the entire reduced support, leading to fourfold abundance after nonvanishing; Manuscript 4 (family 034, 2026-10-04): Assuming the stated logarithmic Iitaka inequality, OpenAI claims semiampleness of nef rational log adjoints on compact Kähler log-canonical spaces in arbitrary dimension. Manuscript 5 (family 034, 2026-09-24): In characteristic zero, nef rational log adjoints of projective log-canonical pairs are claimed to be semiample in every dimension; smooth canonical nonvanishing is another stated consequence; Manuscript 6 (family 034, 2026-09-27): Minimal semipositive metrics of nef klt adjoints are claimed to have no positive Lelong numbers, and zero-Lelong endpoint metrics are claimed to give the stated interior-boundary $H^1$ injection; Manuscript 7 (family 034, 2026-09-26): For log-canonical fourfolds with coefficients from a fixed finite rational set, a uniform rounded adjoint system is claimed to generate the full Iitaka field; a uniform klt Calabi--Yau canonical index is also claimed; Manuscript 8 (family 036, 2026-09-24): A smooth complex projective variety with canonical section-growth numerical dimension one is claimed to admit a terminal factorial minimal model, with good-model consequences using the separate abundance input. Manuscript 9 (family 036, 2026-10-03): For projective characteristic-zero klt pairs, a nef ordinary adjoint plus a nef rational divisor is claimed to be numerically equivalent to a semiample divisor whenever the ordinary adjoint is pseudo-effective; Manuscript 10 (family 036, 2026-10-04): On the stated compact Kähler pairs, a nef adjoint with an additional nef rational line is claimed to have a semiample representative of its Bott--Chern class, allowing a flat change of bundle.

Mathematical statement of the claimed result

Manuscript 1.
Let $X$ be a compact complex space, required to be Kähler, normal and connected, of dimension four and let $\Delta\ge0$ be rational, with $(X,\Delta)$ Kawamata log terminal and actual adjoint $D=K_X+\Delta$ rational Cartier. Assume $D$ is analytically nef: after fixing a Cartier multiple and a Kähler form, its curvature has smooth representatives bounded below by $-\varepsilon$ times that form for every $\varepsilon>0$. Assume also $\kappa(X,D)\ge0$, meaning a nontrivial section exists for at least one positive Cartier multiple. OpenAI claims \[ \exists m>0:\ mD\text{ Cartier},\qquad H^0(X,\mathcal O_X(mD))\otimes\mathcal O_X\twoheadrightarrow\mathcal O_X(mD). \] If $\kappa(X,D)=0$, the multiple may be chosen with $\mathcal O_X(mD)\simeq\mathcal O_X$. This theorem requires nonvanishing as a hypothesis.
Manuscript 2.
This is explicitly conditional on three global premises: $1$ orbifold Iitaka subadditivity in all dimensions for smooth compact manifolds in Fujiki class $\mathcal C$, namely $\kappa(X,K_X+\Delta)\ge\kappa(F,K_F+\Delta|_F)+\kappa(f\mid\Delta)$ for connected-fiber holomorphic fibrations with rational simple-normal-crossing boundary; $\kappa(f\mid\Delta)$ is the Kodaira dimension of the neat orbifold base, using infimum fiber multiplicities; $2$ existence and termination of every negative-extremal-ray minimal model program for pseudo-effective rational klt adjoints on globally strongly rational-factorial compact Kähler fourfolds, with projective divisorial contractions or log-canonical-positive flips and a nef endpoint; $3$ semiampleness of a nef rational klt fourfold adjoint already having nonnegative Iitaka dimension. Given these premises, take a normal connected compact Kähler fourfold $X$ with effective rational $B$, klt pair $(X,B)$, and rational Cartier analytically pseudo-effective $D=K_X+B$. Assume every global rank-one coherent reflexive sheaf has an invertible positive reflexive power. OpenAI claims a bimeromorphic contraction $\phi:X\dashrightarrow Y$ to a fourfold of the same category such that, writing $B_Y=\phi_*B$, \[ D_Y=K_Y+B_Y\text{ is nef and semiample},\qquad a(E;Y,B_Y)\ge a(E;X,B) \] for every prime divisor over the models, strictly for every prime on $X$ contracted by $\phi$. The map extracts no divisor. The three premises are part of the theorem, rather than conclusions proved without assumptions.
Manuscript 3.
Let $(V,C)$ be a normal projective rational-factorial divisorial-log-terminal pair over a characteristic-zero field assumed algebraically closed, with rational effective boundary $C$. Put $A=K_V+C$. Suppose a nonzero effective Cartier divisor $G$ satisfies $G\sim qA$ for a sufficiently divisible integer $q>0$, its support lies in $\operatorname{Supp}\lfloor C\rfloor$, and its line bundle restricted to the whole reduced divisor $G_{\mathrm{red}}$ is semiample. OpenAI claims \[ \kappa(V,A)>0. \] Another theorem asserts that for complex projective pairs $(X,\Delta)$ that are log canonical and have dimension at most four with effective rational boundary and nef rational Cartier $D=K_X+\Delta$, nonvanishing $\kappa(X,D)\ge0$ implies semiampleness, and $\kappa(X,D)=0$ implies $D\sim_{\mathbb Q}0$. The all-dimensional version is conditional: if every smooth connected projective complex $W$ with pseudo-effective $K_W$ has some $H^0(W,mK_W)\ne0$, then every nef rational log canonical adjoint of a projective pair in characteristic zero is semiample.
Manuscript 4.
Assume logarithmic Iitaka subadditivity for every connected-fiber surjective morphism $f:U\to V$ of smooth connected projective complex varieties with reduced simple-normal-crossing divisors $D_U,D_V$ satisfying $\operatorname{Supp}(f^*D_V)\subseteq\operatorname{Supp}D_U$: for a very general smooth fiber $F$ and $D_F=D_U|_F$, \[ \kappa(U,K_U+D_U)\ge\kappa(F,K_F+D_F)+\kappa(V,K_V+D_V). \] Under that premise, OpenAI claims that every normal irreducible compact Kähler complex space $X$ with effective rational $\Delta$, log canonical pair $(X,\Delta)$ and analytically nef rational Cartier $D=K_X+\Delta$ has a positive Cartier multiple generated everywhere by its global sections: \[ H^0(X,\mathcal O_X(mD))\otimes\mathcal O_X\twoheadrightarrow\mathcal O_X(mD). \] The premise covers all dimensions, uses $\kappa=-\infty$ for empty positive systems, and is a hypothesis of this Kähler result.
Manuscript 5.
Let $k$ be algebraically closed of characteristic zero and $(X,B)$ any normal projective pair over $k$, log canonical and with rational effective boundary. For the actual rational Cartier adjoint $D=K_X+B$, OpenAI asserts \[ D\cdot C\ge0\text{ for every curve }C\subset X \quad\Longrightarrow\quad \exists m>0:\ mD\text{ Cartier and }\mathcal O_X(mD)\text{ globally generated}. \] There is no dimension bound and no nonvanishing premise in this main theorem.
Manuscript 6.
For every normal connected projective complex klt pair $(H,\Theta)$ with effective rational $\Theta$ and nef rational Cartier $D_H=K_H+\Theta$, and every projective log resolution $\pi:V\to H$, OpenAI claims that $N=\pi^*D_H$ admits a semipositive singular Hermitian metric of minimal singularities and \[ \nu(h,x)=0\quad\text{for every }x\in V \] for every such metric $h$. Here $\nu$ is the Lelong number; minimal singularities means its weights are no more singular, up to bounded additive error, than the weights of any other semipositive metric on $N$. The components $K_H$ and $\Theta$ need not be separately rational Cartier.
Manuscript 7.
Fix a finite rational set $\Phi\subset[0,1]$. OpenAI claims an integer $m=m(\Phi)>0$ working for every normal integral projective complex fourfold $X$ with effective rational $\Delta$ whose nonzero coefficients lie in $\Phi$, log canonical pair $(X,\Delta)$, and pseudo-effective rational Cartier $D=K_X+\Delta$. Namely, \[ |\lfloor mD\rfloor|\ne\varnothing,\qquad \mathbb C\left(s/t:\ 0\ne s,t\in H^0(X,\mathcal O_X(\lfloor mD\rfloor))\right)=K(D)\subset\mathbb C(X). \] Here $K(D)$ is the field generated by ratios from all positive reflexive systems $H^0(X,\mathcal O_X(\lfloor jD\rfloor))$. Equality of fields asserts the Iitaka fibration itself, rather than only the dimension of its image. The integer is independent of the chosen canonical divisor.
Manuscript 8.
Let $X$ be smooth connected projective over $\mathbb C$, $n=\dim X\ge3$, and suppose $K_X$ is pseudo-effective. Define the section-growth numerical dimension by \[ \kappa_\sigma(X,D)=\max\{k\in\{0,\ldots,n\}:\ \exists\text{ ample Cartier }A,\ \limsup_{m\to\infty}h^0(X,mD+A)/m^k>0\}. \] Assume $\kappa_\sigma(X,K_X)=1$. OpenAI claims finitely many divisorial contractions or flips, each $K$-negative to a normal projective rational-factorial terminal $Y$ with $K_Y$ nef. The inverse map contracts no divisor, and on a common smooth resolution $p:W\to X$, $q:W\to Y$, \[ p^*K_X=q^*K_Y+E,\qquad E\ge0,\quad q_*E=0, \] with the strict transforms of all contracted prime divisors in $\operatorname{Supp}E$. A separate corollary, using the companion log-abundance theorem, claims $K_Y$ semiample. The invariant retains the fixed ample twist $A$; it is not an assumed growth bound for untwisted pluricanonical sections.
Manuscript 9.
For a rational projective pair with klt singularities $(X,B)$ over a characteristic-zero field assumed algebraically closed, assume $K_X+B$ pseudo-effective and $M$ nef rational Cartier. If $D=K_X+B+M$ is nef, OpenAI claims \[ \exists\text{ semiample rational Cartier }L\text{ on }X:\quad D\equiv L. \] Numerical equivalence is equality of degrees on every integral curve. Semiampleness is claimed for a numerically equivalent replacement, not necessarily for the original actual divisor $D$.
Manuscript 10.
Let $X$ be smooth connected compact Kähler, $B\ge0$ rational simple-normal-crossing with coefficients below one, and $M\in\operatorname{Pic}(X)\otimes\mathbb Q$. Assume pseudo-effectivity of $K_X+B$, and nefness of both $M$ and $D=K_X+B+M$, OpenAI claims \[ \exists\text{ semiample rational line bundle }L:\quad c_1(L)=c_1(D)\text{ in }H^{1,1}_{\mathrm{BC}}(X,\mathbb R). \] Nef means the Chern class lies in the closed Kähler cone; pseudo-effective means it contains a positive current. Semiampleness means an actual positive integral multiple is globally generated. Equality is of Bott--Chern classes, not of line bundles.

Outline of the argument

Manuscript 1 (family 034, 2026-09-27): Contractions of threefold null loci and analytic thickenings prepare the boundary components for a fourfold birational reduction. The comparison on floors and lower strata preserves the actual holomorphic line identity, allowing boundary-generated sections to descend to the nef adjoint. Manuscript 2 (family 034, 2026-10-05): Nonvanishing is obtained by fibrations when they exist and by minimal singular metrics in algebraic dimension zero. Spherical transverse structures, reduced-boundary models, gluing of pluricanonical sections, and multiplier-ideal extension address the remaining zero-dimensional-algebraic case before returning to the assumed nef endpoint. Manuscript 3 (family 034, 2026-09-27): Finite-cover invariants lift successive neighborhood layers into divisorial sheaves. Their positive twists provide increasing section counts while the cohomological loss stays bounded, forcing growth without assuming convergence of an infinite formal neighborhood. Manuscript 4 (family 034, 2026-10-04): The induction retains both the nef line and the effective negative part through a restricted birational program. Boundary generation, signed-support root constructions, Hodge-module lifting, and compact cycle deformations produce nonvanishing; affine changes in negative multiplicities make the selected wall crossings finite.

Manuscript 5 (family 034, 2026-09-24): A dimension induction establishes nonvanishing before constructing good models in that same dimension. Cyclic-cover eigenspaces and logarithmic Hodge modules control the boundary lifting obstruction, while the companion subadditivity result supplies the fibration step; descent of the evaluation map recovers the actual line bundle over the original field. Manuscript 6 (family 034, 2026-09-27): An extremal normalized-section argument contradicts a positive Lelong number by improving the minimal weight locally. For injectivity, harmonic forms and local primitives are compared with both endpoint weights; weak convergence in a fixed Hilbert space retains the ordinary coherent-cohomology class. Manuscript 7 (family 034, 2026-09-26): Marked curve chains control degrees of rational quotient leaves with compatibility under projections and finite quotients. Scalar bounds and Frobenius estimates handle the index step; a semiample contraction and the relative big-base theorem then bring uniform floor systems back to the original pair. Manuscript 8 (family 036, 2026-09-24): Positive-boundary programs first reduce the possible exceptional loci. Algebraic codimension-two cycle classes give a finite homological decrease after discrepancy control; a final surface criterion eliminates the remaining obstruction without assuming general minimal-model existence.

Manuscript 9 (family 036, 2026-10-03): Ordinary log abundance and terminating programs supply the birational reductions. The remaining full-nef-dimension Calabi--Yau case is treated with jet and Frobenius estimates, after which the positive part descends as a semiample numerical representative; flat nontorsion bundles explain why numerical equivalence is the conclusion. Manuscript 10 (family 036, 2026-10-04): The induction uses projective positive-part results when algebraic reduction has positive-dimensional image. In algebraic dimension zero, decomposition into torus and generically symplectic pieces, extension estimates, and compact cycle spaces provide the numerical representative while preserving the distinction from an actual line identity.

Formal verification

Family 034 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

Family 036 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources