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Math-198-A2 · revision r1 · on Math-198

OpenAI claim summary for Math-198

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, 2026-10-05): 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 2 (family 034, 2026-09-27): The smooth canonical theorem proves first-section existence in dimension four and the lc corollary treats nef adjoints in dimension at most four; the catalogue asks for arbitrary-dimensional pseudo-effective lc adjoints.

Manuscript 3 (family 034, 2026-09-27): The general lifting reduction assumes smooth canonical nonvanishing in every dimension and needs specified supported-boundary data; it is a conditional reduction, not the catalogue unconditional first-section assertion.

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): Log abundance gives a nonzero Cartier-multiple section for nef lc adjoints; our nonvanishing target assumes only pseudo-effectivity, so the nef hypothesis is an additional restriction.

Manuscript 6 (family 034, 2026-09-24): The smooth canonical-good-model theorem covers zero-boundary smooth complex canonical divisors, including nonvanishing and abundance there. It does not cover arbitrary log-canonical boundaries or all slc varieties.

Manuscript 7 (family 034, 2026-09-26): The log-Iitaka main theorem gives a nonempty system for every pseudo-effective lc rational adjoint on a complex fourfold, which is the desired conclusion only in dimension four.

Manuscript 8 (family 034, 2026-10-05): Uniform trivial multiples are obtained for the log-Calabi–Yau subclass. This does not cover general nef slc adjoints, or all pseudo-effective projective lc adjoints.

Manuscript 9 (family 036, 2026-09-24): A good minimal model gives canonical nonvanishing for smooth projective varieties of numerical dimension one; our target ranges over all pseudo-effective lc adjoints and numerical dimensions.

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

Manuscript

Contents
  1. Nonvanishing conjecture for projective log canonical pairs
  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. Outline of the argument
  14. Formal verification
  15. Status caveat
  16. Sources

Nonvanishing conjecture for projective log canonical pairs

OpenAI's claim

Manuscript 1 (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 2 (family 034, 2026-09-27): A smooth projective complex fourfold with pseudo-effective canonical divisor is claimed to possess a nonzero pluricanonical section; 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, 2026-09-24): The Campana--Peternell Kodaira inequality and the specified Schnell fiber-space conclusion are claimed from a pseudo-effective difference between a canonical multiple and the chosen effective or pulled-back divisor. 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 034, 2026-10-05): A dimension- and coefficient-dependent multiple is claimed to trivialize the log canonical line of every connected projective semi-log-canonical Calabi--Yau pair, independently of its number of components; Manuscript 9 (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.

Mathematical statement of the claimed result

Manuscript 1.
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 2.
For every smooth connected projective complex fourfold $X$ with pseudo-effective canonical class, OpenAI claims \[ \exists m\in\mathbb Z_{>0}:\quad H^0(X,\mathcal O_X(mK_X))\ne0. \] Pseudo-effective means lying in the closure of the cone of effective divisor classes. This is nonvanishing in dimension four; it does not by itself assert semiampleness or nonvanishing in every dimension. The cited companion metric input says that for a projective complex pair that is normal, connected and klt $(H,\Theta)$ with effective rational boundary and nef rational Cartier $D_H=K_H+\Theta$, every projective log resolution $\pi:V\to H$ gives a rational line bundle $\pi^*D_H$ with a semipositive singular metric having minimal singularities; every metric satisfying this condition has vanishing Lelong numbers at all points and trivial multiplier ideals at every positive exponent.
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.
Let $f:X\to Y$ be a connected-fiber surjective morphism of smooth connected projective complex varieties, with smooth geometric generic fiber $F$ satisfying $\kappa(F)=0$. If some ample Cartier divisor $H$ on $Y$ and integer $m_0>0$ have $m_0K_X-f^*H$ pseudo-effective, OpenAI claims \[ \kappa(X)=\dim Y. \] The manuscript also invokes its companion's canonical good-model theorem: every smooth connected projective complex $T$ with pseudo-effective $K_T$ has a $K_T$-negative birational contraction to a normal projective rational-factorial klt $V$ with $K_V$ semiample, and on a common smooth resolution $p:W\to T$, $q:W\to V$, \[ p^*K_T=q^*K_V+E,\qquad E\ge0,\quad q_*E=0. \] The good-model statement is explicitly credited to the companion theorem, not presented as an independent proof here.
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 $k$ have characteristic zero and be algebraically closed; over $k$, fix $d\ge4$ and a finite $\Phi\subset[0,1]\cap\mathbb Q$. OpenAI asserts an integer $a=a(d,\Phi)>0$ such that every connected projective equidimensional semi-log-canonical pair $(X,B)$ of dimension $d$, with nonzero boundary coefficients in $\Phi$ and $K_X+B\sim_{\mathbb Q}0$, satisfies \[ a(K_X+B)\text{ is Cartier},\qquad \mathcal O_X(a(K_X+B))\simeq\mathcal O_X. \] Semi-log-canonical means $X$ is reduced, $S_2$ and nodal in codimension one, $B$ avoids its nonnormal locus, and the normalization with the conductor boundary is log canonical. The hypothesis is rational linear triviality, not merely numerical triviality.
Manuscript 9.
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.

Outline of the argument

Manuscript 1 (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 2 (family 034, 2026-09-27): Minimal metrics and moving centers reduce the problem to compatible jet-order and rank estimates on two factors. Frobenius diagonal filtrations convert ordinary jets to truncated monomials, while instability control keeps the numerical separation needed to force nonvanishing. 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, 2026-09-24): The total-space canonical model, with its good-model property, supplies the geometric reduction. Cartier intersection theory and proper pushforward compare the base with the canonical image without requiring either map to factor through the other; transfer of sections then yields the Kodaira and fiber formulas.

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 034, 2026-10-05): The normal components first receive a uniform pluricanonical exponent. Residues along conductor strata turn descent into a scalar-gluing problem, whose possible characters are bounded using Hodge-norm rank estimates and moving-subvariety fibrations; this produces a common exponent for the glued pair. Manuscript 9 (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.

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