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Math-2375 · mathematics · Algebraic geometry

Log Abundance for Semi-Log-Canonical Pairs

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $(X,\Delta)$ be a projective semi-log-canonical pair with $\Delta$ a $\mathbb Q$-divisor. If $K_X+\Delta$ is nef, must $K_X+\Delta$ be semi-ample?

Formulation references: [S1], [S2].

Short English statement

Let $(X,\Delta)$ be a projective semi-log-canonical pair with $\Delta$ a $\mathbb Q$-divisor. If $K_X+\Delta$ is nef, must $K_X+\Delta$ be semi-ample?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2375-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2375

Manuscript 1 (family 034, 2026-09-27): Semiampleness after nonvanishing for nef klt adjoints on compact Kähler fourfolds; normal lc/klt conclusions or conditional fourfold conclusions do not supply unconditional pluricanonical descent and generation for all nonnormal semi-log-canonical pairs as this record requires.

Manuscript 2 (family 034, 2026-10-05): Conditional good minimal models for strongly Q-factorial compact Kähler fourfolds; normal lc/klt conclusions or conditional fourfold conclusions do not supply unconditional pluricanonical descent and generation for all nonnormal semi-log-canonical pairs as this record requires.

Manuscript 3 (family 034, 2026-09-27): Canonical nonvanishing for smooth projective complex fourfolds, with an lc nef-adjoint consequence; normal lc/klt conclusions or conditional fourfold conclusions do not supply unconditional pluricanonical descent and generation for all nonnormal semi-log-canonical pairs as this record requires.

Manuscript 4 (family 034, 2026-09-27): Supported-boundary lifting and fourfold abundance after nonvanishing; normal lc/klt conclusions or conditional fourfold conclusions do not supply unconditional pluricanonical descent and generation for all nonnormal semi-log-canonical pairs as this record requires.

Manuscript 5 (family 034, 2026-10-04): Kähler log abundance conditional on the stated logarithmic Iitaka assumption; normal lc/klt conclusions or conditional fourfold conclusions do not supply unconditional pluricanonical descent and generation for all nonnormal semi-log-canonical pairs as this record requires.

Manuscript 6 (family 034, 2026-09-24): Semiampleness of nef rational adjoints of projective log canonical pairs in characteristic zero; normal lc/klt conclusions or conditional fourfold conclusions do not supply unconditional pluricanonical descent and generation for all nonnormal semi-log-canonical pairs as this record requires.

Manuscript 7 (family 034, 2026-09-26): Effective log-Iitaka systems for pseudo-effective rational lc adjoints on complex fourfolds; normal lc/klt conclusions or conditional fourfold conclusions do not supply unconditional pluricanonical descent and generation for all nonnormal semi-log-canonical pairs as this record requires.

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.

Log Abundance for Semi-Log-Canonical Pairs

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): A smooth projective complex fourfold with pseudo-effective canonical divisor is claimed to possess a nonzero pluricanonical section. Manuscript 4 (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 5 (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 6 (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 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.

Mathematical statement of the claimed result

Mathematical claim 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.
Mathematical claim 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.
Mathematical claim 3.
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.
Mathematical claim 4.
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.
Mathematical claim 5.
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.
Mathematical claim 6.
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.
Mathematical claim 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.
Mathematical claim 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.

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): 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 4 (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 5 (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 6 (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 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.

Formal verification

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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

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