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Math-2588 · mathematics · Algebraic geometry

Lifting an adjoint section from its reduced support

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let (V,C) be a normal projective Q-factorial dlt pair in characteristic zero with effective rational C, and put $A=K_V+C$. Suppose a nonzero effective Cartier divisor G is linearly equal to qA for a sufficiently divisible positive q and $Supp(G)\subset Supp(\lfloor C\rfloor )$. If $O_V(G)$ restricted to $G_red$ is semiample, must $\kappa (V,A)>0$?

Short English statement

Let (V,C) be a normal projective Q-factorial dlt pair in characteristic zero with effective rational C, and put $A=K_V+C$. Suppose a nonzero effective Cartier divisor G is linearly equal to qA for a sufficiently divisible positive q and $Supp(G)\subset Supp(\lfloor C\rfloor )$. If $O_V(G)$ restricted to $G_red$ is semiample, must $\kappa (V,A)>0$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2588-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2588

Manuscript 1 (family 034, 2026-09-27): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Lifting an adjoint section from its reduced support

OpenAI's claim

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. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
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.

Outline of the argument

Manuscript 1 (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.

Formal verification

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

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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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