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Math-2532 · mathematics · Algebraic geometry

Arithmetic Stein-degree bounds for log Calabi–Yau boundaries

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each $d\geq 1$ and real $t>0$, is there a finite integer N(d,t) bounding $dim_k$ $H^{0}(S^\nu ,O_{S^\nu })$ whenever (X,B) is a normal integral projective lc rational pair over a characteristic-zero field k with dim $X=d$, $H^{0}(X,O_X)=k$ and $K_X+B$ Q-linearly trivial, and S is a prime boundary component of coefficient at least t? This must also bound the Stein degree of $S\to Spec$ k.

Short English statement

For each $d\geq 1$ and real $t>0$, is there a finite integer N(d,t) bounding $dim_k$ $H^{0}(S^\nu ,O_{S^\nu })$ whenever (X,B) is a normal integral projective lc rational pair over a characteristic-zero field k with dim $X=d$, $H^{0}(X,O_X)=k$ and $K_X+B$ Q-linearly trivial, and S is a prime boundary component of coefficient at least t? This must also bound the Stein degree of $S\to Spec$ k.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2532-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2532

Manuscript 1 (family 034, 2026-09-25): 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.

Arithmetic Stein-degree bounds for log Calabi–Yau boundaries

OpenAI's claim

For a projective log Calabi--Yau pair over a characteristic-zero field, the algebraic constants of a boundary component of coefficient at least $t$ are claimed to have degree bounded solely by dimension and $t$. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For every $d\ge1$ and real $t>0$, OpenAI claims $N(d,t)$ valid over every characteristic-zero field $k$. Let $(X,B)$ be a normal integral projective log canonical rational pair of dimension $d$ with $B\ge0$, $H^0(X,\mathcal O_X)=k$ and $K_X+B\sim_{\mathbb Q}0$. For a prime boundary component $S$ of coefficient at least $t$, it asserts \[ [k_S:k]\le N(d,t),\qquad k_S=\text{relative algebraic closure of }k\text{ in }k(S). \] In particular the normalized component has $\dim_kH^0(S^\nu,\mathcal O_{S^\nu})=[k_S:k]$ bounded by the same integer, controlling its arithmetic Stein degree as well as its geometric components.

Outline of the argument

Manuscript 1 (family 034, 2026-09-25): A bounded polarization power descends across a controlled field extension, and strata with integral weights bound Galois orbits of divisorial valuations. If the distinguished divisor becomes vertical in a Mori fibration, a horizontal coefficient-one component and adjunction reduce the dimension.

Formal verification

Family 034 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

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