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Math-2570 · mathematics · Algebraic geometry

The ordinary-double-point normalized-volume gap

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For any singular point x on a normal complex algebraic n-fold, $n\geq 2$, with zero boundary, Q-Cartier $K_X$ and klt singularities near x, must the normalized volume be at most $2(n-1)^n$, attaining that value exactly for the analytic germ of $\Sigma _{i=0}^n$ $z_i^{2}=0$?

Short English statement

For any singular point x on a normal complex algebraic n-fold, $n\geq 2$, with zero boundary, Q-Cartier $K_X$ and klt singularities near x, must the normalized volume be at most $2(n-1)^n$, attaining that value exactly for the analytic germ of $\Sigma _{i=0}^n$ $z_i^{2}=0$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2570-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2570

Manuscript 1 (family 037, 2026-09-24): 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.

The ordinary-double-point normalized-volume gap

OpenAI's claim

Every singular boundary-zero complex klt germ of dimension $n$ is claimed to have normalized volume at most $2(n-1)^n$, with equality exactly for an analytic ordinary double point. 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 $x$ be a closed point that is singular on a normal algebraic complex $n$-fold, $n\ge2$, with rational Cartier $K_X$ and boundary-zero klt singularity near $x$. Define normalized volume by \[ \widehat{\operatorname{vol}}(x,X)=\inf_{v\text{ centered at }x}A_X(v)^n\operatorname{vol}(v), \] where $A_X$ is log discrepancy and $\operatorname{vol}(v)=\limsup_{t\to\infty}n!\,\operatorname{length}(\mathcal O_{X,x}/\{f:v(f)\ge t\})/t^n$. OpenAI claims $\widehat{\operatorname{vol}}(x,X)\le2(n-1)^n$, with equality exactly when the analytic germ is the ordinary double point $\{z_0^2+\cdots+z_n^2=0\}\subset\mathbb C^{n+1}$. All singular boundary-zero klt germs are quantified, not only hypersurfaces.

Outline of the argument

Manuscript 1 (family 037, 2026-09-24): The four-dimensional companion result anchors an induction using minimizing cones and integral approximations of their Reeb gradings. Cyclic stabilizer characters are retained through orbifold quotients, and same-dimensional singular slices compare the remaining volume bounds and equality cases.

Formal verification

Family 037 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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