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Math-2550 · mathematics · Algebraic geometry

Gepner-type numerical stability on quintic threefolds

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every smooth complex quintic $X\subset P^{4}$, is there a locally finite numerical Bridgeland stability condition (Z,P) with the full numerical support property, normalized by $Z(O_x)=-1$ for each point, such that $\Phi =T_{O_X}\circ (-\otimes O_X(1))$ sends $P(\varphi )$ to $P(\varphi +2/5)$ and $Z(\Phi E)=\exp (2\pi i/5)Z(E)$?

Short English statement

For every smooth complex quintic $X\subset P^{4}$, is there a locally finite numerical Bridgeland stability condition (Z,P) with the full numerical support property, normalized by $Z(O_x)=-1$ for each point, such that $\Phi =T_{O_X}\circ (-\otimes O_X(1))$ sends $P(\varphi )$ to $P(\varphi +2/5)$ and $Z(\Phi E)=\exp (2\pi i/5)Z(E)$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2550-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2550

Manuscript 1 (family 055, 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.

Gepner-type numerical stability on quintic threefolds

OpenAI's claim

Every smooth complex quintic threefold is claimed to carry a numerical Bridgeland stability condition with the normalized Gepner charge and phase shift $2/5$ under the specified autoequivalence. 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 smooth quintic hypersurface $X\subset\mathbb P^4_{\mathbb C}$, define on $D^b(\operatorname{Coh}X)$ the autoequivalence \[ \Phi=T_{\mathcal O_X}\circ(-\otimes\mathcal O_X(1)),\qquad T_{\mathcal O_X}(E)=\operatorname{Cone}\left(R\operatorname{Hom}(\mathcal O_X,E)\otimes\mathcal O_X\to E\right). \] OpenAI claims a numerical Bridgeland stability condition $(Z,\mathcal P)$ with support property on the full real numerical Grothendieck group such that \[ Z(\Phi E)=e^{2\pi i/5}Z(E),\quad \Phi(\mathcal P(\varphi))=\mathcal P(\varphi+2/5),\quad Z(\mathcal O_x)=-1 \] for every object $E$, real phase $\varphi$ and closed point $x$. Support means a norm and constant $C$ bound the numerical class of every semistable object by $C|Z(E)|$.

Outline of the argument

Manuscript 1 (family 055, 2026-09-24): Two independent perturbations of the second slope tilt control rank and first Chern character before semistability is defined. The charge bounds the remaining numerical coordinates, enabling finite refinements and Harder--Narasimhan filtrations even for a nondiscrete charge image; compatibility of the tilted aisles supplies the Gepner rotation.

Formal verification

Family 055 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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