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Math-2410 · mathematics · Algebraic geometry

Bogomolov–Gieseker Inequality for Tilt-Semistable Threefold Complexes

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.
BG inequality conjecture: let $X$ be a smooth projective threefold over $\mathbb C$, let $B,\omega\in\operatorname{NS}(X)_{\mathbb Q}$ with $\omega$ ample, and let $E$ be tilt-semistable with $\nu_{\omega,B}(E)=0$. Then $\operatorname{ch}^{B}_{3}(E)\leq\frac{1}{18}\operatorname{ch}^{B}_{1}(E)\cdot\omega^{2}$. The conjecture is known for $X=\mathbb P^{3}$, smooth quadrics, and étale quotients of abelian threefolds.

Formulation references: [S1], [S2].

Short English statement

BG inequality conjecture: let $X$ be a smooth projective threefold over $\mathbb C$, let $B,\omega\in\operatorname{NS}(X)_{\mathbb Q}$ with $\omega$ ample, and let $E$ be tilt-semistable with $\nu_{\omega,B}(E)=0$. Then $\operatorname{ch}^{B}_{3}(E)\leq\frac{1}{18}\operatorname{ch}^{B}_{1}(E)\cdot\omega^{2}$. The conjecture is known for $X=\mathbb P^{3}$, smooth quadrics, and étale quotients of abelian threefolds.

Sources

Catalogue entry

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

Contributions 1

Math-2410-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2410

Manuscript 1 (family 055, 2026-09-24): The primary introduction gives the uncorrected tilt inequality only in a large-parameter tail (and elsewhere permits an additive correction); the problem asks the strong coefficient-one-eighteenth inequality throughout its tilt parameters. Its large-volume conclusion cannot remove the tail threshold.

Bogomolov–Gieseker Inequality for Tilt-Semistable Threefold Complexes

OpenAI's claim

On a smooth projective threefold with trivial canonical bundle, the ordinary and Todd-corrected large-volume charges are claimed to define full numerical stability conditions uniformly in a real twist and ample-direction sector. 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 smooth projective complex threefold, $H$ ample integral and $B_0\in N^1(X)_{\mathbb Q}$. With $\operatorname{ch}^B=e^{-B}\operatorname{ch}$, define $\mathcal B_{H,B}$ by tilting coherent sheaves at slope $H^2\operatorname{ch}_1^B/(H^3\operatorname{ch}_0)$, using positive versus nonpositive Harder--Narasimhan slopes. Put \[ \nu_{a,b}(E)=\frac{H\operatorname{ch}_2^{B_0+bH}(E)-a^2H^3\operatorname{ch}_0(E)/2}{H^2\operatorname{ch}_1^{B_0+bH}(E)}. \] OpenAI claims constants $k\in\mathbb Q_{\ge0}$ and $R_*>0$, depending only on $(X,H,B_0)$, such that, simultaneously for all $a>0$ and $b\in\mathbb R$, every finite-slope $\nu_{a,b}$-semistable object of slope zero satisfies \[ \operatorname{ch}_3^{B_0+bH}(E)\le(a^2/6+k)H^2\operatorname{ch}_1^{B_0+bH}(E), \] and the same inequality with $k=0$ holds whenever $a>R_*$, uniformly in real $b$. If $K_X\simeq\mathcal O_X$, then for each $(B_*,H_*)\in N^1(X)_{\mathbb R}\times\operatorname{Amp}(X)$ it claims open neighborhoods $V$ of $B_*$ and $W$ of $H_*$, together with a single $T>0$, such that all $B\in V$, $H\in W$ and $t\ge T$ admit geometric numerical Bridgeland stability conditions supported on the full real numerical Grothendieck space, with exact charges \[ Z_{B,tH}(E)=-\int_Xe^{-B-itH}\operatorname{ch}(E),\qquad Z^{\mathrm{LV}}_{B,tH}(E)=-\int_Xe^{-B-itH}\operatorname{ch}(E)(1+c_2(X)/24). \] For the first charge the heart is the second tilt at slope $[tH\operatorname{ch}_2^B-t^3H^3\operatorname{ch}_0/6]/[t^2H^2\operatorname{ch}_1^B]$ with the same positive/nonpositive cuts. Geometric means point sheaves are stable of phase one.

Outline of the argument

Manuscript 1 (family 055, 2026-09-24): Product embeddings and restriction-deformation methods control the full Grothendieck group, beyond the four polarized Chern degrees. A separate fixed-volume tilt estimate bounds the third-character error in proportion to the discriminant defect, and slope truncation preserves a threshold uniform in the object and twist.

Formal verification

Family 055 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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