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A curve-class corrected Bogomolov-Gieseker inequality on every polarized threefold

Bayer, Macri and Toda conjectured an inequality ch3B(E)≤a26H2ch1B(E)\mathrm{ch}_3^B(E)\le\frac{a^2}{6}H^2\mathrm{ch}_1^B(E) for tilt-semistable objects of tilt slope zero on threefolds, which would produce Bridgeland stability conditions; Schmidt and Martinez-Schmidt found counterexamples (blow-ups, contracted divisors, Weierstrass fibrations). Bernardara, Macri, Schmidt and Zhao (2017) proved a corrected form for Fano threefolds and asked whether on every smooth polarized threefold (X,H)(X,H) there is a curve class Γ\Gamma such that ch3B(E)≤a26H2ch1B(E)+Γ⋅ch1B(E)\mathrm{ch}_3^{B}(E)\le\frac{a^2}{6}H^2\mathrm{ch}_1^{B}(E)+\Gamma\cdot\mathrm{ch}_1^{B}(E) for all tilt-semistable EE of slope zero; Martinez and Schmidt (2019) asked the version with a fixed rational transverse twist B0B_0. Does such a corrected inequality hold on every polarized threefold?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Algebraic geometry; tilt stability and Bogomolov-Gieseker inequalities
Posed by
Marcello Bernardara, Emanuele Macri, Benjamin Schmidt and Xiaolei Zhao; twisted version by Cristian Martinez and Benjamin Schmidt
Year posed
2017
Years open
9y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
22 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.3: for every smooth projective complex threefold XX, ample integral HH and B0∈N1(X)QB_0\in N^1(X)_{\mathbb Q} there are k≥0k\ge0 rational and R∗>0R_*>0 with ch3B0+bH(E)≤(a2/6+k)H2ch1B0+bH(E)\mathrm{ch}_3^{B_0+bH}(E)\le(a^2/6+k)H^2\mathrm{ch}_1^{B_0+bH}(E) for every tilt-semistable EE of slope zero, and the uncorrected bound for a>R∗a>R_*; no canonical-bundle hypothesis. The same manuscript constructs, on threefolds with trivial canonical bundle, stability conditions with the exact ordinary and square-root-Todd large-volume charges on an open set of twists and polarizations. It does not give explicit constants or prove the uncorrected BMT inequality at all volumes, which is false in general.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. This entry's principal manuscript (September 24, 2026) is the large-volume companion of the quintic Gepner paper; its uniform inequality is proved independently of its charge constructions.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.3 (uniform strong inequality) of the manuscript was read against the two cited questions; it gives, for any smooth projective complex threefold, ample integral HH and rational B0B_0, constants k∈Q≥0k\in\mathbb Q_{\ge0} and R∗R_* such that every finite-slope νa,b\nu_{a,b}-semistable object of slope zero satisfies the inequality corrected by kH2⋅ch1kH^2\cdot\mathrm{ch}_1, for all a>0a>0 and bb, and the uncorrected inequality once a>R∗a>R_*. The paper itself says it answers these corrected-inequality questions affirmatively. Note the correction class is a multiple of H2H^2. Not refereed. No Lean formalization in the release.

Sources

Changelog1 change

Discussion