Prescribed large-volume charges on threefolds with trivial canonical bundle

OpenAI

Mathematics

For each smooth projective complex threefold X, ample integral H and rational B₀, the manuscript gives k≥0 and R*>0 depending only on (X,H,B₀) such that every finite-slope, zero-tilt-slope semistable E satisfies ch₃^(B₀+bH)(E)≤(a²/6+k)H²ch₁^(B₀+bH)(E) for all a>0 and real b. The k correction vanishes uniformly in b for a>R*.

What was already known

Earlier corrected inequalities covered certain Fano threefold polarizations. An uncorrected bound for every volume is false.

AI involvement AI-led

The theorem appears in an OpenAI-authored model-produced manuscript; no named human checker or explanation of this numerical argument is documented.

OpenAI is credited as author of the manuscript and attributes repository results to its internal model; autonomy details are absent.

Supporting source (openai.com) — October 6 release, model origin

Importance Notable

Would extend a corrected Bogomolov–Gieseker inequality to every polarized smooth complex threefold.

Limitations

The all-volume estimate and uniform large-volume tail are manuscript claims without a documented complete human or machine check. No condition on K_X is assumed.

Technical verification Reported

The manuscript states a proof strategy and theorem, but no completed human or machine check of this precise all-threefold inequality was documented.

Human understanding Few

Default for missing evidence: no named human explanation of the central fixed-volume estimate and wall argument was found.

Correction and publication history

2026-10-18 · publication