Corrected Chern-character inequalities for tilt-stable threefold objects
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For a smooth projective complex threefold X, ample integral H and rational B₀, can the third Chern character of every finite-slope zero-tilt-slope semistable E be bounded uniformly by ch₃^(B₀+bH)(E)≤(a²/6+k)H²ch₁^(B₀+bH)(E), for a>0 and real b, with k depending only on (X,H,B₀)? A second target removes k uniformly in b for a above a fixed R*>0.
Why it matters and what counts as progress
Why it matters: Corrected Chern bounds support threefold stability constructions while avoiding an uncorrected all-volume assertion known to fail.
What counts as progress: Prove the corrected bound and its uncorrected large-volume tail for new threefolds or all polarized threefolds; the uncorrected all-volume assertion is false.
How the problem developed
- 2014 · Conjecture [1]
Arend Bayer, Emanuele Macrì and Yukinobu Toda Introduced the double-tilt framework and proposed the third-Chern-character inequality underlying threefold Bridgeland stability constructions.
- 2017 · Partial result [1]
Pietro Bernardara, Emanuele Macrì, Benjamin Schmidt and Xiaolei Zhao Proved a corrected inequality for Fano threefolds polarized by a divisor proportional to −K_X and asked for a curve-class correction on general polarized threefolds.
- 2019 · Partial result [1]
Andrés Martínez and Benjamin Schmidt Formulated a version with a fixed rational transverse twist and gave examples showing the uncorrected inequality can fail at finite parameters.
- 24 Sept 2026 · Reported claim [1]
OpenAI The manuscript claims a uniform corrected inequality on every polarized smooth projective complex threefold and an uncorrected bound beyond a fixed volume threshold.
AI contribution
Prescribed large-volume charges on threefolds with trivial canonical bundle
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*.
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