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

Theorem 1.3: for every smooth projective complex threefold $X$, ample integral $H$ and $B_0\in N^1(X)_{\mathbb Q}$ there are $k\ge0$ rational and $R_*>0$ with $\mathrm{ch}_3^{B_0+bH}(E)\le(a^2/6+k)H^2\mathrm{ch}_1^{B_0+bH}(E)$ for every tilt-semistable $E$ of slope zero, and the uncorrected bound for $a>R_*$; 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.

Theorem 1.3: for every smooth projective complex threefold $X$, ample integral $H$ and $B_0\in N^1(X)_{\mathbb Q}$ there are $k\ge0$ rational and $R_*>0$ with $\mathrm{ch}_3^{B_0+bH}(E)\le(a^2/6+k)H^2\mathrm{ch}_1^{B_0+bH}(E)$ for every tilt-semistable $E$ of slope zero, and the uncorrected bound for $a>R_*$; 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.