The Bogomolov–Gieseker conjecture for stable sheaves and arbitrary weights

Let Setup be fixed, let $v \colon P \to \mathbb{R}_{>0}$ be a weight, and let $\mathcal{E}$ be a reflexive sheaf of rank $r$ that is $v$-stable. Bogomolov–Gieseker conjecture. The inequality

$$\left((2rc_2^{\mathbb{T}}(\mathcal{E})-(r-1)c_1^{\mathbb{T}}(\mathcal{E})^2)\cdot v(\alpha_{\mathbb{T}})\right)\ge 0$$

holds. If equality holds, then $\mathcal{E}$ is locally free and projectively Hermitian flat. This is the arbitrary-weight analogue of the Bogomolov–Gieseker inequality; the statement does not provide a resolution.