The Miyaoka–Yau inequality and equality criterion for arbitrary weights

Assume Setup, with $X$ a Fano manifold and $\omega \in c_1(X)$. Let $v \colon P \to \mathbb{R}_{>0}$ be a weight, let $\mathscr{V}$ be a canonical extension sheaf with extension class $\lambda c_1(X)$ for some $\lambda>0$, and let $\delta'_v(X)$ denote the greatest Ricci lower bound associated with $v$. For a saturated subsheaf $\mathcal{F}\subset\mathscr{V}$ of rank $r$, write $c_1^{\mathbb{T}}(\mathcal{F})$ for its equivariant first Chern class. Miyaoka–Yau conjecture for arbitrary weights. There exists such a canonical extension sheaf $\mathscr{V}$ for which

$$(c_1^{\mathbb{T}}(\mathcal{F})\cdot v(c_1^{\mathbb{T}}(X)))\le\left(1-\delta'_v(X)\left(1-\frac{r}{n+1}\right)\right)(c_1^{\mathbb{T}}(X)\cdot v(c_1^{\mathbb{T}}(X)))$$

for every saturated subsheaf $\mathcal{F}\subset\mathscr{V}$ of rank $r$. This is proposed as an equivariant Miyaoka–Yau inequality with an equality criterion analogous to the known result for the relevant special weight; the arbitrary-weight case remains open.