Slope of a holomorphic bundle on a Kähler threefold

Let $X$ be a compact Kähler complex threefold with complex structure $J$ and a specified Kähler form $\omega$; orient $X$ by its complex structure and write $[\omega ]\in H_{\mathrm{d}\mathrm{R}}^2(X;\mathbb{R})$ for its de Rham class. For a locally free holomorphic vector bundle $E\to X$ of rank $r>0$, let $c_1^{\mathrm{t}\mathrm{o}\mathrm{p}}(E)\in H^2(X;\mathbb{Z})$ be its topological first Chern class as in def-chern-classes-from-the-projective-bundle-relation. Write $c_1^{\mathbb{R}}(E)\in H_{\mathrm{d}\mathrm{R}}^2(X;\mathbb{R})$ for the inverse de Rham image of its coefficient change $c_1^{\mathrm{t}\mathrm{o}\mathrm{p}}(E)\mapsto (c_1^{\mathrm{t}\mathrm{o}\mathrm{p}}(E){)}_{\mathbb{R}}$, using the natural de Rham isomorphism specified in thm-chern-weil-forms-represent-the-at-characteristic-classes-over-the-reals. Assume full AC as in that supplier. Define its $\omega$-degree and $\omega$-slope by ${\mathrm{deg}}_{\omega }(E):=⟨c_1^{\mathbb{R}}(E)⌣[\omega {]}^2,[X]⟩,{\mu }_{\omega }(E):=\frac{{\mathrm{deg}}_{\omega }(E)}{r},$ where $[X]$ is the complex-orientation fundamental class. A rescaling $\omega \mapsto t\omega$ for $t>0$ multiplies both degree and slope by $t^2$. This convention alone does not assert (semi)stability or existence of a Hermitian–Yang–Mills connection.

Kähler class, canonical bundle, and Ricci form

Chern classes from the projective-bundle relation

Characteristic forms represent topological characteristic classes over the reals