PomeDefinition 11.18.3.3B58CDB44607 For a Gauduchon form $\omega $ on $X$, define
\[ \deg _\omega E=\deg _{\omega ^{n-1}}\det E =\int _ X\frac{\mathrm i\operatorname {tr}F_ h}{2\pi }\wedge \omega ^{n-1}, \qquad \mu _\omega (E)=\frac{\deg _\omega E}{r}. \]
For a nonzero torsion-free coherent sheaf $\mathcal F$, use the determinant line bundle $\det \mathcal F=(\bigwedge ^{\operatorname {rank}\mathcal F}\mathcal F)^{**}$ to define its degree and slope. The bundle $E$ is $\omega $-slope-stable if $\mu _\omega (\mathcal F)<\mu _\omega (E)$ for every coherent subsheaf $\mathcal F\subset E$ with $0<\operatorname {rank}\mathcal F<r$ and torsion-free quotient.
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