Let $(X,\mathcal O_X)$ be a ringed space and $\mathcal F$ be a coherent sheaf of $\mathcal O_X$-modules on $(X,\mathcal O_X)$. Are there the definitions of rank, degree and slope of $\mathcal F$ in this general situation? I've seen only definitions for different particular cases. For example, if $X$ is a projective algebraic variety together with an embedding $X \to \mathbb P^N$ and with $\dim X = \dim \mathcal F = d$ then one defines first the Hilbert polynomial $$ P(\mathcal F,m) = \chi(\mathcal F \otimes \mathcal O(m)) \equiv \sum_{i=0}^{d} \alpha_i(\mathcal F) \frac{m^i}{i!}, $$ where $\mathcal O(m)$ is the bundle on $X$ induced by $\mathcal O_{\mathbb P^N}(m)$, and then puts $\DeclareMathOperator{\rank}{rank} \rank \mathcal F = \frac{\alpha_d(\mathcal F)}{\alpha_d(\mathcal O_X)}$ and $\deg \mathcal F = \alpha_{d-1}(\mathcal F)- \rank(\mathcal F) \alpha_{d-1}(\mathcal O_X)$ and finally $\DeclareMathOperator{\slope}{slope} \slope \mathcal F = \frac{\deg \mathcal F}{\rank \mathcal F}$. Another special case is when $X$ is a compact Kahler manifold with the Kahler form $\omega$ and $\mathcal F$ is a torsion-free holomorphic vector bundle, when $\rank \mathcal F$ is the rank of the vector bundle and $$ \deg \mathcal F = \int\limits_X c_1(\mathcal F) \wedge \omega^{n-1}. $$
But what are the definitions of these quantities for example in the case of a complex manifold $X$ (possibly, Kahler) but when $\mathcal F$ is a general coherent sheaf?