Here is another question from the paper "Second Chern class and Riemann-Roch for vector bundles on resolutions of surfaces singularities" by J. Wahl. This is at the beginning of the section $2$.
$(\tilde{X},E) \to (X,o)$ is a good resolution of a normal singularity. $\mathcal{F}$ is a rank $2$ vector bundle on $\tilde{X}$. A subbundle is a line bundle $L$ with invertible quotient. Note that $$L \cdot (\det{\mathcal{F}} - L)= -(L - \det{\mathcal{F}}/2)^2 + \frac{1}{4}(\det{\mathcal{F}})^2 \geq \frac{1}{4}(\det{\mathcal{F}})^2 .$$ Then we can define $$c(\mathcal{F})=\min\{L \cdot (\det{\mathcal{F}} - L) | L \subset \mathcal{F} \text{ a subbundle} \} $$ and $$c_2(\mathcal{F}) = \inf\{c(\tilde{f}^* \mathcal{F})/\deg \tilde{f} | \tilde{f}: \tilde{Y} \to \tilde{X} \text{ proper, generically finite} \} .$$
The question is this means $$(L - \det{\mathcal{F}}/2)^2 \leq 0.$$ If the exact sequence is given by $$ 0 \to L \to \mathcal{F} \to Q \to 0$$ and if $\tilde{X}$ is smooth, we have $$(L - Q)^2 \leq 0 \implies c_1^2(\mathcal{F}) -4 c_2(\mathcal{F}) \leq 0.$$ Then by Bogomolov's theorem, we know that this excludes unstable bundles. But this section mentions nothing about stability so far. I am confused if this definition of second Chern class works for all bundles. I don't think $(L - \det{\mathcal{F}}/2)^2 \leq 0$ is true for all rank $2$ bundles.
The paper also said in $(2.1.1)$ that for this definition of $c_2$ using $\inf$, $$c_1^2(\mathcal{F}) \leq 4 c_2(\mathcal{F}).$$
