Suppose that $X$ is a smooth algebraic K3-surface, and let $0 \neq \mathcal{F} \in \operatorname{Coh}(X)$ be a sheaf supported in dimension $0$. Let $\omega \in \operatorname{Pic}(X) \otimes \mathbb{R}$ be an ample class, and let $\beta \in \operatorname{Pic}(X)$ be arbitrary. I want to verify the following claim by Bridgeland in Stability conditions on K3 surfaces: $$c_1(\mathcal{F}) \cdot \beta - \operatorname{rk}(\mathcal F) - \frac 1 2 (c_1(\mathcal{F})^2 + c_2(\mathcal{F})) < 0$$ Clearly $\operatorname{rk}(\mathcal{F}) = 0$, and I think that $c_1(\mathcal F) = 0$ as well, because the support is $0$-dimensional, but I'm not sure about the argument.
Question 1: Is $c_1(\mathcal F) = 0$ true, and how to see this?
If this is settled, then the claim reduces to $$c_2(\mathcal F) > 0, $$ but I don't see any reason for this.
Question 2: Why does this hold?