Let $X$ be a smooth projective variety with Picard number 1 over $\mathbb{C}$. Let $F$ be a coherent sheaf on $X$ such that $c_1(F)$ is algebraically trivial, and hence numerically trivial. Also rank $F=0$.

So $F$ is supported on a proper closed subscheme of $X$. Can $F$ be supported on a divisor, or is the codimension of Support $ F\geq 2$.

Since the Picard number of $X$ is 1, any non-zero effective divisor is ample. Since $c_1(F)$ is numerically trivial, if $F$ is supported on a divisor, it is non-effective. Is this possible?

Note, I posted this question on MathSE, but have not received any answers.