Stable, semistable and simple sheaves on a $K3$

Question

It is well known that every stable sheaf on a K3 surface $S$ is simple but the contrary is not true. Moreover, if $M$ denotes the (coarse) moduli space of stable sheaves on $S$ with fixed Chern classes and $Spl$ the moduli space of simple sheaves on $S$ with the same Chern classes, $M$ is an open subscheme of $Spl$. Can there exist a component of $Spl$ not intersecting $M$ or is $M$ dense in $Spl$? What about the relation between semistable and simple sheaves? Is a simple sheaf on a K3 surface always semistable? If not, can you give me a counterexample?

Answer 1

$M(c_1, c_2)$ is not always dense in $Spl(c_1, c_2)$, in other words irreducible components of $Spl(c_1, c_2)$ not intersecting $M(c_1, c_2)$ may actually exist.

In fact, in his paper

"Moduli of simple rank-$2$ sheaves on $K3$-surfaces",

Manuscripta Math. 79 (1993), no. 3-4, 253–265,

Z. Qin constructs irreducible components of $Spl(c_1, c_2)$, for suitable values of $c_1$, $c_2$, in which no sheaf is stable. Moreover, he proves the following

Theorem. Let $X$ be a $K3$-surface. Assume that $(4c_2-c^2_1)>16$, and that $S$ is an irreducible component in $Spl(c_1,c_2)$ such that no sheaf in $S$ is stable. Then $S$ is birational to either ${\rm Hilb}^d(X)$ or $X\times{\rm Hilb}^{d-1}(X)$.

I did not check whether these components contain some strictly semistable sheaves.