Sheaves with zero Chern classes on a $K3$ surface.

Question

Let $S$ be a $K3$ surface. Is it true that any sheaf on $S$ with zero Chern classes is isomorphic to $\mathcal{O}_S^{\oplus n}$ for some $n$? If not, do you have any counterexample?

Answer 1

The answer is no. Here is a counterexample.

Take an ample divisor $L$ on $S$ and let $Z \subset S$ be a zero-dimensional subscheme of length $\ell(Z)=L^2$.

Now consider the coherent sheaf $$\mathscr{F}=\mathscr{O}_S(-L) \oplus \mathscr{O}_S(L) \otimes \mathscr{I}_Z.$$

Straightforward computations show that $$c_t(\mathscr{O}_S(-L))=1-Lt, \quad c_t(\mathscr{O}_S(L)\otimes \mathscr{I}_Z)=1+Lt + \ell(Z) t^2,$$

hence $c_t(\mathscr{F})=1$.

So $\mathscr{F}$ has zero Chern classes, but it is not isomorphic $\mathscr{O}_S^{\oplus 2}$ because it is not locally free.

Note that this construction holds for any smooth projective surface $S$, in fact the assumption $K3$ is not used here.