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?



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


