Stable vector bundles with \(c_1=0\) on K3 surfaces (Q1973444)

Let \(S\) be a complex projective K3 surface and \(H\) an ample line bundle on \(S\). Denote by \(M_H(r,0,c_2)\) the moduli space of \(H\)-stable vector bundles \(E\) on \(S\) of rank \(r\) with Chern classes \(c_1(E)=0\), \(c_2(E)=c_2\). In case \(r=2\), these moduli spaces have been studied intensively. In this paper the author studies stable bundles with trivial first Chern class on a K3 surface. A necessary and sufficient condition for the existence of such bundles is proved and it is shown that Mukai's reflection functor gives a new compactification of the moduli space. A relation to the notion of \(T\)-duality in theoretical physics is also discussed. The main result is: Let \((S,H)\) be a polarized K3 surface and let \(r\geq 2\), then \(M_H(r,0,c_2)\) is non-empty if and only if \(c_2\geq 2r\).