Correspondences of K3 surface via moduli of sheaves

K3 surfaces is one of the most remarkable classes of algebraic surfaces which are very important in contemporary Mathematics and Physics. During last 40 years there was a big progress in their understanding, but we still discover some their new, unexpected and important properties. , elements of the Picard lattice (generated by algebraic curves) of a K3 surface deliver remarkable two-dimensional algebraic cycles on the product of the K3 surface with itself, and a correspondence of the K3 surface with itself. It happens when moduli of coherent sheaves over a K3 surface with a given Mukai vector coincide with this K3 surface. We want to study this phenomenon in details.