Correspondences of K3 surface via moduli of sheaves

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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.