Correspondences of K3 surface via moduli of sheaves

K3曲面是一类非常重要的代数曲面，在当代数学物理中占有重要地位。在过去的40年里，对它们的理解有了很大的进步，但我们仍然发现了它们的一些新的、意想不到的和重要的性质。K3曲面的Picard点阵的元素(由代数曲线生成)在K3曲面与其自身的乘积以及K3曲面与其自身的对应关系上产生显著的二维代数循环。当K3曲面上具有给定Mukai向量的相干滑层的模数与该K3曲面重合时，就会发生这种情况。我们想对这一现象进行详细的研究。

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.