Rigid Vector Bundles on Low Dimensional Varieties

This thesis consists of three of my projects that are in the same direction during my PhD study. First I develop an algorithm to compute the cohomology of stable rigid vector bundles on K3 surfaces in terms of their Chern classes. As its applications, I give the first nonexistence results of exceptional bundles on P3 with certain Chern classes, and classify stable rigid sheaves on certain singular K3 surfaces.