A note on spherical bundles on K3 surfaces

In this talk, I will discuss some properties of spherical bundles on K3 surfaces. Let S be a K3 surface with the bounded derived category D^b(S). Let E be a spherical object in D^b(S). Then there always exists a non-zero object F satisfying RHom(E,F)=0. Further, there exists a spherical bundle E on some K3 surfaces that is unstable with respect to all polarization on S. If time permits, I will also discuss a way to “count” spherical bundles with a fixed Mukai vector. These provide (partial) answers to some questions of Huybrechts. This is a joint work with Chunyi Li.