A Note on Spherical Bundles on K3 Surfaces

Abstract

Some questions are posted at the end of Chapter 16 of Huybrechts' book 'Lectures on K3 Surfaces', concerning the bounded derived category of a K3 surface $D^b(S)$. Let $E$ be a spherical object in $D^b(S)$. The first question asks if there always exists a non-zero object $F$ satisfying RHom$(E,F)=0$. Further, let $E$ be a spherical bundle. The second question is whether $E$ is always semistable with respect to some polarization on $S$ and if there is a way to `count' spherical bundles with a fixed Mukai vector. In this note, we provide (partial) answers to these two questions. In the appendix, Genki Ouchi shows that any spherical twist associated to an $n$-spherical object on a smooth projective $n$-dimensional variety is not conjugate to a standard autoequivalence.