A note on spherical bundles on K3 surfaces (with an appendix by Genki Ouchi)

In this note we provide (partial) answers to two questions posed in Chapter 16 of D. Huybrechts' book Lectures on K3 surfaces, concerning the bounded derived category D b (S) of a K3 surface S. The first question asks if for a spherical object E in D b (S), there always exists a non-zero object F satisfying RHom(E, F ) = 0.The second question asks if a spherical bundle 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 with an n-spherical object on a smooth projective n-dimensional variety is not conjugate to a standard autoequivalence. IntroductionLet S be a smooth K3 surface and E be a spherical object in D b (S); in other words, dim Hom(E, E[i]) = 1 when i = 0, 2 , 0 otherwise .The first question at the end of [Huy16, Chapter 16] asks whether E = 0; in other words, does there exist a non-trivial object F such that Hom(E, F [i]) = 0 for all i Z.In the case where the K3 surface S is of Picard number 1, the question is answered in [Ouc20, Appendix A] by Bayer using tools and results from [BB17].Here, we provide an affirmative answer to this question for all K3 surfaces and some more general cases.Proposition 1.1 (Proposition 2.3).Let X be a smooth proper n-dimensional variety over C and E be an n-spherical object in D b (X).Then there exists a non-trivial object F orthogonal to E; in other words, Hom(E,Proposition 1.1 has some immediate corollaries.For example, [Ouc20, Theorem 3.1] now holds for all spherical objects in D b (X) without any extra assumptions.Another corollary is pointed