A note on deformations of moduli spaces of sheaves on K3 surfaces

Abstract

In this paper we study deformation classes of moduli spaces of sheaves on a projective K3 surface. More precisely, let $(S1,H1)$ and $(S2,H2)$ be two polarized K3 surfaces, $m\in \mathbb{N}$, and for $i=1,2$ let $m{v}_i$ be a Mukai vector on $S_i$ such that $H_i$ is $m{v}_i-$generic. Moreover, suppose that the moduli spaces $M_{m{v}_1}(S_1,H_1)$ of $H_1-$semistable sheaves on $S_1$ of Mukai vector $m{v}_1$ and $M_{m{v}_2}(S_2,H_2)$ of $H_2-$semistable sheaves on $S_2$ with Mukai vector $m{v}_2$, have the same dimension. The aim of this paper is to prove that $M_{m{v}_1}(S_1,H_1)$ is deformation equivalent to $M_{m{v}_2}(S_2,H_2)$, showing a conjecture of Z. Zhang contained in [18].

The paper is now part of arXiv:1802.01182

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