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

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 $mv_{i}$ be a Mukai vector on $S_{i}$ such that $H_{i}$ is $mv_{i}-$generic. Moreover, suppose that the moduli spaces $M_{mv_{1}}(S_{1},H_{1})$ of $H_{1}-$semistable sheaves on $S_{1}$ of Mukai vector $mv_{1}$ and $M_{mv_{2}}(S_{2},H_{2})$ of $H_{2}-$semistable sheaves on $S_{2}$ with Mukai vector $mv_{2}$, have the same dimension. The aim of this paper is to prove that $M_{mv_{1}}(S_{1},H_{1})$ is deformation equivalent to $M_{mv_{2}}(S_{2},H_{2})$, showing a conjecture of Z. Zhang contained in [18].