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

TL;DR

This paper proves that moduli spaces of semistable sheaves on different polarized K3 surfaces with the same dimension are deformation equivalent, confirming a conjecture by Z. Zhang and advancing understanding of their deformation classes.

Contribution

It establishes the deformation equivalence of moduli spaces of sheaves on different K3 surfaces with matching dimensions, confirming a conjecture by Z. Zhang.

Findings

Moduli spaces of sheaves on different K3 surfaces are deformation equivalent when they have the same dimension.

The paper confirms Z. Zhang's conjecture on deformation classes of these moduli spaces.

Provides new insights into the deformation theory 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 $mv_{i}$ 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].