Reading 2007 paper [A tour of theta dualities on moduli spaces of sheaves](http://arxiv.org/abs/0710.2908) by [Alina Marian](http://arxiv.org/find/math/1/au%3a+Marian_A/0/1/0/all/0/1) and [Dragos Oprea](http://arxiv.org/find/math/1/au%3a+Oprea_D/0/1/0/all/0/1).



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> Why is any moduli space of coherent sheaves on a K3 surface deformation equivalent to a moduli space of sheaves on an elliptic K3?
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(The authors consider a space of "Gieseker H-semistable sheaves", if that is important)



Moduli spaces of coherent sheaves on K3s
This follows from a result of Yoshioka. In Theorem 8.1 of this [paper](http://arxiv.org/abs/math/0009001) Yoshioka showed that every moduli space of coherent sheaves on a K3 surface $X$ is deformation equivalent to an appropriate Hilbert scheme of points of $X$. Since every K3 is deformation equivalent to an elliptic K3 it follows that their Hilbert schemes are deformation equivalent and so you get the statement that you wanted.


