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Reading 2007 paper A tour of theta dualities on moduli spaces of sheaves by Alina Marian and Dragos Oprea.

Why is any moduli space of coherent sheaves on a K3 surface deformation equivalent to a moduli space of sheaves on an elliptic K3?

(The authors consider a space of "Gieseker H-semistable sheaves", if that is important)

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This follows from a result of Yoshioka. In Theorem 8.1 of this paper 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.

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