Several explicit constructions of locally complete families of polarized HK varieties are known, starting with K3 surfaces (mostly by Mukai), and ending with moduli of Bridgeland semistable objects in the Kuznetsov component of cubic fourfolds. All HK's in such families are of Type $K3^{[n]}$. We will discuss work in progress which aims to construct locally complete families of polarized HK varieties of different Types, in particular of Kummer Type.

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