From logarithmic Hilbert schemes to degenerations of hyperkähler varieties

We construct the first examples of good type III degenerations of hyperkähler varieties in dimension greater than 2. These are presented as moduli of 0-dimensional subschemes on expansions of a degeneration of K3 surfaces. We prove projectivity for our expanded degenerations and compute the dual complexes of the special fibre for two specific degenerations of hyperkahler fourfolds. Moreover, we explain the correspondence between geometric strata of the special fibre and simplices in its dual complex.