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.

💡 Research Summary

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The paper by Shafi Q. and Tschan­z C. presents the first explicit constructions of good type III degenerations of hyperkähler (irreducible holomorphic symplectic, IHS) varieties in dimensions greater than two. The authors start from a type III degeneration of K3 surfaces (X\to C) whose central fibre (X_0) is a normal‑crossing divisor consisting of three coordinate planes. The relative Hilbert scheme (\operatorname{Hilb}^m(X/C)) has smooth general fibres (the Hilbert scheme of (m) points on a K3 surface) but its central fibre is highly singular, preventing it from being a minimal degeneration.

To overcome this, the authors employ the “expanded degeneration” technique introduced in their earlier work