From logarithmic Hilbert schemes to degenerations of hyperk\"ahler varieties

TL;DR

This paper constructs the first known good type III degenerations of hyperk"ahler varieties in higher dimensions, using moduli of subschemes on degenerations of K3 surfaces, and analyzes their geometric and combinatorial properties.

Contribution

It introduces new examples of degenerations of hyperk"ahler varieties and studies their dual complexes and geometric stratification.

Findings

Constructed the first examples of good type III degenerations in dimension > 2.

Proved projectivity of the expanded degenerations.

Computed dual complexes for specific hyperk"ahler fourfold degenerations.

Abstract

We construct the first examples of good type III degenerations of hyperk\"ahler 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.