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From logarithmic Hilbert schemes to degenerations of hyperkähler varieties

Qaasim Shafi, Calla Tschanz

TL;DR

This work constructs the first explicit good type III degenerations of hyperkähler varieties in dimension $>2$ by formulating them as moduli spaces of length $m$ subschemes on expansions of degenerating K3 surfaces. The authors prove projectivity of the expanded degenerations and, for $m=2$, compute the dual complex of the special fibre in two explicit K3 degenerations (quartic and cube), obtaining detailed combinatorial counts for the cells. Their method combines expanded degenerations (CT) with tropical geometry to globalize the local models, yielding a Deligne–Mumford, proper moduli space $\mathfrak{M}_{\mathrm{LW}}^m$ that provides good semistable degenerations of hyperkähler varieties with general fibre $Hilb^m(K3)$. The results illuminate the relationship between geometric strata and the dual complex, offering precise topological invariants of the degenerations and suggesting directions for understanding integral affine structures on dual complexes and moduli compactifications in higher dimensions.

Abstract

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.

From logarithmic Hilbert schemes to degenerations of hyperkähler varieties

TL;DR

This work constructs the first explicit good type III degenerations of hyperkähler varieties in dimension by formulating them as moduli spaces of length subschemes on expansions of degenerating K3 surfaces. The authors prove projectivity of the expanded degenerations and, for , compute the dual complex of the special fibre in two explicit K3 degenerations (quartic and cube), obtaining detailed combinatorial counts for the cells. Their method combines expanded degenerations (CT) with tropical geometry to globalize the local models, yielding a Deligne–Mumford, proper moduli space that provides good semistable degenerations of hyperkähler varieties with general fibre . The results illuminate the relationship between geometric strata and the dual complex, offering precise topological invariants of the degenerations and suggesting directions for understanding integral affine structures on dual complexes and moduli compactifications in higher dimensions.

Abstract

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.
Paper Structure (28 sections, 16 theorems, 17 equations, 11 figures)

This paper contains 28 sections, 16 theorems, 17 equations, 11 figures.

Table of Contents

  1. Results
  2. Future directions
  3. Organisation
  4. Background
  5. Expanded degenerations of surfaces
  6. Semistable families of surfaces
  7. Expanded degenerations
  8. Torus action and moduli of points
  9. Hyperkähler manifolds and degenerations
  10. Hyperkähler varieties
  11. Degenerations of hyperkähler varieties
  12. Two type III degenerations of K3 surfaces
  13. Quartic degeneration
  14. Cube degeneration
  15. Global construction
  16. ...and 13 more sections

Key Result

Theorem A

The stack of stable, length $m$ zero dimensional subschemes $\mathfrak{M}_{\mathrm{LW}}^m$ in $\mathfrak{X}$ over $C$, constructed in Section Section : Globalising, is a Deligne--Mumford stack, proper over $C$. Moreover, the construction gives a good semistable degeneration of irreducible holomorphi

Figures (11)

  • Figure 1: Geometric and tropical pictures of the special fibre $X_0$.
  • Figure 2: Geometric and tropical picture at $t_1=t_2=0$ in $X[1]$.
  • Figure 3: Flattened picture of the triangulated sphere with 4 vertices.
  • Figure 4: Tropical pictures of our chosen subdivision.
  • Figure 5: A subdivision that does not glue.
  • ...and 6 more figures

Theorems & Definitions (51)

  • Theorem A
  • Theorem B
  • Definition 1.1.1
  • Definition 1.1.2: Base codimension
  • Definition 1.1.3
  • Definition 1.1.4
  • Definition 1.1.5
  • Definition 1.2.1
  • Definition 1.2.2
  • Definition 1.2.3
  • ...and 41 more