Degenerations of generalized Kummer varieties

Abstract: We present a method to construct explicit degenerations of higher-dimensional generalized Kummer varieties. We start with a simple degeneration $f: \mathcal Y \to C$ of abelian surfaces. Then $\mathcal{Y} \setminus \mathcal{Y}<em>0$ is an abelian scheme over $C \setminus 0$ and we can form the relative generalized Kummer variety $$K^{n-1}</em>{\circ} = \mathrm{Kum}^{{n-1}(\mathcal{Y}} \setminus \mathcal{Y}<em>0) \to C \setminus 0$$. This is naturally a closed subscheme of the relative Hilbert scheme $$\mathrm{Hilb}^{{n}(\mathcal{Y}} \setminus \mathcal{Y}_0) \to C \setminus 0$$. In previous work (joint with Gulbrandsen) we had constructed a compactification $Iⁿ</em>{\mathcal{Y}/C}$ over $C$ of the latter scheme. The closure $K<sup>{n-1}_{\mathcal{Y}/C}$ of $K<sup>{n-1}_{\circ}$ inside $I<sup>n_{\mathcal{Y}/C}$ yields a canonical way to degenerate the family of generalized Kummer varieties, and is the degeneration we propose. This paper contains a detailed study of the geometry of the scheme $K<sup>{n-1}_{\mathcal{Y}/C}$ and its natural stratification. For $n=2$ we obtain a projective Kulikov model of Kummer surfaces, whereas already for $n=3$ new phenomena occur. We study in detail the dual complex of $K<sup>{2}_{\mathcal{Y}/C}$ and show that this is PL-homeomorphic to the standard $2$-simplex.

• The paper presents an explicit construction of degenerations for generalized Kummer varieties using a relative Hilbert scheme and GIT-stability techniques.
• It introduces novel combinatorial tools, including 'line charts', to stratify the degeneration and encode the distribution of points for the special fiber.
• It demonstrates that for n=2 and n=3, the degenerations yield well-structured dual complexes, providing insights into mirror symmetry and moduli compactifications.

Degenerations of Generalized Kummer Varieties: Technical Summary and Implications

Introduction and Context

The paper "Degenerations of generalized Kummer varieties" (2604.14890) develops an explicit framework for the degeneration of generalized Kummer varieties associated to simple degenerations of abelian surfaces. Generalized Kummer varieties, $\mathrm{Kum}^n(A)$, are central examples of hyperkähler manifolds, constructed as the kernel of the summation map from the Hilbert scheme $\mathrm{Hilb}^{n+1}(A)$ to the abelian surface $A$. While degenerations of hyperkähler manifolds of K3$^{[n]}$-type have been previously studied (notably via expanded degenerations and GIT techniques), explicit constructions and analyses have been largely absent in the Kummer case, especially for dimension greater than two.

The authors address this gap by constructing families of degenerations for generalized Kummer varieties, analyzing their geometry, stratification, and dual complex, and comparing the combinatorics and geometry to known essentials from the theory of degenerations of K3 surfaces and Hilbert schemes.

Construction of Degenerations

The starting point is a simple (type II Kulikov) degeneration $f: \mathcal{Y} \to C$ of abelian surfaces, such that over the complement $C \setminus \{0\}$, $\mathcal{Y}$ is an abelian scheme. Utilizing Jun Li's theory of expanded degenerations, the authors lift this to a relative Hilbert scheme $\mathrm{Hilb}^n(\mathcal{Y}[n]/C[n])$ equipped with a $\mathbb{G}_m^n$-action. The GIT-stable locus of this Hilbert scheme is compactified to a projective dlt-model $I^n_{\mathcal{Y}/C}$, known from prior work.

Crucially, the generalized Kummer variety is realized as a closed subscheme $\mathrm{Hilb}^{n+1}(A)$0 (the locus of degree $\mathrm{Hilb}^{n+1}(A)$1 zero-cycle subschemes whose sum under the group law is zero) inside the family of Hilbert schemes. Its closure $\mathrm{Hilb}^{n+1}(A)$2 inside the dlt model provides the canonical degeneration of generalized Kummer varieties under consideration.

A significant technical advance lies in the study of the Kummer locus in the expanded degenerations framework, requiring an understanding not only of GIT-stability conditions but also of induced group structures and their interaction with the summation constraints that define the Kummer varieties.

Stratification and Combinatorial Analysis

The Kummer locus, both generically and in the degenerate fiber, is shown to admit a stratification indexed by combinatorial data encoding the distribution of points among components in the special fiber of the degeneration, constrained by numerical and group-theoretic conditions. The main structural theorem states that, for all $\mathrm{Hilb}^{n+1}(A)$3 and for each unobstructed combinatorial type, the corresponding stratum is smooth, irreducible, and $\mathrm{Hilb}^{n+1}(A)$4-stable.

A key innovation is the introduction of "line charts," combinatorial gadgets that encode how points distribute over the components and inserted $\mathrm{Hilb}^{n+1}(A)$5-bundles in the special fiber. These charts determine not only whether a combinatorial type yields an admissible Kummer stratum (solving certain integer equations) but also the smoothings (strata in higher-dimensional expansions containing them in closure), facilitating the complete description of the dual complex.

Main Results for $\mathrm{Hilb}^{n+1}(A)$6 and $\mathrm{Hilb}^{n+1}(A)$7

• For $\mathrm{Hilb}^{n+1}(A)$8 (Kummer surfaces): The GIT quotient $\mathrm{Hilb}^{n+1}(A)$9 realizes a projective Kulikov model with strict normal crossing central fiber, and the dual complex is a chain. This confirms and refines the results for degenerations of Kummer surfaces, showing congruence with type II models in the Kulikov-Persson-Pinkham classification.
• For $A$0 (fourfolds): The expanded Kummer locus exhibits new phenomena compared to the Hilbert scheme case. The dual complex of the degeneration's central fiber is explicitly computed to be PL-homeomorphic to the standard $A$1-simplex, with cells corresponding to certain combinatorial types classified entirely via the authors' framework. There are intricate non-transverse intersections and a combinatorially rich gluing of local dual complexes, controlled by the line chart machinery.

Strong and Contradictory Claims

The paper asserts—supported by explicit combinatorial, geometric, and group-theoretic calculations—that for $A$2, the dual complex of the special fiber of the degeneration is always a standard $A$3-simplex, regardless of auxiliary choices, and that for $A$4 the degeneration yields a projective Kulikov model without triple intersections. This is a strong assertion contrasting the complexity and pathologies observed in general degenerations of hyperkähler varieties.

Furthermore, the paper claims that for $A$5, the obtained degenerations cannot, in general, be dlt without further modification (e.g., via additional blowups), providing a counterpoint to the relative simplicity in $A$6 and motivating future work.

Implications and Future Directions

The construction provides explicit dlt (and in some cases Kulikov) models for degenerations of generalized Kummer varieties, with controlled and computable dual complexes. This has direct implications for:

1. Mirror symmetry and Berkovich analytification: Since the essential skeleton agrees with the dual complex for minimal dlt models, these results provide concrete models for comparison to results in non-archimedean analytic geometry and mirror symmetry, extending recent work [Brown and Mazzon].
2. Compactifications of moduli spaces: The controlled construction and stratification offer new tools for developing compactifications of the moduli of hyperkähler or Kummer-type varieties with explicit understanding of boundary phenomena.
3. Special fiber geometry: The method translates the complexity of the degeneration into combinatorial terms directly amenable to computation, allowing new types of intersection-theoretic and arithmetic questions to be posed and answered explicitly.
4. Extension to higher $A$7 and general degenerations: The analysis suggests further research is necessary to handle the $A$8 case, possibly requiring systematic blowups or modifications, as well as the extension to type III degenerations and beyond, which remain open.

Potential extensions of the line chart combinatorics and group action analysis may also impact the study of automorphisms and dynamics on degenerations of hyperkähler manifolds.

Conclusion

This work provides the first comprehensive, explicit, and computationally accessible theory of degenerations for generalized Kummer varieties of arbitrary dimension (with full analysis for $A$9), setting a new standard for what can be achieved in explicit algebraic geometry for higher-dimensional hyperkähler degenerations. The line chart and stratification framework, as well as the explicit computation of dual complexes, open technical avenues in both the practical geometry of degenerations and the theoretical landscape of moduli, mirror symmetry, and non-archimedean geometry.