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arxiv: 2211.14524 · v2 · pith:J5MJ7AC3new · submitted 2022-11-26 · 🧮 math.AG

Thirty-three deformation classes of compact hyperk\"ahler orbifolds

Gr\'egoire Menet This is my paper

Pith reviewed 2026-05-24 10:57 UTC · model grok-4.3

classification 🧮 math.AG
keywords hyperkähler orbifoldsirreducible symplectic varietiesK3 surfacesterminalizationsdeformation classesquotientsFujiki construction
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The pith

Quotients of K3 surface products yield thirty-three deformation classes of compact hyperkähler orbifolds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper generalizes the Fujiki construction by taking terminalizations of quotients of the n-fold product of a K3 surface S. These terminalizations are presented as irreducible symplectic varieties whose smooth locus is simply connected. In dimension four the singularities of twenty-nine such examples are computed explicitly, and the resulting data indicate that the examples lie in independent deformation classes. Four further examples are constructed in dimension six. A reader would care because these objects serve as the elementary pieces in the singular version of the Bogomolov decomposition theorem.

Core claim

By generalizing the Fujiki construction, the terminalizations of quotients of S^n for a K3 surface S produce irreducible symplectic varieties with simply connected smooth locus. In dimension 4, 29 such orbifolds appear independent under deformation as determined by their singularities, with 4 additional examples provided in dimension 6, for a total of 33 deformation classes.

What carries the argument

The terminalization of a quotient of the product of K3 surfaces, which resolves singularities while preserving the symplectic form and whose singularity type distinguishes the deformation class.

If this is right

  • The twenty-nine dimension-four examples lie in distinct deformation classes.
  • The four dimension-six examples constitute additional distinct classes.
  • These varieties function as irreducible symplectic summands in the generalized Bogomolov decomposition.
  • Singularity computations suffice to separate the deformation classes in these dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Varying the finite group or the dimension n may produce further independent classes in higher dimensions.
  • The same terminalization technique could be applied to products of other hyperkähler surfaces to generate new families.
  • Direct comparison of Hodge numbers or second Betti numbers across the thirty-three classes could reveal unexpected isomorphisms or relations.

Load-bearing premise

The terminalizations of quotients of S^n always yield irreducible symplectic varieties whose smooth locus is simply connected.

What would settle it

An explicit flat family over a connected base whose general fiber is one of the claimed dimension-4 examples and whose special fiber is another.

read the original abstract

As their smooth analogue the irreducible symplectic varieties appear as elementary bricks in the generalizations of the Bogomolov decomposition theorem (arXiv:math/0402243, arXiv:2012.00441). Let $S$ be a K3 surface; generalizing the Fujiki construction, we investigate the irreducible symplectic varieties with simply connected smooth locus that can be obtained as terminalizations of quotients of the product $S^{n}$. In dimension 4, we compute the singularities for 29 orbifolds examples which appear to be independent under deformation. We also provide 4 additional orbifolds examples in dimension 6.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript generalizes the Fujiki construction by considering terminalizations of quotients of products of K3 surfaces S^n / G. It computes the singularities of 29 such orbifolds in dimension 4, which are asserted to appear independent under deformation, and provides 4 additional examples in dimension 6. These are presented as irreducible symplectic varieties with simply connected smooth locus.

Significance. If the deformation independence of the 29 classes can be rigorously established, the work would supply a substantial collection of explicit examples of compact hyperkähler orbifolds, contributing concrete data toward the classification of irreducible symplectic varieties in low dimensions and their moduli spaces.

major comments (2)
  1. [Abstract and dimension-4 examples section] Abstract and the section computing the 29 dimension-4 examples: the claim that the examples 'appear to be independent under deformation' rests solely on singularity computations. No explicit deformation invariant (e.g., the Beauville-Bogomolov-Fujiki quadratic form on the second cohomology of the smooth locus, the Fujiki constant, or the lattice of algebraic classes) is exhibited that would provably separate all 29 classes in the moduli space of irreducible symplectic varieties with simply connected smooth locus.
  2. [Setup generalizing the Fujiki construction] Setup generalizing the Fujiki construction: the statement that terminalizations of quotients of S^n yield irreducible symplectic varieties with simply connected smooth locus is taken as given, but requires either a self-contained proof or a precise reference establishing this property for the groups G under consideration.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on our manuscript. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract and dimension-4 examples section] Abstract and the section computing the 29 dimension-4 examples: the claim that the examples 'appear to be independent under deformation' rests solely on singularity computations. No explicit deformation invariant (e.g., the Beauville-Bogomolov-Fujiki quadratic form on the second cohomology of the smooth locus, the Fujiki constant, or the lattice of algebraic classes) is exhibited that would provably separate all 29 classes in the moduli space of irreducible symplectic varieties with simply connected smooth locus.

    Authors: We agree that the current wording relies on the distinct singularity types computed for the 29 examples without exhibiting a numerical deformation invariant to separate the classes rigorously. The phrase 'appear to be independent' was chosen to reflect the observational nature of the claim. In revision we will compute the Fujiki constant for each of the 29 examples (which is a deformation invariant) and include these values in the dimension-4 section; the abstract will be updated to reference this additional data. revision: yes

  2. Referee: [Setup generalizing the Fujiki construction] Setup generalizing the Fujiki construction: the statement that terminalizations of quotients of S^n yield irreducible symplectic varieties with simply connected smooth locus is taken as given, but requires either a self-contained proof or a precise reference establishing this property for the groups G under consideration.

    Authors: The property is inherited from the general framework of terminalizations of finite quotients of hyperkähler manifolds. We will insert a precise citation to arXiv:2012.00441 together with a short paragraph in the setup section explaining why the smooth locus remains simply connected for the finite groups G appearing in our constructions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit constructions and computations

full rationale

The paper derives its 29 deformation classes in dimension 4 (and 4 in dimension 6) via explicit terminalizations of quotients S^n/G for a K3 surface S, followed by direct computation of the resulting singularities. These computations are presented as distinguishing the examples, with the independence claim resting on the observed singularity configurations rather than any equation or parameter that is defined in terms of the output. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the setup or abstract. The cited prior works (Fujiki construction generalizations) are external and do not overlap with the present author in a way that imports uniqueness theorems or ansatzes. The derivation chain is therefore self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard facts about K3 surfaces, their products, group actions, and terminalizations; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption K3 surfaces and their products admit quotients whose terminalizations are irreducible symplectic varieties with simply connected smooth locus
    Invoked in the opening setup generalizing the Fujiki construction.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Terminalizations of quotients of compact hyperk\"ahler manifolds by induced symplectic automorphisms

    math.AG 2024-01 unverdicted novelty 7.0

    Classification of terminalizations of symplectic quotients of K3^{[n]} and generalized Kummer varieties yields at least nine new deformation types of irreducible symplectic varieties of dimension four.

  2. Automorphisms of Nikulin-type orbifolds

    math.AG 2024-11 unverdicted novelty 5.0

    Monodromy group of Nikulin-type orbifolds is maximal; finite order symplectic automorphisms classified up to deformation via action on second integral cohomology.

Reference graph

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