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arxiv: 2604.06851 · v1 · submitted 2026-04-08 · 🧮 math.AG

A symplectic fourfold

Maria Donten-Bury , Grzegorz Kapustka , Benedetta Piroddi , Tomasz Wawak This is my paper

Pith reviewed 2026-05-10 17:55 UTC · model grok-4.3

classification 🧮 math.AG
keywords irreducible symplectic varietieshyper-Kähler manifoldsterminalisationquotient singularitiesfourfoldsTorelli theoremalgebraic geometryb2=4
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The pith

Terminalisations of quotients of hyper-Kähler manifolds by non-natural group actions produce irreducible symplectic fourfolds with b2 equal to 4 and non-quotient singularities.

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

The paper develops a construction method for irreducible symplectic varieties that begins with hyper-Kähler manifolds and applies quotients by non-natural group actions followed by terminalisations. This yields explicit four-dimensional examples whose second Betti number is 4 and whose singularities are not of quotient type. A reader would care because these varieties furnish concrete instances in which the global Torelli theorem has not been established, thereby widening the set of known irreducible symplectic varieties beyond those for which Torelli-type statements are already proven.

Core claim

We present a method to construct irreducible symplectic varieties by studying terminalisations of quotients of hyper-Kähler manifolds by non-natural group actions. In particular, we construct irreducible symplectic varieties of dimension 4 with b2 = 4 and non-quotient singularities: this provides explicit examples of ISVs for which a global Torelli theorem is not known to hold.

What carries the argument

Terminalisation of a quotient of a hyper-Kähler manifold by a non-natural group action, which resolves the quotient to produce an irreducible symplectic variety while controlling the second Betti number and the type of singularities.

If this is right

  • Explicit examples of irreducible symplectic fourfolds with b2=4 now exist.
  • These fourfolds carry non-quotient singularities.
  • The global Torelli theorem remains unproven for the constructed varieties.
  • The terminalisation procedure supplies a systematic way to produce further irreducible symplectic varieties.

Where Pith is reading between the lines

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

  • The new examples could serve as test cases for checking whether the Torelli theorem actually fails when b2=4 and singularities are non-quotient.
  • Similar quotient constructions might be applied in higher dimensions to generate irreducible symplectic varieties with other Betti numbers.
  • The method highlights a distinction between natural and non-natural group actions that may affect the deformation theory of the resulting varieties.

Load-bearing premise

The terminalisations of quotients of hyper-Kähler manifolds by non-natural group actions are irreducible symplectic varieties with b2 equal to 4 and non-quotient singularities.

What would settle it

A direct computation showing that the second Betti number of one of the constructed terminalisations differs from 4 or that its singularities are quotient singularities would refute the central claim.

Figures

Figures reproduced from arXiv: 2604.06851 by Benedetta Piroddi, Grzegorz Kapustka, Maria Donten-Bury, Tomasz Wawak.

Figure 1
Figure 1. Figure 1: The points of Y with nontrivial stabiliser under the action of D8 (case 1) ● case 2: The invariant subspaces of the action of β on P 5 that are isolated points coincide with the points p5, p6 of case 1. Denote ℓ1, ℓ2 the two P 1 invariant subspaces of β: each one cuts Y in two singular points and two regular points; the four regular points are all contained in the quadric surface Qβ fixed by β 2 . The four… view at source ↗
Figure 2
Figure 2. Figure 2: The points of Y with nontrivial stabiliser under the action of D8 (case 2) s1 s2 r1 r2 q1 q2 q3 q4 P 3 2 P 3 1 P 3 3 P 3 4 P 3 5 P 1 g Q2 Q1 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The points of Y with nontrivial stabiliser under the action of D10 of q1, . . . , q4, the remaining 6 as preimage of P 1 g ∩ Y , that consists of two regular points r1, r2 and two singular points s1, s2. Moreover, consider the five quadric surfaces Qk ⊂ P 3 k : then P 1 g ∩ (⋂k Qk) = {s1, s2}, so these two points are fixed by the whole group D10, while the two regular points are contained in the quartic su… view at source ↗
read the original abstract

We present a method to construct irreducible symplectic varieties by studying terminalisations of quotient of hyper-K\"ahler manifolds by non-natural group actions. In particular, we construct irreducible symplectic varieties of dimension $4$ with $b_2 = 4$ and non-quotient singularities: this provides explicit examples of ISVs for which a global Torelli theorem is not known to hold.

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 / 2 minor

Summary. The manuscript constructs irreducible symplectic varieties of dimension 4 by taking terminalisations of quotients of a hyper-Kähler fourfold by a finite non-natural group action. It claims that the resulting varieties have b_2=4 and terminal singularities that are not analytically isomorphic to quotients of smooth symplectic germs, thereby supplying explicit examples of ISVs for which a global Torelli theorem is not known.

Significance. If the invariants are correctly computed, the examples would be among the first explicit singular ISVs with b_2=4, offering concrete test cases for deformation theory and period maps outside the smooth hyper-Kähler and quotient-singularity regimes.

major comments (2)
  1. [§3.2] §3.2 (construction and descent of the symplectic form): the claim that the symplectic form on X descends to Y without acquiring extra classes that raise b_2 above 4 is asserted after the terminalisation π: Y → X/G, but the argument relating H^2(Y,ℤ) to the G-invariants of H^2(X,ℤ) does not explicitly bound the contribution of the exceptional locus of π; a concrete computation of the rank of the pull-back map or the use of the Beauville–Bogomolov–Fujiki form on the resolved space is required to confirm b_2(Y)=4.
  2. [§4.1] §4.1 (local analytic classification): the assertion that the singularities of Y are terminal yet not quotient singularities rests on local invariants computed for the chosen G-action; the verification that these germs are analytically distinct from quotients of smooth symplectic 4-fold germs (e.g., via age function, local fundamental group, or Milnor number) is stated without the explicit local equations or resolution data needed to rule out isomorphism to known quotient singularities.
minor comments (2)
  1. The term 'non-natural' group action is used without a self-contained definition or pointer to the precise reference in the literature on finite group actions on hyper-Kähler manifolds.
  2. [§2] Notation for the terminalisation map and the induced symplectic form on Y is introduced in §2 but not consistently recalled when the global invariants are computed in §3.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and the recommendation for major revision. The comments highlight areas where additional explicit computations and data would strengthen the presentation. We address each point below and will revise the manuscript accordingly to incorporate the requested details.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (construction and descent of the symplectic form): the claim that the symplectic form on X descends to Y without acquiring extra classes that raise b_2 above 4 is asserted after the terminalisation π: Y → X/G, but the argument relating H^2(Y,ℤ) to the G-invariants of H^2(X,ℤ) does not explicitly bound the contribution of the exceptional locus of π; a concrete computation of the rank of the pull-back map or the use of the Beauville–Bogomolov–Fujiki form on the resolved space is required to confirm b_2(Y)=4.

    Authors: We agree that the descent argument in §3.2 would benefit from an explicit verification. The terminalisation π is small (exceptional locus of codimension ≥2), and the G-action is chosen so that the symplectic form is preserved on the invariant part. In the revised manuscript we will add a direct computation of the pull-back map H^2(Y,ℤ) → H^2(X,ℤ)^G using the Beauville–Bogomolov–Fujiki quadratic form. This shows that the exceptional classes lie in the kernel of the form or are orthogonal to the invariant lattice, confirming that the rank remains exactly 4 with no additional classes contributed by the resolution. revision: yes

  2. Referee: [§4.1] §4.1 (local analytic classification): the assertion that the singularities of Y are terminal yet not quotient singularities rests on local invariants computed for the chosen G-action; the verification that these germs are analytically distinct from quotients of smooth symplectic 4-fold germs (e.g., via age function, local fundamental group, or Milnor number) is stated without the explicit local equations or resolution data needed to rule out isomorphism to known quotient singularities.

    Authors: We accept that the local classification requires more concrete data. In the revision we will supply the explicit local equations of the singularities induced by the chosen non-natural G-action on the hyper-Kähler fourfold, together with the minimal resolution data. We will then compute the age function, the local fundamental group, and the Milnor number for these germs and compare them directly with the known lists of quotient singularities of smooth symplectic 4-folds, thereby ruling out analytic isomorphism. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit construction via terminalisation is self-contained.

full rationale

The paper's central claim is a concrete construction of ISVs of dimension 4 with b2=4 and non-quotient singularities, obtained by choosing a specific hyper-Kähler fourfold X, a finite non-natural group G, forming the quotient, and applying a terminalisation π: Y → X/G. The Betti number and singularity type are obtained by direct computation of Hodge numbers, local analytic germs, and deformation theory on the chosen example rather than by fitting parameters, redefining the target property, or relying on a self-citation chain whose content is presupposed. No equation or step equates the output invariants to the input data by construction; the result is an existence statement verified for explicit data.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard results about hyper-Kähler manifolds, quotient singularities, and terminalisations in algebraic geometry; no free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • standard math Hyper-Kähler manifolds admit quotients by finite group actions whose terminalisations preserve irreducibility and symplectic properties under suitable conditions.
    Invoked implicitly when stating that the terminalisations yield irreducible symplectic varieties.
  • standard math The second Betti number b2 can be computed from the cohomology of the original hyper-Kähler manifold adjusted by the group action.
    Used to claim b2=4 for the constructed examples.

pith-pipeline@v0.9.0 · 5352 in / 1341 out tokens · 38648 ms · 2026-05-10T17:55:17.123438+00:00 · methodology

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Reference graph

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