Rank 4 stable vector bundles on hyperk\"ahler fourfolds of Kummer type

Kieran G. O'Grady

arxiv: 2203.03987 · v5 · pith:SQILD7A4new · submitted 2022-03-08 · 🧮 math.AG

Rank 4 stable vector bundles on hyperk\"ahler fourfolds of Kummer type

Kieran G. O'Grady This is my paper

Pith reviewed 2026-05-24 11:21 UTC · model grok-4.3

classification 🧮 math.AG

keywords hyperkähler fourfoldsKummer typestable vector bundlesslope stabilityrigid bundlesChern classesmoduli spaces

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The pith

A unique slope stable rank-4 vector bundle with c1 equal to the polarization exists on general polarized hyperkähler fourfolds of Kummer type and is rigid.

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

The paper shows that for a general polarized hyperkähler fourfold M of Kummer type satisfying one of two numerical conditions on the polarization h, there is exactly one slope stable vector bundle F of rank 4 with c1(F) equal to h and discriminant equal to c2(M). This bundle is moreover rigid. The result extends earlier existence and uniqueness statements for stable rigid bundles from hyperkähler varieties of K3 type to the Kummer case. The work is motivated by the aim of using such bundles to give an explicit description of locally complete families of polarized Kummer-type fourfolds.

Core claim

Let (M,h) be a general polarized HK fourfold of Kummer type such that q_M(h) ≡ -6 mod 16 and the divisibility of h is 2, or q_M(h) ≡ -6 mod 144 and the divisibility of h is 6. Then there exists a unique up to isomorphism slope stable vector bundle F on M such that r(F)=4, c1(F)=h, Δ(F)=c2(M). Moreover F is rigid.

What carries the argument

The slope stable rank-4 vector bundle F with c1(F)=h and Δ(F)=c2(M), whose existence, uniqueness, and rigidity are established under the stated numerical conditions on M.

If this is right

The bundle F supplies a concrete object that can be used to parametrize a locally complete family of polarized hyperkähler fourfolds of Kummer type.
Existence and uniqueness results for stable rigid bundles that were known for K3^{[n]} varieties extend at least partially to fourfolds of Kummer type.
Rigidity of F implies that its deformation space consists of a single point.

Where Pith is reading between the lines

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

Similar uniqueness statements might hold for other ranks or other values of the discriminant on the same class of fourfolds.
The bundle F could serve as a starting point for constructing explicit moduli spaces or period maps for these polarized fourfolds.

Load-bearing premise

M must be a general polarized hyperkähler fourfold of Kummer type satisfying the stated congruence and divisibility conditions on the quadratic form of h.

What would settle it

An explicit computation or construction on one concrete such fourfold M that produces either zero or more than one non-isomorphic slope stable rank-4 bundle with the given Chern classes would disprove the uniqueness statement.

read the original abstract

We partially extend to hyperk\"ahler fourfolds of Kummer type the results that we have proved regarding stable rigid vector bundles on hyperk\"ahler (HK) varieties of type $K3^{[n]}$. Let $(M,h)$ be a general polarized HK fourfold of Kummer type such that $q_M(h)\equiv -6\pmod{16}$ and the divisibility of $h$ is $2$, or $q_M(h)\equiv -6\pmod{144}$ and the divisibility of $h$ is $6$. We show that there exists a unique (up to isomorphism) slope stable vector bundle $\cal F$ on $M$ such that $r({\cal F})=4$, $ c_1({\cal F})=h$, $\Delta({\cal F})=c_2(M)$. Moreover $\cal F$ is rigid. One of our motivations is the desire to describe explicitly a locally complete family of polarized HK fourfolds of Kummer type.

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.

Desk Editor's Note private letter to a colleague

O'Grady extends his earlier existence-uniqueness result for a rigid rank-4 stable bundle to general polarized Kummer fourfolds, but only under two specific arithmetic conditions on q(h) and divisibility.

read the letter

The new content is the adaptation of the K3^[n] argument to the Kummer case. Under the stated hypotheses (general polarized M of Kummer type with q_M(h) ≡ -6 mod 16 and div(h)=2, or the mod-144 version with div=6), there is a unique slope-stable rank-4 bundle F with c1(F)=h and Δ(F)=c2(M), and F is rigid. The statement is precise and the motivation—to get an explicit locally complete family of polarized Kummer fourfolds—is stated plainly. The extension is non-trivial because the Kummer lattice and the possible polarizations differ from the K3^[n] case, so the numerical checks and the deformation arguments have to be redone. That part looks like honest incremental work building on the author's previous papers. The main limitation is the narrow setting: only general members satisfying those two congruence/divisibility pairs. Outside that slice the result says nothing, and even inside it the bundle is only shown to exist for general M. The abstract gives no indication of hidden fitting or circularity; the claim is presented as a direct extension. Without the full proof one cannot check the details of the stability or rigidity arguments, but nothing in the statement contradicts itself or the cited prior results. This is for specialists already following moduli problems on hyperkähler fourfolds. A reader who knows the K3^[n] work will find the comparison useful. It is solid enough to go to referees; the restrictions are explicit so a referee can assess whether the extension is worth the effort.

Referee Report

0 major / 2 minor

Summary. The paper partially extends prior results on stable rigid vector bundles from hyperkähler varieties of K3^[n] type to those of Kummer type. For a general polarized HK fourfold (M,h) of Kummer type satisfying either q_M(h) ≡ -6 mod 16 with divisibility of h equal to 2, or q_M(h) ≡ -6 mod 144 with divisibility 6, it claims there exists a unique (up to isomorphism) slope stable rigid vector bundle F with rank 4, c1(F)=h, and Δ(F)=c2(M).

Significance. If the result holds, it supplies a concrete existence-uniqueness statement for rank-4 bundles on Kummer-type fourfolds under explicit arithmetic conditions on the polarization, building directly on independent K3^[n] work. This could support explicit descriptions of locally complete families of polarized HK fourfolds, as stated in the motivation, and adds to the catalog of rigid stable bundles on hyperkähler fourfolds.

minor comments (2)

Abstract: the two arithmetic conditions on q_M(h) and the divisibility of h are stated clearly but would benefit from a one-sentence reminder of the definition of divisibility in this context (or a reference to the relevant section) for readers outside the immediate subfield.
The manuscript should include an explicit statement of how the Kummer-type case reduces to or differs from the K3^[n] arguments, even if only by citing the precise lemmas that carry over.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript, including the recommendation for minor revision. The report provides a concise summary of the main result but lists no specific major comments under the MAJOR COMMENTS section.

Circularity Check

0 steps flagged

No significant circularity; derivation builds on independent prior results

full rationale

The paper states it partially extends prior results (on stable rigid bundles for K3^[n] varieties) to the Kummer-type case under explicit arithmetic conditions on q_M(h) and divisibility of h. The central claim of existence, uniqueness up to isomorphism, slope stability, and rigidity of F is presented as a new application rather than a self-referential fit or redefinition. No equations or steps in the provided abstract reduce the target statement to its inputs by construction, and the cited prior work is treated as external independent input. This is the normal case of a self-contained extension against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard theory of hyperkähler varieties, the Beauville-Bogomolov-Fujiki quadratic form, and slope stability; no new free parameters or invented entities are introduced.

axioms (1)

domain assumption Existence and basic properties of polarized hyperkähler fourfolds of Kummer type with the stated numerical conditions on the Beauville-Bogomolov-Fujiki form.
Invoked in the statement of the main theorem.

pith-pipeline@v0.9.0 · 5708 in / 1234 out tokens · 27252 ms · 2026-05-24T11:21:33.917292+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1. ... there exists one and only one slope stable vector bundle F on M ... rp(Fq=4, c1(F)=h, Δ(F)=c2(M). ... H1(M,End0(F))=0.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 5.5. ... the restriction of E(L) to a smooth fiber ... is slope stable.

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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