Pith. sign in

REVIEW 2 major objections 5 minor 23 references

The hyperk\"ahler period-index conjecture is false

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper constructs smooth projective hyperkähler fourfolds, in both known deformation types, carrying Brauer classes whose index exceeds the square of the period, disproving the strengthened period-index conjecture for hyperkähler…

desk verdict The period-2 counterexamples are real and mostly self-contained, but the advertised period-5 case depends on an unproven transfer from an external preprint; referee the paper, send the authors back on Lemma 4. read the letter →

arxiv 2608.09436 v1 pith:O6J3THWF submitted 2026-08-10 math.AG

Pieter Belmans , James Hotchkiss This is my paper · ORCID
classification math.AG MSC 14J4214F22
keywords hyperkählerfourfoldsperiod-indexconjectureBrauergroupHodge-theoreticindexgeneralizedKummerfourfoldHilbertschemeofpointstwistedderivedcategoriesclass
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to disprove a stronger form of the period-index conjecture proposed for hyperkähler varieties, which predicted that for every Brauer class α on a hyperkähler variety X, the index of α divides the period raised to half the dimension. It does so by constructing smooth projective hyperkähler fourfolds, in both the K3^[2] and Kum² deformation types, together with Brauer classes of period 2 whose index is divisible by 8, and, for the K3^[2] type, classes of period 5 whose index is divisible by 125. In each case the index therefore fails to divide the square of the period. This matters because the conjecture was tailored to the special Hodge structure of hyperkähler varieties, and the counterexamples show where that special structure stops enforcing the bound.

What carries the argument

The key machinery is the Hodge-theoretic index ind_H(α): the positive generator of the image, under the rank homomorphism, of the integral Hodge classes in the twisted Mukai structure K_0^top(X)_B, where B is a B-field lift of α. Since ind_H(α) divides the usual index ind(α), a lower bound for ind_H disproves the conjecture. The argument runs through two obstruction lemmas. The 2-torsion obstruction uses an auxiliary cohomology class y that pairs trivially with all Hodge classes in the relevant degrees but nontrivially with the square of the B-field, ruling out twisted Hodge classes of rank 2 and 4 and hence forcing the rank to be divisible by 8. The 5-torsion obstruction, quoted from the literature, says that if ind_H(α) divides 25 for a period-5 class, then certain polynomial expressions P1, P2, P3 built from the B-field and Hodge classes must be integral; the paper shows that on a very general K3^[2]-type fourfold with a polarization of square 10 this integrality fails, giving the contradiction.

What would settle it

Compute the Hodge-theoretic index ind_H(α) for the period-5 Brauer class defined in Section 5 on an explicit K3^[2]-type fourfold (not necessarily very general) using the twisted Mukai structure; if any integral Hodge class of rank dividing 25 exists, the contradiction in Theorem 12 would be overturned, and the paper's conclusion for that class would fail.

Watch

Extended reading notes

Core claim

The central claim, stated as Theorem B, is that there exist smooth projective hyperkähler fourfolds X of Kum²-type and of K3^[2]-type, together with Brauer classes α of period 2, such that 8 divides ind(α); in particular ind(α) does not divide per(α)^2. Additionally, there are hyperkähler fourfolds of K3^[2]-type with Brauer classes α of period 5 such that 125 divides ind(α). The proof does not exhibit Azumaya algebras of the required degree directly. Instead it works with the Hodge-theoretic index ind_H(α), defined in terms of integral Hodge classes in the B-field-twisted Mukai structure on topological K-theory, which always divides the actual index; lower bounds on ind_H therefore give lower bounds on ind. On very general fourfolds with a chosen polarization, the authors use numerical conditions involving the Beauville–Bogomolov–Fujiki form to show that no twisted Hodge class of small rank can exist for the chosen Brauer classes, forcing the divisibilities.

Load-bearing premise

The period-5 counterexample relies on an imported lemma, quoted but not proved here, asserting that a bound on the Hodge-theoretic index of the form ind_H(α) | 25 forces certain integral cohomology classes to exist; if that lemma does not extend to the Hodge-theoretic index as the paper assumes, the period-5 construction collapses.

Editorial extensions

If this is right

  • The strengthened period-index conjecture for hyperkähler varieties is false in dimension 4, so bounds on Brauer-class indices on hyperkähler fourfolds must be weaker than per(α)^2.
  • The failure occurs in both known deformation types of hyperkähler fourfolds, indicating that it is a general feature of the deformation families, not a lattice-specific artifact.
  • The original period-index conjecture, which allows the exponent dim X − 1, remains consistent with these examples, since the constructed indices still divide higher powers of the periods.
  • For period-5 classes on K3^[2]-type fourfolds, the index can exceed the square of the period by a factor of 125, showing the gap between the period and the index can be substantial.
  • The Hodge-theoretic index method supplies a practical tool for testing period-index bounds on hyperkähler varieties without constructing Azumaya algebras.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same 2-torsion obstruction may be adaptable to further hyperkähler fourfold deformation types beyond the two considered here, provided their cohomology rings admit an auxiliary class y with the required pairing properties; that would test how universal the failure is.
  • For higher-dimensional hyperkähler varieties, the conjecture's bound is per(α)^(dim/2); extending the obstruction construction to dimension 6 or higher would require classes in H^6 and analogous integrality lemmas, which the paper's methods do not yet supply.
  • The paper's very-general construction suggests the counterexamples form a positive-dimensional locus in moduli, so the conjecture may fail on an open neighborhood of these fourfolds rather than only on isolated examples; this is not established in the paper.
  • One could attempt to lift the Hodge-theoretic index lower bounds to explicit Azumaya algebras or to control the actual index via deformation arguments, which would turn the numerical obstructions into constructive counterexamples.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper aims to disprove the hyperkähler period-index conjecture of Huybrechts (Conjecture A) in dimension 4. It constructs, on very general hyperkähler fourfolds of Kum2-type and K3[2]-type, Brauer classes of period 2 whose Hodge-theoretic index is divisible by 8, and on K3[2]-type a Brauer class of period 5 whose Hodge-theoretic index is divisible by 125. In all cases this gives ind(α) not dividing per(α)^2, contradicting the conjecture. The proofs use the Hodge-theoretic index from [6], two obstruction lemmas (a 2-torsion obstruction and a 5-torsion obstruction), and explicit classes in the cohomology of the relevant deformation types. The integrality of auxiliary classes is shown either by citing known lattice results or, for the Hilbert square case, by a Nakajima-operator computation.

Significance. If correct, the paper gives a definitive counterexample to a conjecture of Huybrechts, showing that the stronger period-index bound fails already in dimension 4. The constructions are explicit and the intersection computations are concrete, which strengthens the result. The period-2 counterexamples (Theorems 6 and 9) appear to be well supported by the written arguments and standard cited facts. The period-5 counterexample (Theorem 12) is a compelling contradiction argument, but it rests on an imported lemma whose logical adaptation is not fully documented. The paper also benefits from the recent context of Perry's disproof of the original period-index conjecture in high dimensions.

major comments (2)
  1. [§2.2, Lemma 4] The proof of Lemma 4 is not self-contained and the logical direction of the transfer from ind(α) to ind_H(α) is not demonstrated. The hypothesis of Lemma 4 is ind_H(α)|25, whereas the cited [11, Theorem 1.9] is stated to concern ind(α). The paper asserts that [11, Lemma 5.12] transfers a lower bound for ind(α) to ind_H(α), but no precise statement of that lemma is given, and it is not shown how the contrapositive of such a transfer yields the existence of h1,h2,h3 under the hypothesis ind_H|25. Since ind_H|25 does not formally imply ind|25, this is a load-bearing gap: Theorem 12 relies entirely on Lemma 4 to obtain the integrality of the P_i. The authors should either prove Lemma 4 directly or state the exact form of [11, Theorem 1.9] and [11, Lemma 5.12] and spell out the logical deduction.
  2. [§2.2, Lemma 4, second remark] The claim that the fourth obstruction P4 'provides no further constraint on the index' is only asserted in one sentence. Since the paper's period-5 argument depends on the precise form of the obstructions from [11, Theorem 1.9], the authors should explain this reduction carefully: for example, why one may choose the fourth Hodge class so that P4=0, and why this choice does not alter the validity of the other P_i. As written, a reader cannot verify this point without consulting the cited paper.
minor comments (5)
  1. [§3–§5] The phrase 'standard isotropic basis of U^⊕3' is ambiguous; the convention used in the computations is that (u1,u2)=(v1,v2)=(w1,w2)=1, not the perhaps more common convention (u_i,v_i)=1. Please state this pairing explicitly at first use to avoid confusion.
  2. [Title and abstract] The title 'The hyperkähler period-index conjecture is false' could be misread as referring to the usual period-index conjecture for hyperkähler varieties rather than Huybrechts's stronger conjecture. Consider a more specific title such as 'Huybrechts's strong period-index conjecture is false in dimension 4'.
  3. [§5, Eq. (32)] The expression '∫ P3 a ≡ 92·6/5 .0 mod Z' appears to contain a typo; it should read either '≡ 92·6/5 mod Z' or '≠ 0 mod Z'.
  4. [§5, proof of Theorem 12] After deriving 5|k, the text says 'Hence 24/5 b h1 is integral, and so is h2'. It may be worth adding one sentence explaining that h2 is a Hodge class of type (2,2), so Lemma 14 applies to it; this is implicit but helpful.
  5. [§2.1, Lemma 2] The proof of Lemma 2 is cited to [6, Lemma 5.8(4)]. For the paper to be readable as a standalone document, a one-sentence indication of why per(α)|ind_H(α)|ind(α) holds would be useful, though the citation is acceptable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the counterexamples are explicit computations, and the cited Hodge-theoretic index results are independent support rather than equivalent inputs.

full rationale

The derivation chain is not circular. The period-2 counterexamples (Theorems 6 and 9) are built from explicit cohomology classes and verified by direct intersection computations (Lemmas 7 and 10), together with external integrality results (Lemmas 5 and 11). No parameter is fitted to the target divisibility, and no target equation is used as an input. The Hodge-theoretic index ind_H and the inequality per | ind_H | ind are imported from [6]; although this is a self-citation (Hotchkiss is a coauthor), Lemma 2 is a parameter-free general divisibility theorem whose assumptions do not include the counterexample statement, so it is independent support rather than a circular premise. The period-5 case (Theorem 12) is not self-contained because Lemma 4 relies on [11, Theorem 1.9] and [11, Lemma 5.12] for the ind-to-ind_H transfer; this is a completeness and correctness concern about an external cited result, not a reduction of the claim to its own input. No equation in the paper is definitionally equal to a fitted parameter or a renamed version of the conjecture being disproved.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data and no new entities are postulated. The central claim rests on the Hodge-theoretic index theory from [6], the period-index obstruction from [11], standard lattice and intersection-form computations, and parallel transport/integrality results for auxiliary classes. The five-torsion case leans most heavily on [11], so the ledger is dominated by external theorems rather than by new assumptions.

assumptions (6)
  • domain assumption The Hodge-theoretic index ind_H(alpha) divides the actual index ind(alpha), and both share prime factors with the period (Lemma 2, from [6, Lemma 5.8(4)]).
    Invoked in every theorem to convert ind_H lower bounds into ind lower bounds. Not proved in this paper; taken from the first author's prior preprint.
  • domain assumption The twisted Mukai Hodge structure K^top_0(X)_B exists, is integral, and depends only on the Brauer class (Section 2.1, [6, Definitions 4.9 and 5.7, Corollary 4.14]).
    Underpins Definition 1 and Lemma 3. Accepted from [6] without reproduction.
  • domain assumption For very general marked fourfolds of the chosen families, the Hodge decomposition is as stated in (13) and (18): Picard rank 1 and H^{2,2} spanned by lambda^2 and q^vee, with the extra V summand in Kum2-type.
    Used in the proofs of Theorems 6, 9, and 12 to evaluate all possible Hodge classes; cited from [5], [13], [2], and [23].
  • domain assumption The Beauville-Bogomolov-Fujiki pairings and quartic intersection formulas (10), (11), (12), (15), (16), (17) hold, and the auxiliary classes y are integral via parallel transport and Nakajima operator results (Lemmas 5, 8, and 11).
    All numerical checks in Sections 3 and 4 rely on these identities; integrality is cited from [13, Proposition 6.30] and [21, Theorem 5.4] plus a parallel transport argument.
  • domain assumption Lemma 4: if ind_H(alpha) divides 25 for a period-5 class, then integral Hodge classes h1, h2, h3 exist making P1, P2, P3 integral, and the lower bound from [11, Theorem 1.9] transfers from ind to ind_H.
    The entire 5-torsion counterexample in Theorem 12 rests on this external theorem and on [11, Lemma 5.12], neither proved in the text.
  • domain assumption The integral cohomology of K3^[2]-type and Kum2-type fourfolds is torsion-free, so every Brauer class is topologically trivial and admits a B-field lift (Section 2, [16, Theorem 1] and [12, Theorem 1]).
    Needed for all B-field considerations and for the explicit form alpha = exp(2 pi i b / l).

how reviews work

0 comments
Cite this review

Pith. "Pith review of The hyperk\"ahler period-index conjecture is false." pith.science (2026). https://pith.science/paper/O6J3THWF

@misc{pith2026260809436,
  author       = {Pith},
  title        = {Pith review of: The hyperk\"ahler period-index conjecture is false},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O6J3THWF}},
  note         = {Machine review of arXiv:2608.09436}
}
abstract

Huybrechts conjectured that for every Brauer class $\alpha$ on a hyperk\"ahler variety $X$ we have that $\mathop{\rm ind}(\alpha)\mid\mathop{\rm per}(\alpha)^{\dim(X)/2}$, strengthening the usual period-index conjecture. We show that this fails on certain hyperk\"ahler fourfolds, both in type $\mathrm{K}3^{[2]}$ and $\mathrm{Kum}^2$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 10 canonical work pages

  1. [2]

    On the cohomology ring of a simple hyper-Kähler manifold (on the results of Verbitsky)

    F. A. Bogomolov. “On the cohomology ring of a simple hyper-Kähler manifold (on the results of Verbitsky)”. In:Geom. Funct. Anal.6.4 (1996), pp. 612–618.doi: 10.1007/BF02247113. MR: 1406665

    work page doi:10.1007/bf02247113 1996

  2. [11]

    Aise Johan de Jong and Alexander Perry.The period-index problem and Hodge theory. 2022. arXiv: 2212.12971v1 [math.AG]

    arXiv 2022

  3. [6]

    James Hotchkiss.Hodge theory of twisted derived categories and the period-index problem. 2022. arXiv:2212.10638v1 [math.AG]

    arXiv 2022

  4. [7]

    James Hotchkiss, Davesh Maulik, Junliang Shen, Qizheng Yin, and Ruxuan Zhang.The period- index problem for hyper-Kähler varieties via hyperholomorphic bundles. 2025. arXiv: 2502.09774v2 [math.AG]

    arXiv 2025

  5. [1]

    Variétés Kähleriennes dont la première classe de Chern est nulle

    Arnaud Beauville. “Variétés Kähleriennes dont la première classe de Chern est nulle”. In:J. Differential Geom.18.4 (1983), pp. 755–782. MR: 730926.url: http://projecteuclid.org/ euclid.jdg/1214438181

    arXiv 1983

  6. [3]

    Alessio Bottini and Daniel Huybrechts.The period-index problem for hyperkähler varieties: Lower and upper bounds. 2025. arXiv:2512.15131v1 [math.AG]

    arXiv 2025

  7. [4]

    Exposant et indice d’algèbres simples centrales non ramifiées

    Jean-Louis Colliot-Thélène. “Exposant et indice d’algèbres simples centrales non ramifiées”. In: Enseign. Math. (2)48.1-2 (2002). With an appendix by Ofer Gabber, pp. 127–146. MR:1923420

    work page 2002

  8. [5]

    On a conjecture of Matsushita

    Bert van Geemen and Claire Voisin. “On a conjecture of Matsushita”. In:Int. Math. Res. Not. IMRN 10 (2016), pp. 3111–3123.doi:10.1093/imrn/rnv230. MR:3551831

    work page doi:10.1093/imrn/rnv230 2016

Show all 23 references
  1. [8]

    Daniel Huybrechts.The period-index problem for hyperkähler manifolds. 2025. arXiv: 2411 . 17604v2 [math.AG]

    2025

  2. [9]

    Equivalences of twisted𝐾3surfaces

    Daniel Huybrechts and Paolo Stellari. “Equivalences of twisted𝐾3surfaces”. In:Math. Ann.332.4 (2005), pp. 901–936.doi:10.1007/s00208-005-0662-2. MR:2179782

    2005 doi

  3. [10]

    The period-index problem for the Brauer group of an algebraic surface

    A. J. de Jong. “The period-index problem for the Brauer group of an algebraic surface”. In:Duke Math. J.123.1 (2004), pp. 71–94.doi:10.1215/S0012-7094-04-12313-9. MR:2060023. 10

    2004 doi

  4. [12]

    Corrigendum: “Integral cohomology of the generalized Kummer fourfold

    Simon Kapfer and Grégoire Menet. “Corrigendum: “Integral cohomology of the generalized Kummer fourfold” (Algebraic Geometry 5, no. 5 (2018), 523–567)”. In:Algebr. Geom.7.4 (2020), pp. 482–485.doi:10.14231/AG-2020-014. MR:4156412

    2018 doi

  5. [13]

    Integral cohomology of the generalized Kummer fourfold

    Simon Kapfer and Grégoire Menet. “Integral cohomology of the generalized Kummer fourfold”. In:Algebr. Geom.5.5 (2018), pp. 523–567.doi:10.14231/ag-2018-014. MR:3847205

    2018 doi

  6. [14]

    Hodge-theoretic obstruction to the existence of quaternion algebras

    Andrew Kresch. “Hodge-theoretic obstruction to the existence of quaternion algebras”. In:Bull. London Math. Soc.35.1 (2003), pp. 109–116.doi:10.1112/S0024609302001546. MR:1934439

    2003 doi

  7. [15]

    A survey of Torelli and monodromy results for holomorphic-symplectic varieties

    Eyal Markman. “A survey of Torelli and monodromy results for holomorphic-symplectic varieties”. In:Complex and differential geometry. Vol. 8. Springer Proc. Math. Springer, Heidelberg, 2011, pp. 257–322.isbn: 978-3-642-20299-5.doi:10.1007/978-3-642-20300-8\ _15. MR:2964480

    2011 doi

  8. [16]

    Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces

    Eyal Markman. “Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces”. In:Adv. Math.208.2 (2007), pp. 622–646.doi: 10.1016/j.aim.2006.03.006. MR:2304330

    2007 doi

  9. [17]

    Hiraku Nakajima.Lectures on Hilbert schemes of points on surfaces. Vol. 18. University Lecture Series. American Mathematical Society, Providence, RI, 1999, pp. xii+132.isbn: 0-8218-1956-9. doi:10.1090/ulect/018. MR:1711344

    1999 doi

  10. [18]

    Hodge classes of type(2, 2) on Hilbert squares of projective K3 surfaces

    Simone Novario. “Hodge classes of type(2, 2) on Hilbert squares of projective K3 surfaces”. In: Kyoto J. Math.66.1 (2026), pp. 187–221

    2026

  11. [19]

    Irreducible symplectic 4-folds numerically equivalent to(𝐾 3)[2]

    Kieran G. O’Grady. “Irreducible symplectic 4-folds numerically equivalent to(𝐾 3)[2] ”. In:Com- mun. Contemp. Math.10.4 (2008), pp. 553–608.doi: 10.1142/S0219199708002909. MR: 2444848

    2008 doi

  12. [20]

    Alexander Perry.The period-index conjecture is false. 2026. arXiv:2608.03684v1 [math.AG]

    2026 arXiv

  13. [21]

    Integral operators and integral cohomology classes of Hilbert schemes

    Zhenbo Qin and Weiqiang Wang. “Integral operators and integral cohomology classes of Hilbert schemes”. In:Math. Ann.331.3 (2005), pp. 669–692

    2005

  14. [22]

    On the Beauville form of the known irreducible symplectic varieties

    Antonio Rapagnetta. “On the Beauville form of the known irreducible symplectic varieties”. In: Math. Ann.340.1 (2008), pp. 77–95.doi:10.1007/s00208-007-0139-6. MR:2349768

    2008 doi

  15. [23]

    Cohomology of compact hyper-Kähler manifolds and its applications

    M. Verbitsky. “Cohomology of compact hyper-Kähler manifolds and its applications”. In:Geom. Funct. Anal.6.4 (1996), pp. 601–611.doi:10.1007/BF02247112. MR:1406664. Pieter Belmans,p.belmans@uu.nl Mathematical Institute, Utrecht University, Budapestlaan 6, 3584CD Utrecht, Nether...

    1996 doi

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.