{"id":"22f81085-28b8-4ff6-84b1-1241739c520e","arxiv_id":"2608.09436","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Huybrechts's hyperkähler period-index conjecture is false in dimension 4: on some K3^[2] and Kum2 fourfolds, period-2 classes have index divisible by 8, and on some K3^[2] fourfolds a period-5 class has index divisible by 125.","lead":"The paper constructs Brauer classes on hyperkähler fourfolds whose index is too large for the period, disproving Huybrechts's strengthened period-index conjecture. Generalists in algebraic geometry should read it because it settles a recent conjecture and shows Hodge-theoretic index bounds can produce new counterexamples.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Period-5 claim is not self-contained: Lemma 4 imports an ind-to-ind_H transfer from [11] whose logical direction is unverified; if invalid, Theorem 12 collapses, though period-2 counterexamples survive.","rationale":"The reader's weakest_assumption identified the same spot: dependence on [11] for the 5-torsion case. I agree that this is the critical juncture. The period-2 counterexamples are, as far as I can see, correct; they already disprove Huybrechts' conjecture, so the paper's central claim is very likely true. The difference is in the recommended disposition: because the period-5 statement is part of Theorem B as quoted, a claim that cannot be verified from the cited material should not be accepted as established. The right revision is either to supply a complete proof of Lemma 4 or to state Theorem B only for the period-2 cases. Hence CONDITIONAL rather than ACCEPT.","tokens_in":10062,"tokens_out":38894,"duration_ms":300701,"concrete_test":"Open [11] (arXiv:2212.12971) and write a formal derivation of Lemma 4 from Theorem 1.9 and Lemma 5.12. Record exactly which hypothesis (ind(alpha)|25, ind_H(alpha)|25, or existence of a Hodge class of rank 25 in K_0^top(X)_{b/5}) is used to obtain each P_i integral, and which direction of Lemma 5.12 is applied. Then test the resulting implication on a simple case with ind=125 and ind_H=5: if the derivation concludes ind_H|25 implies ind|25, that implication is false and Lemma 4 is not a consequence of the cited results. If the derivation cannot be completed, Theorem 12 should be replaced by an independent proof or removed from Theorem B.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's most substantial self-contained contribution is the two period-2 counterexamples. I checked the computations in Lemmas 7 and 10 using the hyperbolic-pair convention in which (u1,u2), (v1,v2), (w1,w2) are the three standard pairs; with that convention the intersection numbers work and Lemmas 5 and 8 supply the needed integrality. So Conjecture A fails already in dimension 4 for period 2, assuming those cited integrality results are correct. The load-bearing weakness is the period-5 result, Theorem 12, which depends entirely on Lemma 4. Lemma 4 asserts that ind_H(alpha)|25 forces Hodge classes h1,h2,h3 such that P1,P2,P3 are integral. Its proof says [11, Theorem 1.9] is a statement about ind(alpha), not ind_H(alpha), and that [11, Lemma 5.12] transfers a lower bound for ind(alpha) to ind_H(alpha). No derivation is shown that the hypothesis ind_H|25 puts us in the setting of those cited results. The standard relation is ind_H|ind, so ind|25 implies ind_H|25; the reverse implication is not automatic. If [11, Lemma 5.12] only upgrades a lower bound ('ind does not divide 25 implies ind_H does not divide 25'), its contrapositive would need the additional implication ind_H|25 implies ind|25, which is false in general (e.g. ind=125, ind_H=5). The proof of Lemma 4 does not spell out how the cited lower-bound transfer produces the statement as written. If that adaptation fails, the contradiction in Theorem 12 cannot be started and the period-5 counterexample is unsupported. The abstract's headline result does not require period 5, but the full Theorem B as stated does.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10383,"tokens_out":20387,"duration_ms":150865,"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":[{"comment":"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.","section":"§2.2, Lemma 4"},{"comment":"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.","section":"§2.2, Lemma 4, second remark"}],"minor_comments":[{"comment":"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.","section":"§3–§5"},{"comment":"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'.","section":"Title and abstract"},{"comment":"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'.","section":"§5, Eq. (32)"},{"comment":"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.","section":"§5, proof of Theorem 12"},{"comment":"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.","section":"§2.1, Lemma 2"}],"recommendation":"major_revision","confidential_remarks":"The period-5 counterexample depends crucially on the transfer statement in [11, Lemma 5.12]; it would be prudent to have an expert verify that the direction and hypotheses match what the authors claim. The period-2 results appear solid and would already refute Conjecture A in dimension 4, so the paper's main conclusion is likely robust even if the period-5 part requires adjustment."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things. First, the paper's central claim is right for period 2: there are hyperkähler fourfolds of Kum2- and K3[2]-type with Brauer classes of period 2 whose index is divisible by 8, disproving Huybrechts's stronger period-index conjecture in dimension 4. The computations in Lemmas 7 and 10 check out, the integrality lemmas are argued properly (the detour through Nakajima operators for Lemma 8 is legitimate), and the paper is honest about what it takes from elsewhere. Second, the period-5 result in Theorem 12 is not on the same footing: it rests entirely on Lemma 4, which imports a Hodge-theoretic index statement from de Jong–Perry [11]. The stress-tester's worry lands. The proof of Lemma 4 is a two-sentence remark, and the direction of the transfer (from ind_H|25 to the existence of integral P_i) is not shown. If the adaptation is invalid, the 5-torsion counterexample collapses. The period-2 counterexamples survive, and they are enough to refute the conjecture as stated.\n\nWhat the paper does well: it gives explicit, checkable constructions, not soft non-density arguments. The intersection-theoretic computations are all written out, the class y is shown integral, and the 2-torsion Lemma 3 is proven cleanly. The citation of [6] for Lemma 2 is legitimate — that is the first author's prior theorem, and it is used as a black box but with a clear statement. The AI disclosure is refreshingly direct.\n\nWhere the soft spots are: the load-bearing external dependence for period 5, plus a general reliance on several preprints ([6], [7], [11]) that are not yet refereed. Those are not flaws by themselves, but they do mean the paper's full Theorem B as stated is less secure than the abstract suggests. The proof of Lemma 4 should either be expanded with a real argument or replaced by a direct computation.\n\nWho is this for? Anyone working on period-index problems, Brauer groups of hyperkähler varieties, or Hodge-theoretic index obstructions. The period-2 part is a clean counterexample that deserves to be known. The paper deserves a serious referee — the referee should focus on Lemma 4 and verify whether [11, Lemma 5.12] actually implies what the authors claim. If that fails, the period-5 part should be excised or fixed, but the paper still works as a counterexample in dimension 4.\n\nRecommendation: send to peer review. It is a genuine advance, the core math is reproducible, and the main weakness is a fixable gap rather than a fatal flaw.","headline":"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.","tokens_in":10937,"tokens_out":1503,"would_cite":true,"duration_ms":14949,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J42","14F22"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["hyperkähler fourfolds","period-index conjecture","Brauer group","Hodge-theoretic index","generalized Kummer fourfold","Hilbert scheme of points","twisted derived categories","index of Brauer class"],"falsifier":"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.","tokens_in":9837,"feed_emoji":"✖️","tokens_out":9941,"duration_ms":73212,"temperature":0.7,"pith_summary":"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.","feed_headline":"Hyperkähler index conjecture is false in dimension 4","feed_subtitle":"On fourfolds, Brauer classes of period 2 and 5 reach indices 8 and 125, past the conjectured bound.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"introduces the Hodge-theoretic index and proves that it divides the usual index (Lemma 2), the mechanism that makes lower bounds on ind_H disprove Conjecture A.","marker":"[6]"},{"why":"supplies the 5-torsion obstruction quoted as Lemma 4, which converts the assumption ind_H(α)|25 into integrality conditions on Hodge-class polynomials.","marker":"[11]"},{"why":"states the hyperkähler period-index conjecture (Conjecture A) that the paper disproves.","marker":"[8]"},{"why":"provides the integrality of the auxiliary class y and the ring-structure computations used in the Kum²-type section.","marker":"[13]"},{"why":"supplies the creation-operator identities on Hilbert squares used to prove integrality of the auxiliary class in the K3^[2] section.","marker":"[18]"},{"why":"gives integrality of a certain operator expression used in that same integrality proof.","marker":"[21]"},{"why":"provides the parallel transport results that extend the integrality statements from the special fourfolds to very general ones.","marker":"[15]"}],"fun_headline_variants":["Huybrechts' index conjecture fails on hyperkähler fourfolds","Hyperkähler period-index conjecture false for K3^[2] and Kum^2 fourfolds","Fourfold Brauer classes violate Huybrechts' period-index bound","Period 2 and 5 classes refute hyperkähler index conjecture","Counterexamples to hyperkähler period-index conjecture in dimension 4"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Huybrechts' index conjecture fails on hyperkähler fourfolds","Hyperkähler period-index conjecture false for K3^[2] and Kum^2 fourfolds","Fourfold Brauer classes violate Huybrechts' period-index bound","Period 2 and 5 classes refute hyperkähler index conjecture","Counterexamples to hyperkähler period-index conjecture in dimension 4"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001459,"raw_usage":{"total_tokens":5823,"prompt_tokens":850,"completion_tokens":4973,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":466,"completion_tokens_details":{"reasoning_tokens":4866}},"tokens_in":466,"tokens_out":4973,"duration_ms":31497,"temperature":1.0,"reasoning_tokens":4866,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:24:46.672694+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"states the hyperkähler period-index conjecture (Conjecture A) that the paper disproves."},{"cited_title":"Hodge classes of type(2, 2) on Hilbert squares of projective K3 surfaces","cited_arxiv_id":null,"evidence_quote":"supplies the creation-operator identities on Hilbert squares used to prove integrality of the auxiliary class in the K3^[2] section."},{"cited_title":"Integral operators and integral cohomology classes of Hilbert schemes","cited_arxiv_id":null,"evidence_quote":"gives integrality of a certain operator expression used in that same integrality proof."}],"review_version":1}