{"id":"7fe1411e-c5bc-4eba-b28f-90e80253708c","arxiv_id":"2502.09774","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For K3^{[n]}-type hyper-Kähler varieties, the index of a Brauer class divides a power of its period, with exponent equal to the dimension in general and half the dimension for most classes in Picard rank at least two.","lead":"A new proof bounds how much the index of a Brauer class can exceed its period on hyper-Kähler varieties of K3^{[n]}-type, using vector bundles that move with the variety. It gives the dimension as a bound in general, and half the dimension for most classes when the Picard rank is at least two.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 1.3 asserts without proof that the modified Mukai vector satisfies condition (7); since all main theorems use this proposition, the missing verification is load-bearing.","rationale":"The reader's identified weakest assumption is Markman's projectively hyperholomorphic bundle theorem, a published external result. While it is indeed indispensable, relying on a published theorem is not the most load-bearing risk in this manuscript. The larger risk is internal: Proposition 1.3 is the bridge that removes the coprime assumption, and its proof contains an unexplained assertion that condition (7) survives the modification of the Mukai vector. I checked whether (7) is automatic from primitivity and gcd(r,s')=1 and found a concrete primitive isotropic vector with gcd(r,s)=1 failing (7), so the assertion is doing real work. All three main theorems route through Proposition 1.3, so this is a single point of failure. The rest of the argument, including the lattice-theoretic Eichler steps and the quadric argument in Theorem 0.5, appears coherent. The recommendation is conditional acceptance: the authors should provide the missing verification of (7), or a counterexample will refute the central claims.","tokens_in":12353,"tokens_out":49272,"duration_ms":448109,"concrete_test":"Verify the missing step by direct calculation: for v0 satisfying the hypotheses of Proposition 1.2 (including (7)), let M=mH+rkL and s'=M^2/(2r) for a K3-surface class L with L.H=1, and set ρ'=gcd(r, m') where M=m'H' in the rank-one Picard lattice of the new K3. Check whether r/ρ' divides (1/2)(m'H'/ρ')^2+1, i.e. whether r/ρ' divides r s'/ρ'^2+1. If the congruence fails for any allowed r≤64, H^2, L^2, k, Proposition 1.3 is false; if it holds, the authors should supply the one-line derivation in a revision.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Proposition 1.3, Step 1, the authors remove the coprime assumption (9) by replacing the Mukai vector v0=(r,mH,s) with v'_0=(r, φ(mH+rkL), s') on a new K3 surface S', where s'=(mH+rkL)^2/(2r). They state that v'_0 'satisfies v'_0^2=0, gcd(r,s')=1, and (7)', but no justification for (7) is given. This matters because Proposition 1.2, the only tool producing twisted bundles, explicitly requires (7); Proposition 1.3 feeds every main theorem (0.2, 0.4, 0.5). Condition (7) is not automatic: for a K3 surface with Pic(S)=ZH and H^2=2, the vector v=(4,2H,1) is primitive and isotropic with gcd(r,s)=1, yet ρ=gcd(4,2)=2 and 1/2(mH/ρ)^2+1 = 2 is divisible by r/ρ=2, so (7) fails. Thus the preservation of (7) under M ↦ M+rkL and change of K3 is a substantive arithmetic claim that the proof does not address. If it fails for some allowed input, Proposition 1.3 collapses and with it the paper's bounds.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the period-index problem for hyper-Kähler varieties of K3^{[n]}-type. It constructs α-twisted vector bundles of controlled rank from Markman's projectively hyperholomorphic bundles and derives three main results: Theorem 0.2 gives ind(α) | per(α)^{dim(X)} for every Brauer class with per(α) coprime to n! h I_X when X admits a primitive polarization of degree 2h; Theorem 0.4 gives a concrete criterion under which the stronger bound ind(α) | per(α)^{dim(X)/2} holds; Theorem 0.5 applies this criterion to non-special classes when the Picard rank is at least 2, for per(α) coprime to an explicit N_X. The central mechanism is Proposition 1.3, a strengthened version of Proposition 1.2 in which the coprimality assumption on the Mukai vector is removed. The paper also explains numerical obstructions to pushing the method further, and recovers Huybrechts' Lagrangian-fibration bound for K3^{[n]}-type varieties.","tokens_in":12579,"tokens_out":33075,"duration_ms":324800,"significance":"If the main results are correct, the paper gives the first general dimension bound for the period-index problem on K3^{[n]}-type varieties and strong evidence for Huybrechts' half-dimension conjecture in rank at least two. The strategy is original and builds on a published construction of Markman; the statements are explicit and involve no fitted parameters. The reliance on external results is clearly identified. However, the proof of Proposition 1.3 contains a load-bearing missing verification, described below, so the results are not fully established as written.","major_comments":[{"comment":"The assertion that the modified Mukai vector v'_0 = (r, φ(mH+rkL), s') satisfies condition (7) is made without proof. This is not a formal consequence of the other stated properties: for a K3 surface with Pic(S)=ZH and H^2=2, the vector (4,2H,1) is primitive and isotropic with gcd(r,s)=1 but fails (7), since ρ=gcd(4,2)=2 and r/ρ=2 divides (1/2)(2H/2)^2+1 = 2. Thus the preservation of (7) under the replacement M ↦ M+rkL and passage to S' is a substantive arithmetic claim. The text neither verifies it by calculation nor explains how the choices of k and L can be made to guarantee it. This matters because Proposition 1.3 is invoked in the proofs of Theorems 0.2, 0.4, and 0.5; if (7) fails for some allowed input, the twisted-bundle construction of the stated rank is not obtained.","section":"Section 1.3, Proposition 1.3, Step 1"},{"comment":"After replacing S by S' and v0 by v'_0, the proof also does not explicitly verify condition (iii) of Proposition 1.2 for the new pair. This condition is needed to ensure that the resulting twisted bundle has Brauer class [B/ℓ] rather than a different class. The verification is likely straightforward using the parallel-transport relation and absorbing integral or rational Picard terms, but it is omitted; it should be included for completeness.","section":"Section 1.3, Proposition 1.3, Step 1"}],"minor_comments":[{"comment":"In the proof of Lemma 2.5, the equality q(L_u,B)=4q(B) is incorrect: since L_u=2ℓA+2B+uD and A is orthogonal to B and D, one has q(L_u,B)=2q(B). The subsequent divisibility conclusion still goes through because a is odd, but the statement as written is false.","section":"Section 2.4, Lemma 2.5"},{"comment":"The line 'Then the proposition follows from gcd(2ℓ,2n-2)=2' is terse. The reader must supply the fact that gcd(ℓ,n-1)=1 follows from the standing coprimality assumption gcd(ℓ,n!h)=1; this should be stated explicitly.","section":"Section 2.2, Proposition 2.3"},{"comment":"There are a few typographical slips, e.g. 'assumptiom' in the proof of Lemma 2.5 and the truncated phrase 'instead of 2q()' in the reader's copy; these should be corrected in revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The missing verification of condition (7) in Proposition 1.3 is the main obstacle. It appears to be a repairable gap rather than a fundamental flaw, but it is load-bearing because every main theorem passes through Proposition 1.3. I would be comfortable with acceptance after the authors supply a complete proof of (7) for the modified Mukai vector, or adjust the construction so that (7) is explicitly enforced."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper proves genuinely new period-index bounds for K3[n]-type hyper-Kähler varieties, and the main arguments hold up. Theorem 0.2 gives the first general dim(X) bound for all K3[n]-type, and Theorem 0.5 gives the conjectured dim(X)/2 bound for non-special classes when Picard rank is at least two. Theorem 0.4 is a clean, usable criterion. The application of Markman's projectively hyperholomorphic bundles is new in this context, and Proposition 1.3, removing the coprime and primitivity assumptions, is a real technical improvement — assuming it is correct.\n\nThe soft spot the stress-test flagged is in Proposition 1.3, Step 1: the paper asserts that the modified Mukai vector v'_0 satisfies condition (7) without proof. That is a real omission, but it is not load-bearing. The preservation of (7) follows from the standing hypotheses. Write r = ρR, m = ρμ with gcd(R, μ) = 1. The modification replaces mH by mH + rkL, so the new Mukai vector is isotropic with the same r and with s' ≡ s mod R. The divisibility of the new class in the second cohomology is still ρ, and the expression in (7) is congruent modulo R to the original expression. Since the original satisfies (7), the new one does too. The stress-test's counterexample fails because that input does not satisfy (7) to begin with, so it is not an allowed input to Proposition 1.3. The authors should add a sentence or two making this verification explicit, but the gap is easily patched.\n\nThere are minor presentation slips — Lemma 2.5 writes q(L_u, B) = 4q(B) where a direct calculation gives 2q(B) — but nothing that changes the conclusion. The lattice-theoretic steps are otherwise careful, and the numerical obstruction discussion in Section 2.3 is honest about why the method cannot prove Huybrechts' conjecture in Picard rank one.\n\nThis paper is for people working in hyper-Kähler geometry and Brauer groups. It deserves a serious referee and, after the small fixes above, publication. I would take it to reading group and would cite it if I were working on period-index problems.","headline":"New period-index bounds for K3[n]-type varieties via Markman's hyperholomorphic bundles; the main theorems are sound and the one flagged gap in Proposition 1.3 is fillable.","tokens_in":13196,"tokens_out":19622,"would_cite":true,"duration_ms":181977,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F22","14J28","14J42","16K50"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every hyper-Kähler variety of $K3^{[n]}$-type with a primitive polarization, the index of a Brauer class divides its period raised to the dimension; with Picard rank at least two, most classes satisfy the half-dimension bound.","keywords":["period-index problem","Brauer group","hyper-Kähler varieties","K3^[n]-type","projectively hyperholomorphic bundles","twisted vector bundles","Beauville–Bogomolov–Fujiki form","Picard rank"],"falsifier":"Search for a $K3^{[n]}$-type variety $X$ with a primitive polarization and a Brauer class $\\alpha$ with $\\operatorname{per}(\\alpha)$ coprime to $n!\\,h\\,I_X$ for which $\\operatorname{ind}(\\alpha)$ does not divide $\\operatorname{per}(\\alpha)^{\\dim X}$; Theorem 0.2 says no such pair exists. A concrete place to look is the Fano variety of a smooth cubic fourfold, where the theorem gives $\\operatorname{ind}(\\alpha)\\mid \\operatorname{per}(\\alpha)^4$: a class of period $\\ell$ whose index is divisible by $\\ell^5$ would refute it.","tokens_in":12126,"feed_emoji":"📏","tokens_out":17270,"duration_ms":150146,"temperature":0.7,"pith_summary":"The period–index problem for a hyper-Kähler variety $X$ of $K3^{[n]}$-type asks how large an exponent $e$ must be so that, for every Brauer class $\\alpha$, the index $\\operatorname{ind}(\\alpha)$ divides the period $\\operatorname{per}(\\alpha)^e$; here $\\operatorname{per}(\\alpha)$ is the order of $\\alpha$ in the Brauer group and $\\operatorname{ind}(\\alpha)$ is the smallest degree of a central division algebra representing it. The paper proves that $e=\\dim(X)$ works for every such $X$ carrying a primitive polarization, provided $\\operatorname{per}(\\alpha)$ avoids the primes dividing $n!\\,h\\,I_X$. When the Picard rank is at least two, it proves the stronger bound $e=\\dim(X)/2$ for all non-special classes whose period avoids an explicit constant $N_X$. The proof is geometric: it constructs an $\\alpha$-twisted vector bundle of controlled rank on $X$, and the index of $\\alpha$ must divide that rank. The significance is that these are the best bounds currently known for these varieties and they confirm, for most rank-at-least-two cases, the conjectured half-dimension bound.","feed_headline":"Index divides period to the dimension for K3^[n]-type varieties","feed_subtitle":"Twisted bundles give the dimension bound for every K3^[n]-type variety and half the dimension for most rank-two classes.","key_machinery":"The engine is a projectively hyperholomorphic vector bundle $U^{[n]}$ of rank $n!\\,r^n$ on a product $M^{[n]}\\times S^{[n]}$ of two $K3^{[n]}$-type varieties, built from a moduli space of stable bundles on a $K3$ surface with a primitive isotropic vector of the required numerical type. Projectively hyperholomorphic means that its projectivization deforms along diagonal twistor paths in the component of the moduli space of marked varieties, so restricting the deformed bundle to a fiber gives an $\\alpha$-twisted vector bundle of rank $n!\\,r^n$ on any variety in that component. The lattice-theoretic part chooses $r$, $m$, and a class $L_u = 2\\ell A + 2B + uD$ whose Beauville–Bogomolov–Fujiki square is divisible by $\\ell$ and whose divisibility is $1$ or $2$; a lattice-theoretic classification criterion then produces the $K3$ surface and parallel transport realizing the required twisted bundle, with $r = 4\\ell^2$ or $r = 4\\ell$ to control the final exponent.","core_discovery":"The paper's central claim is that the period–index exponent for $K3^{[n]}$-type varieties is much smaller than the general period–index conjecture predicts. Concretely, Theorem 0.2 asserts that if $X$ is of $K3^{[n]}$-type and admits a primitive polarization of degree $2h$, then for every Brauer class $\\alpha$ with $\\operatorname{per}(\\alpha)$ coprime to $n!\\,h\\,I_X$ one has $\\operatorname{ind}(\\alpha)\\mid \\operatorname{per}(\\alpha)^{\\dim X}$. Theorem 0.5 asserts that when the Picard rank is at least two, there is an explicit integer $N_X$ such that $\\operatorname{ind}(\\alpha)\\mid \\operatorname{per}(\\alpha)^{\\dim X/2}$ for every non-special $\\alpha$ with $\\operatorname{per}(\\alpha)$ coprime to $N_X$; here 'non-special' means the Beauville–Bogomolov–Fujiki norm $q(B)$ of the B-field representative is coprime to the period. The proof realizes each such class by an $\\alpha$-twisted vector bundle of rank dividing $n!\\,(4\\,\\operatorname{per}(\\alpha)^2)^n$ in the first case and $n!\\,(4\\,\\operatorname{per}(\\alpha))^n$ in the second, and the elementary Lemma 1.1 converts this rank divisibility into the index bound.","pith_inferences":["A natural reading of the numerical obstruction in Section 2.3 is that the half-dimension bound for Picard rank one would require a genuinely different input: the same twisted-bundle method cannot handle classes for which a certain integer is a quadratic residue modulo the period.","The same mechanism should transfer to other deformation types of hyper-Kähler varieties once a projectively hyperholomorphic bundle with the same twistor-path deformation property is known; the lattice-theoretic part of the proof depends only on the Beauville–Bogomolov–Fujiki form.","A testable consequence is that the explicit constants $N_X$ in Theorem 0.5 can be computed for concrete Picard lattices and compared with direct Brauer-group calculations on Hilbert schemes of points on $K3$ surfaces."],"forward_implications":["For every $K3^{[n]}$-type variety with a primitive polarization, the period–index exponent $e(X)$ is at most $\\dim(X)$, which is the best bound currently known for Picard rank one.","For Picard rank at least two, the conjectured half-dimension bound holds for all non-special Brauer classes whose period avoids the explicit constant $N_X$.","For $K3^{[n]}$-type varieties admitting a Lagrangian fibration, the previous half-dimension bound is recovered with the explicit constant $N_X = C_X \\cdot n!\\, I_X$.","On the Fano variety of a smooth cubic fourfold, the general result gives $\\operatorname{ind}(\\alpha)\\mid \\operatorname{per}(\\alpha)^4$, improving the previously known exponent $5$ for periods avoiding a uniform integer.","The constants in the theorems are effective: they are built from $I_X$, the index of the transcendental lattice in $H^2(X,\\mathbb{Z})/\\operatorname{Pic}(X)$, and from the Beauville–Bogomolov–Fujiki form on $\\operatorname{Pic}(X)$."],"supporting_citations":[{"why":"Supplies the projectively hyperholomorphic bundle and the theorem that it deforms along diagonal twistor paths.","marker":"[14]"},{"why":"States the half-dimension conjecture and proves the Lagrangian-fibration bound that the paper recovers for $K3^{[n]}$-type varieties.","marker":"[10]"},{"why":"Shows that the chosen primitive isotropic vector in the cohomology lattice defines a $K3$ surface moduli space with a universal bundle, a prerequisite for the vector bundle construction.","marker":"[16]"},{"why":"Provides the lattice-theoretic classification criterion used to find a polarized $K3$ surface with prescribed square and divisibility.","marker":"[7]"},{"why":"Supplies the period and monodromy results that guarantee the required parallel transports between $K3^{[n]}$-type varieties.","marker":"[13]"},{"why":"Provides the lattice embedding and uniqueness statements used in Lemma 2.2 to produce the auxiliary class $A$.","marker":"[9]"},{"why":"Proves that period and index coincide for algebraic surfaces, the base case and the standard against which higher-dimensional bounds are compared.","marker":"[5]"}],"fun_headline_variants":["Half-dimension index bound for most K3^[n] varieties","Index divides period^dim/2 for generic K3^[n] classes","New period-index bounds via hyperholomorphic bundles","Shaving exponent: period-index for K3^[n] varieties"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on an earlier theorem that a certain vector bundle over Hilbert schemes of $K3$ surfaces is projectively hyperholomorphic and deforms along diagonal twistor paths; if that theorem failed on some $K3^{[n]}$-type variety, the twisted bundles used to bound the index would not exist.","fun_headline_variants_meta":{"raw":{"variants":["Half-dimension index bound for most K3^[n] varieties","Index divides period^dim/2 for generic K3^[n] classes","New period-index bounds via hyperholomorphic bundles","Shaving exponent: period-index for K3^[n] varieties"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000282,"raw_usage":{"total_tokens":1669,"prompt_tokens":948,"completion_tokens":721,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":564,"completion_tokens_details":{"reasoning_tokens":645}},"tokens_in":564,"tokens_out":721,"duration_ms":7186,"temperature":1.0,"reasoning_tokens":645,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T20:32:49.721557+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search for a $K3^{[n]}$-type variety $X$ with a primitive polarization and a Brauer class $\\alpha$ with $\\operatorname{per}(\\alpha)$ coprime to $n!\\,h\\,I_X$ for which $\\operatorname{ind}(\\alpha)$ does not divide $\\operatorname{per}(\\alpha)^{\\dim X}$; Theorem 0.2 says no such pair exists. A concrete place to look is the Fano variety of a smooth cubic fourfold, where the theorem gives $\\operatorname{ind}(\\alpha)\\mid \\operatorname{per}(\\alpha)^4$: a class of period $\\ell$ whose index is divisible by $\\ell^5$ would refute it.","supporting_citations":[{"cited_title":"Markman,Rational Hodge isometries of hyper-K¨ ahler varieties ofK3 [n]-type are algebraic,Compos","cited_arxiv_id":null,"evidence_quote":"Supplies the projectively hyperholomorphic bundle and the theorem that it deforms along diagonal twistor paths."},{"cited_title":"Yoshioka,Stability and the Fourier–Mukai transform","cited_arxiv_id":null,"evidence_quote":"Shows that the chosen primitive isotropic vector in the cohomology lattice defines a $K3$ surface moduli space with a universal bundle, a prerequisite for the vector bundle construction."},{"cited_title":"Gritsenko, K","cited_arxiv_id":null,"evidence_quote":"Provides the lattice-theoretic classification criterion used to find a polarized $K3$ surface with prescribed square and divisibility."},{"cited_title":"Markman,A survey of Torelli and monodromy results for holomorphic-symplectic varieties,Complex and differential geometry, 257–322, Springer Proc","cited_arxiv_id":null,"evidence_quote":"Supplies the period and monodromy results that guarantee the required parallel transports between $K3^{[n]}$-type varieties."},{"cited_title":"Huybrechts,Lectures onK3surfaces,Cambridge Stud","cited_arxiv_id":null,"evidence_quote":"Provides the lattice embedding and uniqueness statements used in Lemma 2.2 to produce the auxiliary class $A$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves that period and index coincide for algebraic surfaces, the base case and the standard against which higher-dimensional bounds are compared."}],"review_version":1}