{"id":"b0969525-68c6-4ce4-b02a-9be5fbfd68b8","arxiv_id":"1908.05953","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A toric blow-up method computes the integral cohomology of quotients by cyclic prime-order groups and yields new Beauville-Bogomolov lattices for K3[m]-type quotients by order 5 and 7 symmetries.","lead":"This paper computes the full integer-valued cohomology of spaces obtained by dividing a complex manifold by a cyclic symmetry group of prime order, using toric geometry to resolve the singularities created by fixed points. It then uses the result to compute the Beauville-Bogomolov form, a key invariant, for new singular symplectic spaces built from Hilbert schemes of points on K3 surfaces.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorems 1.2/1.3 hinge on the external classification that every order-5/7 symplectic automorphism on K3[m]-type (m≤4/6) is standard; a non-standard automorphism would invalidate equation (40), the proof of ℓ^{2*}_-=0, and the BB-lattice formulas.","rationale":"The reader's weakest_assumption correctly identifies the external classification as the load-bearing condition. My independent pass found no internal contradiction in the toric blow-up sections: Proposition 3.12 is supported by the lens-space computation (real orientation is preserved since the real determinant of a unitary map is 1); Proposition 3.14 and Theorem 3.16 follow from the excision diagrams; the use of Proposition 3.7 is for subfans, so surjectivity is legitimate. Theorem 4.13's proof under hypothesis (3) does not need the unproved ℓ^1_p=0 part of Theorem 4.9, so Remark 4.10 is not a threat. The claim in Corollary 5.2 that Sym^j H^2(S,Fp) has only N1 and N_p blocks is plausible for j

Theorem 4.13 -> Theorem 1.1) is coherent as far as I can see; the soft spot is in the applications. Section 5.3 (statement before Corollary 5.7) invokes Mongardi [24, Thm 7.2.7] and [25, Thm 2.5] to assert that every symplectic automorphism of order 5 (resp. 7) on a hyperkähler manifold of K3[m]-type with m≤4 (resp. m≤6) is standard, i.e. deformation equivalent to a natural automorphism on S[m]. Corollary 5.2 uses equation (40), H^*(S[m],Fp) ≃ ⊕_j (Sym^j H^2(S,Fp))^{⊕m_j}, to prove ℓ^{2*}_-(S[m])=0; Theorem 4.9(3) then gives spectral-sequence degeneration. If a non-standard automorphism of order 5 or 7 existed in the stated m-range, (40) would not be available for it: the Fp[G]-module structure of H^*(X,Fp) could have ℓ^{2*}_-≠0, the degeneration hypothesis would fail, and Theorem 1.2/1.3 would not follow. The paper gives no independent proof of this classification; it is a cited external result. This is a dependency rather than an internal inconsistency, but it is load-bearing for the headline applications.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a toric blow-up technique to compute the integral cohomology of quotients X/G, where X is a compact complex manifold and G is a cyclic group of prime order acting with only isolated fixed points. The main theoretical results are: a description of the cohomology of toric blow-ups of Cn/G and Pn/G (Section 3); a criterion for degeneration of the equivariant cohomology spectral sequence in terms of Boissière–Nieper-Wisskirchen–Sarti invariants and the number of fixed points (Theorem 4.9); and a computation of coefficients of surjectivity, p-torsion in even degrees, and odd-degree torsion of X/G under a degeneration hypothesis (Theorems 1.1 and 4.13). The applications compute Beauville–Bogomolov lattices of quotients of K3[m]-type hyperkähler manifolds by symplectic automorphisms of order 5 and 7 (Theorems 1.2 and 1.3). The central technical chain appears coherent, but the advertised statements omit some standing hypotheses, and the order-5 and order-7 applications rely on an external classification result.","tokens_in":33752,"tokens_out":11538,"duration_ms":108518,"significance":"If the results are correct, they provide a general method for computing integral cohomology of quotients by prime-order cyclic groups with isolated fixed points, and they give the first Beauville–Bogomolov forms for singular primitively symplectic varieties of dimension strictly greater than 4. The paper contains genuinely new technical ingredients: the systematic use of toric blow-ups for isolated quotient singularities, explicit computations of the resulting lattices, and a degeneration criterion expressed through the Boissière–Nieper-Wisskirchen–Sarti invariants. The proofs are detailed and, where they depend on the author's prior work [23], the paper explains which statements are being generalized and how the earlier p≤19 restriction is removed. The main reservations concern completeness of the stated hypotheses and the heavy but legitimate dependence on the external classification of symplectic automorphisms for the headline applications.","major_comments":[{"comment":"The statement of Theorem 1.1 in the introduction, and the abstract, omit the standing hypothesis made at the start of Section 4.1 that H*(X,Z) is p-torsion-free. This hypothesis is used throughout the proof of Theorem 4.13, for instance in Proposition 2.16, Lemma 4.6, and in the definition of the invariants ℓ^k_+(X) on the torsion-free part of cohomology. Without this assumption the conclusions α_k(X)=0 and p-torsion-freeness of H^{2k}(X/G,Z) are not established, so the theorem as printed is stronger than what is proved. The abstract also omits the spectral-sequence degeneration hypothesis that is explicitly required in Theorem 1.1. Please restate Theorem 1.1 and the abstract with the full set of hypotheses.","section":"Theorem 1.1 (Introduction) and Section 4.1"},{"comment":"The introduction and abstract advertise necessary and sufficient conditions for degeneration of the equivariant cohomology spectral sequence, but the equivalence in Theorem 4.9(iii) is proved only under the additional assumption ℓ^1_p(X)=0, and Remark 4.10 states that the author has not been able to remove this condition. As stated, condition (3) alone is not shown to imply degeneration. Please qualify the claim in the abstract and in Section 1.2, and state the status of the extra condition directly in Theorem 4.9. In the proof of Corollary 5.2, Theorem 4.9(iii) is invoked without explicitly checking ℓ^1_p(X)=0; for hyperkähler manifolds H^1(X,Z)=0, so the condition is automatic, but it should be stated.","section":"Theorem 4.9 / Remark 4.10"},{"comment":"Theorems 1.2 and 1.3 depend on the external classification, cited to [24, Theorem 7.2.7 and Section 7.3] and [25, Theorem 2.5], that every symplectic automorphism of order 5 or 7 on a hyperkähler manifold of K3[m]-type with m≤6 is standard. This is a load-bearing dependence: equation (40) is used to prove ℓ^{2*}_-(S[m])=0, and without the classification the F_p[G]-module structure of H^*(X,F_p) could have nonzero ℓ^{2*}_- components, so the spectral-sequence degeneration and the subsequent lattice computations would not follow. The paper should state this dependence explicitly in the statements of Theorems 1.2 and 1.3, and should either give the precise theorem in [24]/[25] covering the full range m≤6 or provide a proof of the needed classification statement.","section":"Section 5.3 / Corollary 5.7"}],"minor_comments":[{"comment":"The torsion-freeness of H^*(S[m],Z) is attributed to [31, Theorem 2.2], but the reference [31] is titled as concerning the Hilbert scheme of two points; the integral basis theorem of [28] already gives torsion-freeness for all m, so the citation should be corrected or supplemented.","section":"Section 5.2, proof of Corollary 5.2"},{"comment":"The abstract says 'We describe the integral cohomology of X/G' without mentioning the degeneration hypothesis of Theorem 1.1 or the p-torsion-freeness assumption; the introduction's bullet list for Theorem 4.9 similarly overstates the result by omitting the condition ℓ^1_p(X)=0.","section":"Abstract and Section 1.2"},{"comment":"There are numerous typos and OCR artifacts that should be cleaned: 'man ifold' in the abstract, 'propostion' in Section 2.4, 'therm' in the proof of Theorem 4.9, and several '/integerdivide/' artifacts in Section 3.2 and in displayed diagrams. These do not affect the mathematics but make the text hard to read.","section":"Throughout"},{"comment":"In the paragraph after Definition 4.3, the term 'coefficient of resolution' is used before its notation β^{2k}(X) is introduced; please make the definition self-contained at first use.","section":"Section 4.4"},{"comment":"When Lemma 2.25 is applied to conclude that all even u^k vanish, the range of k is checked tersely; in particular the cases k=n and k=n+1 deserve an explicit sentence, since the lemma's hypothesis is only stated for k≥n+2.","section":"Section 4.5, proof of Theorem 4.9"}],"recommendation":"major_revision","confidential_remarks":"The technical core of the paper appears sound and the toric blow-up computations are substantial, but the advertised main theorems need to be restated with the correct standing hypotheses and with an explicit qualification of the dependence on the Mongardi classification for the order-5 and order-7 applications. I would be willing to see a revision that fixes these statements; if the authors also give a precise citation or proof for the classification step, acceptance seems appropriate. The unresolved condition in Theorem 4.9 is honestly disclosed and does not by itself undermine the main quotient-cohomology theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know before opening this arXiv: the paper is more honest than its abstract, and the main applications rest on a classification result that is cited rather than proved. The abstract promises an unconditional description of H^*(X/G,Z), but Theorem 1.1 needs the equivariant cohomology spectral sequence to degenerate at E_2. That hypothesis is hidden in the introduction, not in the abstract. The applications do verify it, but still.\n\nThe genuinely new material is good. Section 3 computes the integral cohomology of toric blow-ups of C^n/G, and that machinery is new and clean. The generalization of the Boissière–Nieper-Wisskirchen–Sarti structure theorem to all primes (Theorem 2.10) removes the annoying p≤19 restriction from the author's earlier work. Theorem 4.13, the detailed version of Theorem 1.1, is a substantial conditional description of the cohomology of the quotient. The proofs are detailed; I did not find a gap or an internal contradiction.\n\nThe soft spots are in proportion. Theorem 4.9 claims necessary and sufficient conditions for degeneration, but the sufficiency direction needs ℓ^1_p(X)=0, and Remark 4.10 admits this. So the real theorem is 'necessary, and sufficient with one more condition.' For the main applications the missing condition holds, so this is a presentation problem, not a mathematical one. The applications to order-5 and order-7 quotients rely on Mongardi's classification that all such symplectic automorphisms on K3[m]-type with m≤4/6 are standard. If that classification is wrong, equation (40) fails and the proof of ℓ^{2*}_-=0 collapses. The paper gives no independent proof, and the classification is external. This is a dependency, not a circular step, and the classification is probably right, but it is load-bearing.\n\nThis is a serious paper for the hyperkähler and orbifold cohomology people. The toric blow-up section has independent value. The author should fix the abstract and the Theorem 4.9 claim before publication. I'd send it to a serious referee, and the referee should scrutinize Section 3 and the use of Mongardi's classification. I would bring it to a reading group, but only with the caveat that the headline results are conditional on external input.","headline":"A substantial and mostly coherent paper whose abstract overstates the main theorem; the applications depend on a cited classification that is load-bearing but likely correct.","tokens_in":34291,"tokens_out":3850,"would_cite":true,"duration_ms":35809,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M25","14J50","55N91","14J28","14C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the integral cohomology of a quotient by a prime-order group is completely controlled by the number of fixed points and Jordan-block invariants, once the equivariant cohomology spectral sequence degenerates.","keywords":["integral cohomology of quotients","equivariant cohomology spectral sequence","toric blow-up","Jordan-block invariants","Beauville–Bogomolov form","hyperkähler manifolds of K3[m]-type","symplectic automorphisms","isolated fixed points"],"falsifier":"Test the explicit prediction for a natural order-5 automorphism on $S^{[2]}$: the paper gives $\\eta=14$, $H^3\\oplus H^7=(\\mathbb{Z}/5\\mathbb{Z})^{11}$, and $H^5=(\\mathbb{Z}/5\\mathbb{Z})^4$; an independent computation of these groups that disagrees would refute Theorem 4.13. Likewise, a non-standard symplectic automorphism of order 5 or 7 on a K3$[m]$-type manifold with $m\\le 6$ would break the isomorphism (40) on which the applications rest.","tokens_in":33190,"feed_emoji":"🧮","tokens_out":8404,"duration_ms":78693,"temperature":0.7,"pith_summary":"Prime-order group actions on compact complex manifolds are the basic building block for understanding quotients in integral cohomology. This paper claims that whenever the equivariant cohomology spectral sequence of $(X,G)$ with $\\mathbb{F}_p$ coefficients degenerates at the second page, the integral cohomology of the quotient $X/G$ is completely determined: all surjectivity coefficients vanish, all even cohomology is $p$-torsion-free, and the $p$-torsion in odd degrees is a direct sum of copies of $\\mathbb{Z}/p\\mathbb{Z}$ whose number is the fixed-point count minus a Jordan-block invariant. The proof supplies a new tool, the integral cohomology of toric blow-ups of quotient singularities, and applies the result to compute Beauville–Bogomolov lattices of singular symplectic orbifolds coming from Hilbert schemes of points on K3 surfaces under order-5 and order-7 automorphisms. If correct, this provides the first systematic way to get the full integral cohomology of such quotients rather than only rational or invariant information.","feed_headline":"Fixed points and Jordan blocks fix the integral cohomology of quotients","feed_subtitle":"When the equivariant spectral sequence collapses, even cohomology is torsion-free and odd torsion is an exact count.","key_machinery":"The load-bearing tools are the $\\ell_q$ invariants, which count Jordan blocks of size $q$ for the action of a generator of $G$ on $H^k(X,\\mathbb{F}_p)$, together with the split $\\ell_+,\\ell_-$ of a free $\\mathbb{Z}[G]$-module. These invariants encode the group cohomology of $G$ with coefficients in $H^*(X,\\mathbb{Z})$ and $H^*(X,\\mathbb{F}_p)$, and the paper proves a structure theorem valid for every prime $p$. On the geometric side, the paper uses toric blow-ups of $\\mathbb{C}^n/G$ and $\\mathbb{P}^n/G$: these resolve isolated quotient singularities, have torsion-free even cohomology, and their exceptional cycles generate sublattices of discriminant $p^2$. Poincaré duality of the resolved quotient then forces the coefficients of surjectivity to vanish and computes the torsion.","core_discovery":"The central claim is Theorem 4.13: let $X$ be a compact complex manifold, $G$ a cyclic automorphism group of prime order $p$ with $\\eta(G)$ isolated fixed points, and assume the equivariant cohomology spectral sequence of $(X,G)$ with $\\mathbb{F}_p$ coefficients degenerates at the second page. Then $\\alpha_k(X)=0$ for all $1\\le k\\le 2n$; $H^{2k}(X/G,\\mathbb{Z})$ is $p$-torsion-free for all $0\\le k\\le n$; and for $1\\le k\\le n-1$, $\\operatorname{tors}_p\\big(H^{2k+1}(X/G,\\mathbb{Z})\\oplus H^{2n-2k+1}(X/G,\\mathbb{Z})\\big)=(\\mathbb{Z}/p\\mathbb{Z})^{\\eta(G)-\\ell^{2k}_+(X)}$, where $\\ell^{2k}_+(X)$ counts trivial Jordan blocks in the $\\mathbb{F}_p[G]$-module $H^{2k}(X,\\mathbb{Z})$. The paper also proves that degeneration is equivalent, under a vanishing condition on the first cohomology invariant, to the numerical identity $\\eta(G)=\\ell^{2*}_+(X)+\\ell^{2*+1}_-(X)$, and that odd surjectivity coefficients satisfy a pairing identity. The applications are Theorems 1.2 and 1.3, giving the Beauville–Bogomolov lattices $U(5)\\oplus U^2\\oplus\\langle -10(m-1)\\rangle$ and $U\\oplus\\Lambda\\oplus\\langle -14(m-1)\\rangle$ with explicit Fujiki constants.","pith_inferences":["The same package could be applied to composite-order groups by decomposing actions into prime-order steps; nothing in the main theorem uses cyclicity beyond the Jordan-block description, though the fixed-point hypotheses would need checking.","If the standardness classification cited for orders 5 and 7 is extended to order 11 or 13 under suitable bounds on $m$, the method would immediately produce the next Beauville–Bogomolov lattices for singular symplectic varieties.","The paper leaves open the split of the odd torsion between paired degrees; testing the conjectured split on computed Hilbert-scheme examples, say $m=4$, would be a direct numerical check."],"forward_implications":["The coefficients of surjectivity $\\alpha_k(X)$ vanish in every degree, so the transfer image from $X$ already accounts for the full torsion-free cohomology of the quotient.","The $p$-torsion of odd cohomology is paired symmetrically between degrees $2k+1$ and $2n-2k-1$, with total count $\\eta(G)-\\ell^{2k}_+(X)$.","For hyperkähler manifolds of K3$[m]$-type with symplectic automorphism groups of order 5 ($m\\le 4$) or 7 ($m\\le 6$), the quotient is a singular primitively symplectic variety with Beauville–Bogomolov lattice $U(5)\\oplus U^2\\oplus\\langle -10(m-1)\\rangle$ or $U\\oplus\\Lambda\\oplus\\langle -14(m-1)\\rangle$, and with the stated Fujiki constant.","Degeneration of the equivariant spectral sequence can be detected by a representation-theoretic numerical identity, so no geometric computation is needed once the $\\mathbb{Z}[G]$-module structure of $H^*(X,\\mathbb{Z})$ is known.","Previous results that were restricted to primes $p\\le 19$ are extended to every prime, because the Jordan-block structure theorem holds without restriction."],"supporting_citations":[{"why":"Supplies the Smith transfer push-forward map and its two identities, which underpin the definition of the coefficients of surjectivity.","marker":"[30]"},{"why":"Introduces the Jordan-block invariants and the Smith-theory computations that the paper generalizes to all primes.","marker":"[4]"},{"why":"Establishes the previous quotient-cohomology framework, including exceptional lattices and surjectivity coefficients, which the paper extends.","marker":"[23]"},{"why":"Provides the standard fact that smooth complete toric varieties have torsion-free cohomology concentrated in even degrees.","marker":"[11]"},{"why":"Supplies the smoothness and completeness criteria for toric varieties used to construct toric resolutions.","marker":"[29]"},{"why":"Gives the equivariant cohomology isomorphism for smooth toric varieties used to prove surjectivity of pull-backs between toric resolutions.","marker":"[12]"},{"why":"Provides the integral basis of the cohomology of Hilbert schemes of points on K3 surfaces, used to describe the $\\mathbb{F}_p[G]$-module structure.","marker":"[28]"},{"why":"Contains the classification that symplectic automorphisms of order 5 and 7 on K3$[m]$-type manifolds with $m\\le 6$ are standard.","marker":"[24]"},{"why":"Supplies the deformation-theoretic statement that all such automorphisms are natural deformations, used for the same standardness conclusion.","marker":"[25]"},{"why":"Gives the invariant lattice of a K3 surface under symplectic automorphisms of prime order, which enters the Beauville–Bogomolov lattice computation.","marker":"[14]"}],"fun_headline_variants":["Spectral collapse yields exact torsion in cyclic quotient cohomology","Fixed points and Jordan blocks decide quotient torsion structure","Torsion-free even cohomology when equivariant sequence degenerates","Prime-order quotients: exact torsion under spectral collapse","Beauville-Bogomolov forms via spectral collapse and fixed points"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The order-5 and order-7 applications rest on the cited classification that every symplectic automorphism of those orders on a K3$[m]$-type manifold with $m\\le 6$ is standard; if a non-standard one existed, the symmetric-power description of the cohomology action would fail and the lattice computations would not go through.","fun_headline_variants_meta":{"raw":{"variants":["Spectral collapse yields exact torsion in cyclic quotient cohomology","Fixed points and Jordan blocks decide quotient torsion structure","Torsion-free even cohomology when equivariant sequence degenerates","Prime-order quotients: exact torsion under spectral collapse","Beauville-Bogomolov forms via spectral collapse and fixed points"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000757,"raw_usage":{"total_tokens":3408,"prompt_tokens":1030,"completion_tokens":2378,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":646,"completion_tokens_details":{"reasoning_tokens":2292}},"tokens_in":646,"tokens_out":2378,"duration_ms":19385,"temperature":1.0,"reasoning_tokens":2292,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:59:51.278829+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the explicit prediction for a natural order-5 automorphism on $S^{[2]}$: the paper gives $\\eta=14$, $H^3\\oplus H^7=(\\mathbb{Z}/5\\mathbb{Z})^{11}$, and $H^5=(\\mathbb{Z}/5\\mathbb{Z})^4$; an independent computation of these groups that disagrees would refute Theorem 4.13. Likewise, a non-standard symplectic automorphism of order 5 or 7 on a K3$[m]$-type manifold with $m\\le 6$ would break the isomorphism (40) on which the applications rest.","supporting_citations":[{"cited_title":"Smith, Transfer and ramified coverings , Math","cited_arxiv_id":null,"evidence_quote":"Supplies the Smith transfer push-forward map and its two identities, which underpin the definition of the coefficients of surjectivity."},{"cited_title":"Boissière, M","cited_arxiv_id":null,"evidence_quote":"Introduces the Jordan-block invariants and the Smith-theory computations that the paper generalizes to all primes."},{"cited_title":"Menet, On the integral cohomology of quotients of complex manifold s, Journal de Mathé- matiques pures et appliquées, 119 (2018), no.9, 280-325","cited_arxiv_id":null,"evidence_quote":"Establishes the previous quotient-cohomology framework, including exceptional lattices and surjectivity coefficients, which the paper extends."},{"cited_title":"Danilov, The geometry of toric varieties , (Russian) Uspekhi Mat","cited_arxiv_id":null,"evidence_quote":"Provides the standard fact that smooth complete toric varieties have torsion-free cohomology concentrated in even degrees."},{"cited_title":"Oda, Convex bodies and algebraic geometry , Springer-Verlag, New York Berlin Heidelberg, 1988","cited_arxiv_id":null,"evidence_quote":"Supplies the smoothness and completeness criteria for toric varieties used to construct toric resolutions."},{"cited_title":"Franz, Describing toric varieties and their equivariant cohomology , Colloq","cited_arxiv_id":null,"evidence_quote":"Gives the equivariant cohomology isomorphism for smooth toric varieties used to prove surjectivity of pull-backs between toric resolutions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the integral basis of the cohomology of Hilbert schemes of points on K3 surfaces, used to describe the $\\mathbb{F}_p[G]$-module structure."},{"cited_title":"Mongardi, Automorphisms of hyperkähler manifolds , PhD Thesis, University of Rome 3 (2013)","cited_arxiv_id":null,"evidence_quote":"Contains the classification that symplectic automorphisms of order 5 and 7 on K3$[m]$-type manifolds with $m\\le 6$ are standard."},{"cited_title":"Mongardi, On natural deformations of symplectic automorphisms of manif olds of K3[n] type, C","cited_arxiv_id":null,"evidence_quote":"Supplies the deformation-theoretic statement that all such automorphisms are natural deformations, used for the same standardness conclusion."},{"cited_title":"Garbagnati, A","cited_arxiv_id":null,"evidence_quote":"Gives the invariant lattice of a K3 surface under symplectic automorphisms of prime order, which enters the Beauville–Bogomolov lattice computation."}],"review_version":1}