{"id":"b5112ecb-5875-4eb8-8066-00c231955595","arxiv_id":"2608.13481","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A smooth non-product canonically polarized threefold is constructed whose Kähler-Einstein metric has an 8-dimensional space of infinitesimal Einstein deformations, with exactly a 6-dimensional integrable subspace.","lead":"This paper constructs a smooth threefold with negative curvature whose Einstein metric has a deformation direction that cannot be smoothed into a real family. It settles a reformulated version of a long-standing question about non-integrable Einstein deformations in real dimension six.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.2's IED obstruction rests entirely on Nagy's preprint [8]; if the third-order equation or the obstruction identification fails, the central claim collapses.","rationale":"I read the full construction and proof. The internal steps after the external imports are coherent: the diagonal quotient is smooth, the descent and Künneth arguments in Lemma 2.2 are valid, the bracket formula (2.1) follows from the vanishing of cross-terms and H^2(C,TC)=0, and the integrability of the λ=0 directions via the family of G-covers is sound. The non-productness argument via the Albanese map and monodromy also checks out. The remaining soft spot is external dependence: Proposition 2.1 relies on the Böhning--Graf von Bothmer--Pignatelli surface and, crucially, on the claim H^1(S,T_S)^G=H^1(S,T_S); Proposition 3.2 relies on Nagy's third-order Einstein equation and on the identification of the obstruction class with a nonzero multiple of [ξ,ξ]. Between these, I consider the Nagy dependency the most load-bearing because without it the paper only obtains a non-integrable ICD, not the advertised non-integrable IED. The reader's weakest_assumption already identified both imports, so my concern overlaps substantially. I do not think this changes the verdict from the reader's moderate-confidence ACCEPT, because the internal logic is consistent and the external results are cited precisely; however, independent verification of [8] would substantially raise confidence.","tokens_in":5941,"tokens_out":32286,"duration_ms":295555,"concrete_test":"Independently re-derive the third-order Einstein equation in the fixed-volume normalization of [8, Thm. 1.2] for a compact Kähler-Einstein metric and a trace-free divergence-free h, and verify that the cohomology class of the real bracket [h,h]_c equals a nonzero constant multiple of the Dolbeault class [ξ,ξ] under Koiso's isomorphism. If the computation yields any extra term depending on h_2 or the scalar curvature, or if the constant is zero, then equation (3.2) is incomplete and Proposition 3.2 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the threefold admits a non-integrable IED is not established by the complex obstruction alone; it depends entirely on Proposition 3.2, which imports Nagy's preprint [8, Thm. 1.2(ii)(b)] for the third-order Einstein equation and the assertion that the complexification of the real class [h,h]_c is a nonzero universal scalar multiple of [ξ,ξ]. Neither the equation nor the proportionality constant is derived or checked in the present paper. The proof's fixed-volume normalization is also imported from [8, Thm. 5.12] and asserted to preserve h. If Nagy's equation actually contains additional terms involving h_2 or the scalar curvature in this normalization, or if the proportionality constant is zero, then the contradiction in Proposition 3.2 evaporates and the λ≠0 directions could be IED-integrable even though they are ICD-obstructed. Since [8] is an unpublished preprint and this is the only bridge from complex obstruction to Einstein obstruction, this is the least secure link in Theorem 1.2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a smooth projective threefold X as the diagonal quotient (S×C)/G, where S is the Böhning–Graf von Bothmer–Pignatelli surface with ample canonical bundle and H^1(S,T_S)=Cα, and C is a Galois cover of a genus-2 curve B with Galois group G=(Z/7)^4. It proves H^1(X,T_X)=Cα_X⊕H^1(B,T_B), that [λα_X+β,λα_X+β]≠0 whenever λ≠0, and hence that these directions are non-integrable infinitesimal complex deformations. Using Koiso's isomorphism and a third-order Einstein equation imported from Nagy [8], it concludes that the corresponding infinitesimal Einstein deformations are non-integrable and obstructed at third order, while the λ=0 directions are integrable. It also proves X is not biholomorphic to a product. The abstract states that this answers a suitably generalized version of the Dai–Wang–Wei question in real dimension 6.","tokens_in":6108,"tokens_out":11621,"duration_ms":116412,"significance":"Conditional on the imported results, the paper would give the first example in real dimension 6 of a negative Kähler–Einstein metric with a non-integrable infinitesimal Einstein deformation, and it exhibits a remarkably clean structure: an 8-dimensional real space of IEDs whose integrable directions form exactly a 6-dimensional subspace. The quotient construction in §2 is transparent, the descent and bracket computations in Lemma 2.2 are explicit and checkable, and the non-productness argument in §4.2 is self-contained. The paper introduces no free parameters and does not rely on any circular reasoning. Its main weakness is that the Einstein-obstruction half of the theorem rests on an unreviewed preprint, so the claimed unconditional answer to the question is not yet demonstrated within the manuscript itself.","major_comments":[{"comment":"The central Einstein-obstruction statement is a black-box import from the unpublished preprint [8]. Eq. (3.2) as printed, B(h_2−h_2)+[h,h]_c=0, has a self-cancelling first term and is either a typo or already implies [h,h]_c=0 as a form; in either case the actual third-order equation being used is not stated. The contradiction in Prop. 3.2 then depends on the assertion, also imported from [8, §2.2], that the complexification of the real class [h,h]_c is a nonzero universal scalar multiple of [ξ,ξ]. Neither the equation nor this proportionality is derived or checked in the present paper. Since Prop. 3.2 is the only bridge from the nonzero complex bracket to the Einstein obstruction, Theorem 1.2(iii) is not established unless the authors supply a proof of, or a detailed verification of, the needed third-order equation and the proportionality constant.","section":"§3, Prop. 3.2 and Eq. (3.2)"},{"comment":"The construction inherits its only obstructed direction from the cited surface S of [2]. The invariance assertion H^1(S,T_S)^G=H^1(S,T_S) is essential: without it the diagonal quotient would lose the nonzero bracket and the whole theorem would collapse. The paper's justification is a chain of citations to [2, Cor. 5.4, Lem. 2.11, Thm. 5.5, Thm. 3.19, Cor. 2.13] with a compressed explanation. Because this is a load-bearing point, the authors should state the precise result they need as a lemma and either prove it or reproduce the verification from [2] in enough detail that the reader can check it without reconstructing the cited chain.","section":"§2, Prop. 2.1"},{"comment":"The proof of Prop. 3.2 asserts that the fixed-volume normalization imported from [8, Thm. 5.12] preserves the tangent vector h and that the Einstein constant is E to the required order. These assertions are not demonstrated in the manuscript. If the normalization involved a diffeomorphism with nontrivial first jet, or if the Einstein constant were not E at third order under this normalization, the identification of the direction κ_g(ξ) with the tangent to an actual Einstein curve could fail. The authors should either prove these normalization facts or state them as a lemma with a complete proof.","section":"§3, Prop. 3.2"}],"minor_comments":[{"comment":"The expression h_2−h_2 is identically zero; the equation should be corrected to display the actual second-order tensor and operator that appear in the third-order Einstein equation.","section":"§3, Eq. (3.2)"},{"comment":"In the Künneth decomposition of H^2(S×C,T_{S×C}), the full formula should include the terms H^0(S,T_S)⊗H^2(C,T_C) and H^0(C,T_C)⊗H^2(S,T_S), which vanish because both H^0 spaces are zero; stating the full formula would avoid confusion.","section":"§2, Lemma 2.2"},{"comment":"There is a misspelling: 'Schwann-Semmelmann' should be 'Schwahn–Semmelmann', both in the text and in the reference entry.","section":"§4.2"},{"comment":"The computation g(C)=1+7^4(2−1)=2402 is not used anywhere later; if it is kept for motivation, a one-line derivation from Riemann–Hurwitz would be helpful.","section":"§2"},{"comment":"The notation 'dimRM̸ = 4' in the abstract and at the end of Question 1.1 is corrupted; it should read dim_R M ≠ 4.","section":"§1 and abstract"}],"recommendation":"major_revision","confidential_remarks":"The paper's own contribution, the quotient construction and the descent/bracket computations, is sound and elegant. The decisive issue is the dependence on Nagy's unpublished preprint [8] for the third-order Einstein equation and the proportionality between [h,h]_c and [ξ,ξ]. If the editor is willing to accept a citation to an unreviewed preprint as sufficient for the central claim, the paper could be accepted after the typos are fixed; otherwise the authors should be asked to include a proof of the imported statements. I do not see grounds for rejection, because the missing pieces can in principle be supplied without changing the construction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this paper constructs the first negative Kähler–Einstein threefold whose space of infinitesimal Einstein deformations has non-integrable directions. The construction is explicit and the complex deformation story is solid. The main caveat is that the step from 'non-integrable complex deformation' to 'non-integrable IED' is not proved in the paper; it is a black-box appeal to an unreviewed preprint.\n\nThe good parts: taking the Böhning–Graf von Bothmer–Pignatelli surface (rigid, not infinitesimally rigid, K ample), pulling a (Z/7)^4 cover of a genus 2 curve, and taking the diagonal quotient is a neat way to get a smooth threefold with exactly one obstructed direction. The descent computation in Lemma 2.2 is correct, and the bracket formula (2.1) is transparent. The non-productness argument is careful and works. The precise dimensions—real IED dimension 8, integrable real 6—are new and make the theorem easy to state.\n\nThe weak spot, and it is not minor: Proposition 3.2 is the entire bridge from complex to Einstein integrability, and it is entirely quoted from Nagy [8]. The paper does not derive the third-order Einstein equation, does not check the fixed-volume normalization, and does not verify the proportionality constant that identifies the real bracket with [ξ,ξ]. If any of those imported claims is wrong or has a hidden extra term, the main theorem loses its Einstein conclusion. The stress-test note is right to focus here. The surface dependence on [2] is also essential, but that is published and can be checked; the invariance H^1(S,TS)^G = H^1(S,TS) is still a point a referee should look at.\n\nMy take: the paper deserves a serious referee, not a desk rejection. But the referee should be asked to verify the Nagy dependency, ideally by getting the preprint and checking the relevant theorem in this setting. If that holds, this is a solid, citable counterexample. I wouldn't cite it in my own work before that verification, but I'd definitely read it closely and bring it to a reading group.","headline":"A genuine new counterexample—negative Kähler–Einstein threefold with non-integrable IED—but the Einstein side of the proof is imported wholesale from Nagy's unreviewed preprint, so the referee must verify that bridge.","tokens_in":6607,"tokens_out":3197,"would_cite":false,"duration_ms":28557,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C25","32G05","14J30"],"pacs":[],"model":"deepseek-v4-flash","headline":"A canonically polarized threefold is built whose Kähler–Einstein metric has obstructed infinitesimal Einstein deformations.","keywords":["Kähler-Einstein metrics","infinitesimal Einstein deformations","infinitesimal complex deformations","Kodaira-Spencer bracket","obstructed deformations","canonically polarized varieties","threefolds","Kuranishi spaces"],"falsifier":"Check whether the surface's deformation class $\\alpha$ has a vanishing quadratic bracket in $H^2(S,T_S)$ or an explicit second-order lift in its Kuranishi space; if either happens, $[\\alpha_X,\\alpha_X]$ would vanish and the claimed non-integrability of both the complex and Einstein deformations would collapse. A direct computation of the third-order Einstein obstruction for the descended metric that returns zero would also disprove the claim.","tokens_in":5746,"feed_emoji":"📐","tokens_out":11055,"duration_ms":96260,"temperature":0.7,"pith_summary":"The paper constructs a smooth projective threefold $X$ with ample canonical bundle and a negative Kähler–Einstein metric, $\\mathrm{Ric}(g)=-g$, and proves that this metric carries an infinitesimal Einstein deformation that cannot be extended to an actual curve of Einstein metrics. The same tangent direction is non-integrable as an infinitesimal complex deformation. The threefold is not biholomorphic to a product, although it admits an étale cover that is one. This settles a suitably generalized version of an open question in real dimension six by showing that obstructed Einstein deformations occur for negative Kähler–Einstein metrics in that dimension.","feed_headline":"Obstructed Einstein deformations appear in real dimension six","feed_subtitle":"The construction is a non-product threefold whose Einstein moduli space has exactly six integrable directions out of eight.","key_machinery":"The construction is the equivariant quotient $X=(S\\times C)/G$, where $S$ is a smooth projective surface with $K_S$ ample, $q(S)=0$, and $H^1(S,T_S)=\\mathbb{C}\\alpha$ with $[\\alpha,\\alpha]\\neq 0$, while $C$ is an étale Galois cover of a genus-two curve $B$ with Galois group $G\\cong(\\mathbb{Z}/7)^4$ acting diagonally on $S\\times C$. The action is free, so $X$ is smooth, and the descent lemma shows that pullback identifies $H^1(X,T_X)$ with the $G$-invariant part of the cohomology of $S\\times C$, with the only nontrivial Kodaira–Spencer bracket coming from the surface factor. The bridge to Einstein geometry is a real-linear isomorphism between $H^1(X,T_X)$ and the space of infinitesimal Einstein deformations, together with the third-order Einstein equation of the cited preprint, which turns a nonzero primary complex obstruction into a third-order Einstein obstruction for the corresponding metric deformation.","core_discovery":"The central claim is Theorem 1.2: there exist a smooth projective threefold $X$, a class $\\alpha_X\\in H^1(X,T_X)$, and a Kähler–Einstein metric $g$ with $\\mathrm{Ric}(g)=-g$ such that $K_X$ is ample, $X$ is not a product, $[\\alpha_X,\\alpha_X]\\neq 0$ in $H^2(X,T_X)$, and the infinitesimal Einstein deformation corresponding to $\\alpha_X$ under the standard real-linear isomorphism is non-integrable and obstructed at third Einstein order. More precisely, for a genus-two curve $B$ there is a decomposition $H^1(X,T_X)=\\mathbb{C}\\alpha_X\\oplus H^1(B,T_B)$, and a class $\\lambda\\alpha_X+\\beta$ is integrable as an infinitesimal complex deformation and as an infinitesimal Einstein deformation if and only if $\\lambda=0$. Consequently the real vector space of infinitesimal Einstein deformations has dimension 8, the integrable directions form a real 6-dimensional subspace, and every direction outside that subspace is obstructed.","pith_inferences":["A natural extension is to replace the base curve by a higher-dimensional base whose fundamental group surjects onto a nontrivial group, which would produce analogous obstructed examples in higher real dimensions by the same quotient mechanism.","If a surface with an obstructed deformation space of dimension larger than one could be combined with a suitable free action, the resulting Einstein metrics would have an integrable infinitesimal Einstein deformation subspace of higher codimension, giving a family of moduli spaces with varying obstruction patterns.","The construction suggests a general recipe: any rigid but not infinitesimally rigid surface with a faithful group action can be promoted to a non-product Kähler–Einstein manifold with obstructed deformations by taking a diagonal free quotient over a carefully chosen base curve."],"forward_implications":["A negative Kähler–Einstein metric in real dimension six can have non-integrable infinitesimal Einstein deformations, settling the generalized question for that dimension.","The same example shows that an obstructed infinitesimal complex deformation can remain obstructed as an Einstein deformation, even though every infinitesimal Einstein deformation integrates to second Einstein order.","Near this metric, the Einstein moduli space has an eight-dimensional tangent space of which only a six-dimensional subspace consists of integrable directions, so the moduli point is singular in a precise deformation-theoretic sense.","Because $X$ is canonically polarized and not a product, the phenomenon is intrinsic to a compact manifold with ample canonical bundle and is not produced by taking products of lower-dimensional examples."],"supporting_citations":[{"why":"It supplies the surface with the single obstructed deformation direction and the faithful group action whose invariant cohomology fills $H^1(S,T_S)$.","marker":"[2]"},{"why":"It provides the real-linear isomorphism between $H^1(M,T_M)$ and the space of infinitesimal Einstein deformations for negative Kähler–Einstein metrics.","marker":"[5]"},{"why":"It supplies the third-order Einstein equation used to turn a nonzero primary complex obstruction into a third-order Einstein obstruction.","marker":"[8]"},{"why":"It establishes that every infinitesimal Einstein deformation integrates to second Einstein order, which makes the third-order failure the deciding phenomenon.","marker":"[9]"},{"why":"It provides the existence theorem for Kähler–Einstein metrics with negative Ricci curvature used to normalize the metrics on the surface and the curve.","marker":"[1]"},{"why":"It provides the independent existence theorem for the same Kähler–Einstein metrics on the factors.","marker":"[12]"},{"why":"It formulates the original question about variational stability of Kähler–Einstein metrics that the paper generalizes and answers.","marker":"[3]"},{"why":"It formulates the negative-scalar-curvature version of the question and supplies the survey background on holonomy rigidity.","marker":"[10]"}],"fun_headline_variants":["Non-integrable Einstein deformations in real dimension six","Eight Einstein deformations, six integrable: a threefold answer","Kähler-Einstein threefold with obstructed deformations","Answering Dai-Wang-Wei in real dimension six","Non-product threefold: eight deformations, six integrable"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the cited existence of a surface with exactly one first-order deformation direction whose self-bracket is nonzero, together with a faithful group action whose invariant subspace is the whole deformation space, and on the correctness of the third-order Einstein equation; if any of these fails, the threefold example does not exist.","fun_headline_variants_meta":{"raw":{"variants":["Non-integrable Einstein deformations in real dimension six","Eight Einstein deformations, six integrable: a threefold answer","Kähler-Einstein threefold with obstructed deformations","Answering Dai-Wang-Wei in real dimension six","Non-product threefold: eight deformations, six integrable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000703,"raw_usage":{"total_tokens":3140,"prompt_tokens":884,"completion_tokens":2256,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":2169}},"tokens_in":500,"tokens_out":2256,"duration_ms":16779,"temperature":1.0,"reasoning_tokens":2169,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:06:33.025758+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether the surface's deformation class $\\alpha$ has a vanishing quadratic bracket in $H^2(S,T_S)$ or an explicit second-order lift in its Kuranishi space; if either happens, $[\\alpha_X,\\alpha_X]$ would vanish and the claimed non-integrability of both the complex and Einstein deformations would collapse. A direct computation of the third-order Einstein obstruction for the descended metric that returns zero would also disprove the claim.","supporting_citations":[{"cited_title":"Böhning, H.-C","cited_arxiv_id":null,"evidence_quote":"It supplies the surface with the single obstructed deformation direction and the faithful group action whose invariant cohomology fills $H^1(S,T_S)$."},{"cited_title":"Koiso,Einstein metrics and complex structures, Invent","cited_arxiv_id":null,"evidence_quote":"It provides the real-linear isomorphism between $H^1(M,T_M)$ and the space of infinitesimal Einstein deformations for negative Kähler–Einstein metrics."},{"cited_title":"Third order Einstein deformations for Kaehler-Einstein metrics","cited_arxiv_id":"2606.04501","evidence_quote":"It supplies the third-order Einstein equation used to turn a nonzero primary complex obstruction into a third-order Einstein obstruction."},{"cited_title":"Nagy and U","cited_arxiv_id":null,"evidence_quote":"It establishes that every infinitesimal Einstein deformation integrates to second Einstein order, which makes the third-order failure the deciding phenomenon."},{"cited_title":"Aubin,Équations du type Monge-Ampère sur les variétés kählériennes compactes, Bull","cited_arxiv_id":null,"evidence_quote":"It provides the existence theorem for Kähler–Einstein metrics with negative Ricci curvature used to normalize the metrics on the surface and the curve."},{"cited_title":"Yau,On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Comm","cited_arxiv_id":null,"evidence_quote":"It provides the independent existence theorem for the same Kähler–Einstein metrics on the factors."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It formulates the original question about variational stability of Kähler–Einstein metrics that the paper generalizes and answers."}],"review_version":1}