{"id":"ca93f798-6f00-4ca7-ac37-649a6e57c373","arxiv_id":"2607.26786","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On every compact nilmanifold with left-invariant complex structure, invariant forms induce an isomorphism on Dolbeault cohomology in all bidegrees.","lead":"Left-invariant forms compute the full Dolbeault cohomology of any compact complex nilmanifold, settling a conjecture open for over twenty years. The same argument shows deformations stay invariant and that Bott–Chern and related numbers are lattice-independent.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No internal fracture found beyond the already-flagged dependence on the polynomial C^n-cover theorem [14].","rationale":"The reader correctly isolated the only load-bearing vulnerability: dependence on the overlapping-author polynomial-cover preprint [14]. The internal architecture (polynomial-growth ∂̄-Poincaré → Cartan–Leray zig-zag into Ω^p_pol; unipotent rational exhaustion → Hochschild/Shapiro comparison to Chevalley–Eilenberg) is classical once that model is granted, and the dimension-squeeze strategy is non-circular. I found no independent gap in Lemmas 4.2–4.5, Proposition 4.6, Theorem 5.4, or the five-lemma induction of Lemma 5.5 that would overturn the claim if [14] holds. Consequences (Hasegawa deformation conjecture, Angella Bott–Chern conjecture) inherit the same single external risk and do not introduce new ones. Therefore the ACCEPT / moderate-confidence verdict stands; the concrete check is simply to audit [14] for the polynomial inverse and degree bound the present text treats as given.","tokens_in":22561,"tokens_out":653,"duration_ms":43709,"concrete_test":"Check the proof of [14, Thm 1.2] line-by-line for an explicit polynomial inverse of Φ and a nilpotency-class bound on deg Φ, deg Φ^{-1}; then, for one concrete non-abelian example in complex dimension 4 whose Hodge numbers are already known by other methods, compute dim W_m^{(p)} stabilizations and verify dim H^q(Γ,Ω^p_pol) equals the known h^{p,q}. Mismatch or missing inverse falsifies the reduction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central dimension squeeze (Thm 4.1 surjectivity + Thm 5.1 dimension match + Console–Fino injectivity) is internally coherent and uses standard tools (Hörmander, Cartan–Leray, Hochschild, rational induction). Its single load-bearing external input remains Theorem 3.1/[14]: Φ polynomial with polynomial inverse. Every subsequent device—the uniform degree bound D (Prop. 3.3), the algebraic G_C-action and identification Ω^p_pol ≅ ind^{G_C}_{G_{0,1}} E (Prop. 5.2), local finiteness/unipotence of the W_m exhaustion (Thm 5.4/Construction 5.3), and the growth-control zig-zag that lands in Ω^p_pol (Lemmas 4.2–4.4)—is built on that model. The manuscript itself notes that polynomiality of Φ^{-1} is only implicit in [14], and the degree bound cites the concurrent thesis [31]. If [14] fails to supply a polynomial inverse or a uniform D, both the surjectivity staircase and the colimit comparison collapse. Conditional on [14] standing, no further soft spot in the squeeze appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves that for every compact nilmanifold M=Γ\\G with left-invariant complex structure J, the inclusion of left-invariant forms induces an isomorphism ι: H^{p,q}_∂̄(g,J) → H^{p,q}_∂̄(M) in every bidegree (Theorem 1.1), settling the Console–Fino conjecture. The argument reduces Dolbeault cohomology via a Cartan–Leray double complex to group cohomology H^q(Γ,Ω^p(C^n)), then shows every class has a polynomial representative (Theorem 4.1, Hörmander L^{2} estimates plus a controlled zig-zag) and that polynomial group cohomology matches the Chevalley–Eilenberg cohomology of (g,J) (Theorem 5.1/5.6, via algebraic induction, local unipotence, and Hochschild comparison). Console–Fino injectivity closes the dimension squeeze. Consequences include invariance of small deformations (Theorem 1.2/7.2, Hasegawa) and computation of Bott–Chern, Aeppli, Frölicher, Varouchas, and Bigolin invariants by invariant forms, independently of the lattice (Corollary 7.6, Angella).","tokens_in":22831,"tokens_out":1733,"duration_ms":55669,"significance":"The result closes a conjecture open for more than two decades and known only in special classes (parallelizable, nilpotent, abelian/rational, dim≤3). The deformation consequence settles the second half of Hasegawa’s conjecture and makes the Kodaira–Spencer algebra of nilmanifolds explicitly computable by invariant forms (Theorem 7.1). The E_{1}-isomorphism of double complexes yields lattice-independence of Bott–Chern and related invariants, proving Angella’s conjecture. The proof architecture—polynomial C^n model, growth-controlled Cartan–Leray zig-zag, and unipotent rational-module comparison—is a substantial and reusable contribution to the cohomology of complex homogeneous spaces. Conditional on the companion biholomorphism theorem, the logical skeleton is complete and uses standard tools correctly.","major_comments":[{"comment":"Load-bearing input Theorem 3.1/[14]: the entire reduction (uniform degree bound D in Prop. 3.3, algebraic G_C-action and Ω^p_pol ≑ ind^{G_C}_{G_{0,1}} E in Prop. 5.2, W_m-exhaustion and local unipotence in Constr. 5.3/Thm. 5.4, and the polynomial-growth zig-zag of §4) rests on Φ being polynomial with polynomial inverse. The manuscript states that polynomiality of Φ^{-1} “follows from the proof contained in [14], even if it is not stated explicitly there.” Before acceptance, either [14] must explicitly record polynomiality of the inverse, or this paper must supply a short self-contained verification. As written, a reader cannot check the hypothesis of Lemmas 3.2 and 4.3–4.4 from cited statements alone.","section":"§3, Theorem 3.1 and Lemma 3.2"},{"comment":"Proposition 3.3 asserts a uniform w-degree bound D < ∞ independent of Φ, with the existence of a bound in terms of the nilpotency step referred to the concurrent thesis [31]. The constant D controls the finite-dimensionality of the G_C-spans W_m and W^{(p)}_m (Construction 5.3) and hence the colimit comparison in Theorem 5.6(a)–(b). A brief self-contained degree estimate (or a precise theorem number and statement from [31]/[14]) should appear in the text so that the finiteness used in §5 does not depend on an unpublished thesis for a load-bearing constant.","section":"§3, Proposition 3.3; Construction 5.3"}],"minor_comments":[{"comment":"In the proof of Theorem 4.1, the inhomogeneous coboundary convention is introduced mid-argument; a one-line pointer back to §2.3 at the first use of (δ_Γ β)(γ)=γ·β−β would help the reader track signs in the total-complex identity D_tot(∑ ε_k β_k)=ε_{q-1} c_q − α̃.","section":"§4.4"},{"comment":"Lemma 4.4 claims polynomial growth of all derivatives of Γ-invariant forms by rewriting in the invariant coframe and using polynomial Jacobians of Φ,Φ^{-1}. The inductive step on ∂^{(m)} is sketched; a sentence making precise that a_{kj}, b_{kj} are polynomial (entries of DΦ^{-1}) would remove any ambiguity.","section":"§4.1, Lemma 4.4"},{"comment":"Theorem 5.6(c) invokes exactness of algebraic induction because X is affine, citing [7]. A parenthetical recall that R^t ind=0 for t>0 on affine quotients would make the collapse of the Grothendieck spectral sequence immediate for non-specialists.","section":"§5.3, Theorem 5.6(c)"},{"comment":"Reference [14] is listed as arXiv:2606.31448 and [17] as arXiv:2607.07059; ensure final bibliographic data (and the explicit inverse-polynomiality statement requested above) are aligned before publication. The note that Sillari–Tomassini found the solvmanifold counterexample earlier ([26], work in preparation) should be updated if a public reference becomes available.","section":"References"},{"comment":"Typographical inconsistencies: “Frölicher” appears both with and without the o-umlaut encoding in the abstract/front matter; “Hörmander” is correct in §4 but the running header uses “INV ARIANT” with a spurious space. Normalize.","section":"Front matter / running heads"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a companion to arXiv:2606.31448 by a largely overlapping author set; the two results together resolve both halves of Hasegawa’s program for nilmanifolds. I see no novelty or citation issue, but acceptance of the present manuscript should be coordinated with the status of [14]: if [14] is still under review, the editor may wish to condition final acceptance on [14] explicitly stating polynomiality of Φ^{-1} and a uniform degree bound, or on those facts being written into this text. Conditional on that external input, I regard the argument as sound and the contribution as appropriate for a top specialist journal."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper closes the Console–Fino conjecture: left-invariant forms compute Dolbeault cohomology in every bidegree for any left-invariant complex structure on a compact nilmanifold. That has been open outside special classes (parallelizable, nilpotent, abelian, rational, dim ≤ 3) for twenty-plus years. The same argument gives Hasegawa’s deformation claim and Angella’s Bott–Chern claim for free, plus independence of the lattice for the usual double-complex invariants.\n\nWhat is new is the route. They import the polynomial biholomorphism Φ : (G,J) → C^n from their concurrent preprint, push the deck action to polynomial automorphisms of bounded degree, and identify Dolbeault of M with group cohomology of holomorphic forms on the cover. Hörmander L^{2} estimates plus a Cartan–Leray zig-zag produce a polynomial representative for every class (Thm 4.1). Local finiteness and unipotence of the Γ-action on the polynomial module then match that cohomology to the Chevalley–Eilenberg cohomology of the Lie algebra (Thm 5.1/5.6). Console–Fino injectivity finishes the dimension squeeze. The architecture is standard and readable; the growth-control lemmas and the rational-induction comparison are carefully written.\n\nThe soft spot is real but narrow: everything after Section 3 rides on Φ being polynomial with polynomial inverse and on a uniform degree bound D. The manuscript itself notes that polynomiality of the inverse is only implicit in the companion paper, and the degree bound cites a concurrent thesis. If that input fails, both the surjectivity staircase and the colimit comparison collapse. Conditional on the companion standing, I see no internal fracture in the squeeze. Self-citation is concentrated on that one necessary input; the classical toolkit (Nomizu, Hörmander, Hochschild, Shapiro) is used correctly.\n\nThis is for people who work on non-Kähler homogeneous geometry or deformation theory of nilmanifolds. It deserves a serious referee who will check the growth estimates and the unipotence flags, and who will want the companion result settled or at least tightly cross-referenced. I would send it out.","headline":"Settles the long-open Dolbeault-invariants conjecture for nilmanifolds via a clean dimension squeeze; the only real external risk is the concurrent polynomial C^n-cover result it rests on.","tokens_in":23529,"tokens_out":545,"would_cite":true,"duration_ms":10152,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E25","32G05","17B56","20J06"],"pacs":[],"model":"grok-4.5","headline":"Left-invariant forms compute Dolbeault cohomology on every compact complex nilmanifold.","keywords":["nilmanifold","Dolbeault cohomology","left-invariant complex structure","polynomial automorphisms","deformations","Bott–Chern cohomology","group cohomology","Frölicher spectral sequence"],"falsifier":"Exhibit a compact nilmanifold with left-invariant complex structure whose Dolbeault number $h^{p,q}$ differs from the dimension of the corresponding invariant cohomology, or show that some small deformation is not again left-invariant.","tokens_in":23385,"feed_emoji":"∫","tokens_out":950,"duration_ms":15923,"temperature":0.7,"texified_at":"2026-08-05T21:49:43.021097+00:00","pith_summary":"For more than twenty years it has been conjectured that the Dolbeault cohomology of a compact nilmanifold with a left-invariant complex structure is completely determined by the finite-dimensional complex of left-invariant forms. This paper proves that conjecture in full generality: the inclusion of invariant forms induces an isomorphism in every bidegree. The argument uses a polynomial biholomorphism of the universal cover with $C^n$, reduces cohomology to group cohomology of polynomial forms, and compares that to the Chevalley–Eilenberg complex via local unipotence of the deck action. As direct consequences, small deformations remain left-invariant, and Bott–Chern, Aeppli, and Frölicher-type invariants are likewise computed by invariant forms and do not depend on the choice of lattice.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":3217,"prompt_tokens":559,"completion_tokens":2658,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":2179}},"feed_headline":"Invariant forms settle Dolbeault cohomology on nilmanifolds","feed_subtitle":"A 20-year conjecture is proved: left-invariant forms compute every Dolbeault group, and deformations stay invariant.","key_machinery":"The polynomial model of the universal cover: a biholomorphism $\\Phi: (G, J) \\to C^n$ that is polynomial with polynomial inverse, so the deck action becomes a group of polynomial automorphisms of bounded degree; Dolbeault cohomology is then identified with group cohomology of polynomial holomorphic forms, which is shown isomorphic to the invariant Chevalley–Eilenberg cohomology.","core_discovery":"For every compact nilmanifold $M = \\Gamma\\backslash G$ carrying a left-invariant complex structure $J$, the natural map from the invariant Dolbeault cohomology $H^{p,q}_{\\bar{\\partial}}(g, J)$ into $H^{p,q}_{\\bar{\\partial}}(M)$ is an isomorphism in every bidegree $(p, q)$.","pith_inferences":["The same polynomial-cover technique cannot be expected to salvage the Dolbeault computation for general solvmanifolds, where the paper already notes that surjectivity of the invariant map can fail even with Stein universal covers.","Once deformations are known to stay inside the invariant class, moduli of complex structures on nilmanifolds reduce to algebraic deformation problems on the Lie algebra, opening concrete classification in higher dimensions.","Independence of the lattice suggests that numerical Hodge-type invariants of nilmanifolds are really invariants of the underlying real Lie algebra plus complex structure, not of the discrete quotient."],"forward_implications":["Small deformations of a left-invariant complex structure on a compact nilmanifold remain left-invariant (and are realized by lattices of polynomial automorphisms of uniformly bounded degree).","The inclusion of invariant forms is an E1-isomorphism of double complexes, so both Frölicher spectral sequences and their abutments agree with the invariant ones.","Bott–Chern and Aeppli cohomologies, the six Varouchas groups, and Bigolin–Schweitzer cohomologies are all computed by invariant forms.","These cohomological invariants are independent of the choice of lattice in a fixed simply connected nilpotent Lie group.","The invariant Kodaira–Spencer DGLA is quasi-isomorphic to the full Kodaira–Spencer algebra, so deformation theory is governed by finite-dimensional invariant data."],"fun_headline_variants":["Invariant forms compute all Dolbeault groups on nilmanifolds","Left-invariant forms give full Dolbeault cohomology of nilmanifolds","Nilmanifold Dolbeault cohomology reduces to invariant forms","Invariant forms settle Dolbeault cohomology conjecture for nilmanifolds","Left-invariant forms isomorphic to Dolbeault cohomology in every bidegree"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The entire argument rests on the universal cover being biholomorphic to complex n-space by a map that is polynomial with polynomial inverse; without that polynomial model the reduction to polynomial group cohomology and the unipotence comparison do not run.","fun_headline_variants_meta":{"raw":{"variants":["Invariant forms compute all Dolbeault groups on nilmanifolds","Left-invariant forms give full Dolbeault cohomology of nilmanifolds","Nilmanifold Dolbeault cohomology reduces to invariant forms","Invariant forms settle Dolbeault cohomology conjecture for nilmanifolds","Left-invariant forms isomorphic to Dolbeault cohomology in every bidegree"]},"model":"grok-4.5","effort":"low","cost_usd":0.003141,"raw_usage":{"total_tokens":1024,"prompt_tokens":642,"num_sources_used":0,"completion_tokens":73,"cost_in_usd_ticks":31408000,"prompt_tokens_details":{"text_tokens":642,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":309,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":642,"tokens_out":73,"duration_ms":5344,"temperature":1.0,"reasoning_tokens":309,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T21:07:46.553851+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a compact nilmanifold with left-invariant complex structure whose Dolbeault number $h^{p,q}$ differs from the dimension of the corresponding invariant cohomology, or show that some small deformation is not again left-invariant.","supporting_citations":[],"review_version":1}