{"id":"3b6b2e2c-5891-48e8-a992-8f755ba1770f","arxiv_id":"1908.06356","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The basic Dolbeault cohomology of the canonical foliation on complex moment-angle manifolds and manifolds with maximal torus action admits a Hodge decomposition and is generated by degree (1,1) classes.","lead":"This paper proves that a special cohomology theory attached to the leaves of a natural foliation on complex manifolds with torus symmetry satisfies a Hodge decomposition, with nontrivial classes only in the diagonal bigrading. It answers a conjecture of Battaglia and Zaffran and gives explicit algebraic formulas for the cohomology rings of moment-angle manifolds and related spaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The cone-wise proof of Lemma 4.6 does not ensure that stellar subdivisions of adjacent maximal cones agree on their shared facets, so Theorem 4.9's polytopal subdivision is not established for non-rational fans.","rationale":"I reviewed the full argument. The polytopal case is solidly grounded in [24, Proposition 4.4] and [15, Theorem 8.1], and Lemma 3.2's injectivity argument is standard modulo the positivity checks. The load-bearing step is the extension from polytopal to general fans via stellar subdivision. The reader's weakest_assumption points to Theorem 4.9; my concern is more specific: the proof of Lemma 4.6 is not merely terse but lacks a necessary compatibility argument. A stellar subdivision at a 2-cone lying in a facet affects both adjacent maximal cones, so a cone-wise reduction is only valid if the subdivided facet is refined identically from both sides. No such statement appears, and the cited rational result is not obviously local. This is a genuine gap in the proof as written, though the theorem may well be true and patchable. I do not see an issue severe enough to recommend rejection, and the components not on this critical path are reasonable. Hence I keep the reader's CONDITIONAL verdict and make no further adjustment.","tokens_in":14207,"tokens_out":35370,"duration_ms":376277,"concrete_test":"Check Lemma 4.6 in the minimal global case: let Σ be the fan in R^3 with maximal cones σ+ = cone(e1,e2,e3) and σ− = cone(e1,e2,−e3) sharing the facet τ = cone(e1,e2), and let Σ′ be a rational subdivision that subdivides τ by the ray e1+e2 and subdivides σ− further by an interior ray such as e1+2e2+e3. Apply, to each maximal cone separately, the stellar-subdivision sequence realizing that cone's part of Σ′ as prescribed by the rational result used in Lemma 4.6. If the two sequences do not induce the same subdivision of τ at the end, the cone-wise reduction in Lemma 4.6 is invalid. To settle the theorem, one would need either a compatible choice of sequences or an explicit counterexample; absent that, Theorem 4.9 is unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central reduction is Theorem 4.9: every complete simplicial fan, possibly non-rational, admits a polytopal subdivision by stellar subdivisions whose new rays are integer combinations of the old generators. This is proved via Lemma 4.6, whose proof argues 'cone-wise': for each maximal cone σ, apply the rational de Concini–Procesi result to the rational cone σ and its rational subdivision σ ∩ Σ′. But stellar subdivisions at 2-dimensional cones that lie in a common facet of two adjacent maximal cones subdivide that facet, and the two sides produce different subdivisions unless the sequences are chosen compatibly. The proof gives no gluing argument; applying the rational result independently to σ and to its neighbour σ′ can yield incompatible subdivisions of the shared facet, in which case the union is not a fan. The cited [9, §2.4] is a statement about globally defined rational fans, not about independent maximal cones. Since Theorem 4.10 constructs the foliated surjection to a transversely Kähler moment-angle manifold only from a polytopal fan produced by Theorem 4.9, the Fujiki-foliation step—and hence the Hodge decomposition and diagonal vanishing in Theorem 4.11—rests on this unpatched compatibility.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the basic Dolbeault cohomology of the canonical holomorphic foliation on complex moment-angle manifolds and, more generally, on complex manifolds with a maximal torus action. The main results are: (1) a Hodge decomposition for the basic Dolbeault cohomology of the canonical foliation, with all nonzero groups of type (p,p); (2) an explicit description of the basic Dolbeault algebra as C[v_1,...,v_m]/(I_K+J) with all generators of type (1,1); (3) an extension of this description to all complex manifolds with a maximal torus action; and (4) a DGA model for the ordinary Dolbeault cohomology of moment-angle manifolds. The central mechanism is the introduction of \"Fujiki foliations\" and a reduction, via generalized toric blow-ups and stellar subdivisions of fans, to the transversely Kähler (polytopal) case, where the decomposition is already known by work of El Kacimi-Alaoui and Ishida.","tokens_in":14438,"tokens_out":7908,"duration_ms":81370,"significance":"If the main results hold, the paper resolves a question of Battaglia and Zaffran about the Hodge numbers of the canonical foliation and gives a complete and elegant algebraic description of its basic Dolbeault cohomology. The notion of Fujiki foliation, together with the foliated toric blow-up construction, is a natural and potentially reusable tool for studying foliated cohomology outside the transversely Kähler setting. The extension to all maximal torus actions and the DGA model for ordinary Dolbeault cohomology are substantial additional contributions. The paper is carefully written and builds appropriately on prior work, including the authors' earlier computation of basic de Rham cohomology and Ishida's classification of maximal torus actions.","major_comments":[{"comment":"The proof of Lemma 4.6 is incomplete: it reduces to the rational case cone-wise, asserting that for each maximal cone one can apply the de Concini–Procesi result to the rational fan formed by the cone and its rational subdivision. However, stellar subdivisions performed independently in adjacent maximal cones need not agree on a shared facet; a 2-dimensional cone lying in the common facet would be subdivided from the two sides in potentially incompatible ways unless the sequences of stellar subdivisions are chosen globally and compatibly. The cited rational result applies to a globally rational fan, not to an arbitrary collection of independent maximal cones. Because Theorem 4.10 constructs the required foliated surjection onto a transversely Kähler moment-angle manifold only from the polytopal fan supplied by Theorem 4.9, this gap is load-bearing for the Hodge decomposition and the diagonal vanishing in Theorem 4.11. The authors should supply a global proof of the non-rational stellar-subdivision statement, or cite a theorem that covers it.","section":"Section 4.3, Lemma 4.6 and Theorem 4.9"},{"comment":"The proof of injectivity of f* uses a local trivialization U' ≅ U × W and then asserts that the restriction of the transverse Kähler form ω to each slice {x}×W makes the restricted foliation transversely Kähler. Since the foliation F' is not assumed to be tangent to the slices, the restriction of a foliation to a submanifold requires justification; one must argue that the intersection of the leaves of F' with the fiber is a foliation on the fiber to which El Kacimi–Alaoui's theory applies. A similar issue arises in the claim that f^*(σ) is positive on U×{y}. This is likely standard in the Riemannian foliation setting, but it should be stated explicitly and proved or referenced, as the argument underpins the Hodge decomposition for Fujiki foliations.","section":"Section 3, Lemma 3.2"}],"minor_comments":[{"comment":"There are several typographical errors: \"Dolbealut\" in the abstract, \"to rus\" in the abstract, and inconsistent spacing in \"L VM-\" in the introduction. These should be corrected.","section":"Abstract and Introduction"},{"comment":"The statement that \"polytopality is an open condition for simplicial fans\" after replacing a ray by a rational perturbation would benefit from a reference or a short justification, because moving a ray changes the fan combinatorially in a neighborhood of that ray.","section":"Section 4.3, proof of Theorem 4.9"},{"comment":"The notation α_i ∈ N should specify that N denotes the positive integers, and the fact that the resulting stellar subdivision depends on the chosen α_i (suppressed in the notation) should be stated explicitly for clarity.","section":"Section 4.2, Construction 4.2"},{"comment":"The ideal J in Theorem 4.12 is defined using u ∈ (t/r)^*, while in Theorem 5.1 it is u ∈ (g/r)^*; a brief reminder of the identification of these dual spaces would help the reader compare the two statements.","section":"Sections 4.4 and 5"},{"comment":"The differential d_{Z_K} on the model is described only through its action on W^{1,0} and W^{0,1}; a more explicit description of the quasi-isomorphism, or at least a sentence explaining how the differential is determined, would improve readability.","section":"Section 6, Theorem 6.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is well-organized and the main theorems are significant. My main reservation is the incomplete proof of Lemma 4.6, which is load-bearing for the entire reduction. If the authors provide a correct global proof of the non-rational stellar-subdivision statement, or point to a reference that contains it, the paper should be suitable for publication. The reliance on the authors' earlier work [17] and on Ishida's classification is appropriate and not a cause for concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's main claim is a Hodge decomposition for the basic Dolbeault cohomology of the canonical foliation on complex moment-angle manifolds, for arbitrary (possibly non-rational) fans, together with an explicit ring presentation and a DGA model. That answers a published conjecture by Battaglia and Zaffran, and the new notion of a Fujiki foliation is a reasonable and potentially reusable tool. The polytopal case was already known, so the whole weight falls on the reduction step: Theorem 4.10 constructs a transversely Kähler cover from a polytopal subdivision of the fan. This is a good idea and the surrounding setup is careful.\n\nThe soft spot is real and it is in Lemma 4.6. The proof says we can argue cone-wise: for each maximal cone, apply the rational de Concini–Procesi result to that cone and its rational subdivision. But a stellar subdivision of a 2-dimensional cone in the boundary of one maximal cone also subdivides the adjacent maximal cone sharing that face. Applying the rational result independently to two neighbouring cones can produce incompatible subdivisions of the common facet, and the union may not be a fan. No gluing argument is provided. So Theorem 4.9 is not established as written. This is load-bearing: Theorem 4.11 for non-polytopal fans depends on it. The statement may well be true and patchable, but the proof needs to be supplied or replaced with a citation to a result that covers the non-rational case.\n\nA smaller point: Lemma 3.2 uses the fact that a surjective holomorphic map is a locally trivial bundle on a dense open set; that is standard but should be stated. The positivity argument itself looks fine once that's in place. The paper leans on earlier work by the same authors, but those are independent published results, not circular.\n\nIf I were refereeing this, I would not desk-reject it. The main idea is sound and the answer to the conjecture is significant. But I would send it back with a request for a complete proof of Lemma 4.6 or a precise reference. As it stands, the reduction step is a sketch, and a sketch at the hinge of the paper is not enough.","headline":"A genuinely useful paper that likely proves the right theorem, but the central reduction to the polytopal case has a real proof gap that needs fixing before the main theorem is fully established.","tokens_in":14958,"tokens_out":7170,"would_cite":true,"duration_ms":70951,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32J18","32L05","32M05","32Q55","37F75","57R19","14M25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Complex moment-angle manifolds with maximal torus action have diagonal basic Dolbeault cohomology.","keywords":["basic Dolbeault cohomology","canonical holomorphic foliation","moment-angle manifolds","maximal torus action","Fujiki foliation","Hodge decomposition","stellar subdivisions","transversely Kähler foliations"],"falsifier":"Construct a complete simplicial fan with non-rational rays, form the polytopal fan $\\Sigma_h$ from the hyperplanes through its codimension-one cones, and check whether every perturbation of the new rays to nearby rays rational in each cone's lattice still gives a polytopal fan; the first fan for which no such perturbation and no sequence of stellar subdivisions produces a polytopal fan is a direct counterexample to Theorem 4.9 and breaks Theorem 4.10.","tokens_in":13999,"feed_emoji":"","tokens_out":10584,"duration_ms":94084,"temperature":0.7,"pith_summary":"Complex moment-angle manifolds are non-Kähler, but their canonical holomorphic foliation still has well-behaved cohomology. The paper proves that the basic Dolbeault cohomology of this foliation has a Hodge decomposition, $H^r = \\bigoplus_{p+q=r} H^{p,q}$, and that all non-zero classes sit on the diagonal $p=q$. The proof introduces the notion of a Fujiki foliation and reduces the general case to the polytopal case by a foliated analogue of toric blow-up, where the foliation is transversely Kähler and the result was known. The same argument carries over to every compact complex manifold with a maximal holomorphic torus action, including LVM- and LVMB-manifolds. The payoff is a complete algebraic description of the basic Dolbeault cohomology ring and a DGA model for the ordinary Dolbeault cohomology of these spaces.","feed_headline":"Foliated Hodge decomposition proved for torus-invariant manifolds","feed_subtitle":"The basic Dolbeault ring is generated by (1,1) classes, and all off-diagonal Hodge numbers vanish.","key_machinery":"The machinery has three parts. First, a Fujiki foliation is defined as a holomorphic foliation that is the image, under a surjective holomorphic foliated map, of a transversely Kähler foliation; an injectivity lemma shows that such maps induce injections of basic Dolbeault cohomology, and hence a Hodge decomposition. Second, generalized toric blow-up is defined concretely: a stellar subdivision of the fan at a cone $\\tau$ gives a map $f_\\tau:(U(K_\\tau),\\mathcal F_{\\Sigma_\\tau})\\to(U(K),\\mathcal F_\\Sigma)$ that sends leaves to leaves. Third, Theorem 4.9 supplies the combinatorial bridge: any complete simplicial fan, even with non-rational generators, can be made polytopal by stellar subdivisions whose new rays are integer combinations of the old generators within each cone. The composition is the foliated version of the toric Chow lemma, a surjection from a transversely Kähler moment-angle manifold to the original one.","core_discovery":"The central result is Theorem 4.11: for any complex moment-angle manifold $Z_K$ with its canonical holomorphic foliation $\\mathcal F_h$, the pair $(Z_K,\\mathcal F_h)$ is a Fujiki foliation, so there is a Hodge decomposition $$H^r_{\\mathcal F_h}(Z_K;\\mathbb C)=\\bigoplus_{p+q=r}$H^{{p,q}}$_{\\mathcal F_h}(Z_K)$$ and $H^{p,q}_{\\mathcal F_h}(Z_K)=0$ whenever $p\\neq q$. Consequently the basic Dolbeault cohomology algebra is $$$H^{{*,*}}$_{\\mathcal F_h}(Z_K)\\cong \\mathbb C[v_1,\\dots,v_m]/(I_K+J)$$ with every generator $v_i$ of type $(1,1)$; here $I_K$ is the Stanley–Reisner ideal of the simplicial complex $K$ and $J$ is the ideal of linear relations among the fan generators. Theorem 5.1 extends this description to all compact complex manifolds with maximal holomorphic torus action, a class that includes LVM- and LVMB-manifolds, by a transverse equivalence with moment-angle manifolds. Theorem 6.1 then feeds the basic ring into a DGA model for ordinary Dolbeault cohomology.","pith_inferences":["The foliated blow-up construction is not tied to moment-angle geometry: it suggests a recipe for reducing any holomorphic foliation with a torus-invariant transversal structure to a transversely Kähler one, and would then yield Hodge decompositions in other families of non-Kähler manifolds.","If the polytopal subdivision theorem extends to fans with additional symmetry, the same Fujiki-foliation argument would likely give equivariant Hodge decompositions for the basic cohomology, not just the plain one.","A numerically testable consequence is that ordinary Dolbeault cohomology depends only on the simplicial complex and the fan relations, so changing the complex structure on the same smooth moment-angle manifold cannot change the Dolbeault groups as long as the marked fan is unchanged."],"forward_implications":["The basic Hodge numbers of the canonical foliation vanish off the diagonal, so the basic Dolbeault cohomology of $Z_K$ is completely determined by its diagonal entries.","For every compact complex manifold with maximal holomorphic torus action, the basic Dolbeault ring is $\\mathbb C[v_1,\\dots,v_m]/(I_K+J)$ with all generators of type $(1,1)$.","The ordinary Dolbeault cohomology of a complex moment-angle manifold is quasi-isomorphic to a DGA built from the basic Dolbeault ring and an $R$-invariant subspace $W\\subset\\Omega^1(Z_K)$ of dimension $(m-n)/2$.","A foliated analogue of the toric Chow lemma holds: every complex moment-angle manifold is the holomorphic foliated image of a transversely Kähler one."],"supporting_citations":[{"why":"supplies the polytopal case: the canonical foliation is transversely Kähler when the fan is polytopal.","marker":"[24, Proposition 4.4]"},{"why":"gives the Hodge decomposition for basic cohomology of transversely Kähler foliations.","marker":"[10, Theorem 3.4.6]"},{"why":"shows that in the transversely Kähler case all basic Dolbeault generators have type $(1,1)$, proving diagonal vanishing there.","marker":"[15, Theorem 8.1]"},{"why":"classifies complex manifolds with maximal torus action as quotients of toric varieties, carrying the extension in Section 5.","marker":"[14]"},{"why":"computes basic de Rham cohomology as $\\mathbb C[v_1,\\dots,v_m]/(I_K+J)$, which the Hodge decomposition upgrades to Dolbeault.","marker":"[17, Theorem 3.4]"},{"why":"provides the transverse distribution and quasi-isomorphism used in the DGA model for ordinary Dolbeault cohomology.","marker":"[16]"},{"why":"supplies the rational stellar-subdivision result adapted cone-wise in Lemma 4.6.","marker":"[9, §2.4]"},{"why":"shows that stellar subdivisions of a polytopal fan remain polytopal, used in Lemma 4.7.","marker":"[2, Claim 3]"},{"why":"provides the polytopal fan from the hyperplane arrangement in Theorem 4.9.","marker":"[8, Lemma 6.9.2]"}],"fun_headline_variants":["Hodge decomposition for torus-invariant foliations","Basic Dolbeault ring: all generators are (1,1)","Moment-angle manifolds get foliated Hodge theory","Torus actions yield pure Hodge diamonds","Foliated cohomology: off-diagonal Hodge numbers vanish"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on Theorem 4.9: every complete simplicial fan, even one with non-rational generators, can be made polytopal by stellar subdivisions whose new rays lie in the integer span of each cone's old generators; the proof is sketched cone-wise, citing the rational cases, and the foliated surjection needed for the Fujiki argument does not exist if this fails.","fun_headline_variants_meta":{"raw":{"variants":["Hodge decomposition for torus-invariant foliations","Basic Dolbeault ring: all generators are (1,1)","Moment-angle manifolds get foliated Hodge theory","Torus actions yield pure Hodge diamonds","Foliated cohomology: off-diagonal Hodge numbers vanish"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000204,"raw_usage":{"total_tokens":1379,"prompt_tokens":926,"completion_tokens":453,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":542,"completion_tokens_details":{"reasoning_tokens":372}},"tokens_in":542,"tokens_out":453,"duration_ms":4629,"temperature":1.0,"reasoning_tokens":372,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:48:29.749843+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a complete simplicial fan with non-rational rays, form the polytopal fan $\\Sigma_h$ from the hyperplanes through its codimension-one cones, and check whether every perturbation of the new rays to nearby rays rational in each cone's lattice still gives a polytopal fan; the first fan for which no such perturbation and no sequence of stellar subdivisions produces a polytopal fan is a direct counterexample to Theorem 4.9 and breaks Theorem 4.10.","supporting_citations":[{"cited_title":"Complex manifolds with maximal torus actions","cited_arxiv_id":null,"evidence_quote":"classifies complex manifolds with maximal torus action as quotients of toric varieties, carrying the extension in Section 5."},{"cited_title":"Transversely K¨ ahler structures on central foliations of complex manifolds","cited_arxiv_id":null,"evidence_quote":"provides the transverse distribution and quasi-isomorphism used in the DGA model for ordinary Dolbeault cohomology."}],"review_version":1}