{"id":"d38c2812-1b41-4a35-8199-a6934d6b990f","arxiv_id":"2505.01795","paper_version":1,"verdict":"UNVERDICTED","confidence":"MODERATE","novelty_score":0.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An expository account of non-Kähler metric classes, their incompatibilities, and their stability, with all results cited from previous papers.","lead":"This paper surveys dozens of results about special non-Kähler metrics, such as balanced, pluriclosed, and locally conformally Kähler metrics, on compact complex manifolds. It collects known coexistence, blow-up, and deformation results and cites the original papers without proving them.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.5's broad LCK/balanced incompatibility rests on an unstated positivity result from the authors' own OV25; if that theorem is mis-transcribed or has a gap, the survey's strongest global claim is unsupported.","rationale":"The reader's weakest assumption is that every cited theorem is correct and accurately transcribed, with Theorem 4.5 given as the example. My stress-test agrees with that identification: Theorem 4.5 is the broadest LCK/balanced incompatibility statement in the survey, and its proof is not merely omitted but its key technical input, the positivity of the Lee-form degree for all known LCK manifolds, is only gestured at in Remark 4.6. Because the paper is a survey, the appropriate response is not to reject it but to flag that its central global claim inherits all of its epistemic weight from OV25. The concrete check I propose would settle whether the transcription is faithful and whether the positivity result covers all three known LCK classes. If the check passes, the survey's statement can be trusted; if it fails, the survey overstates the current state of knowledge. Since no original claim in the paper is being accepted or rejected, the reader's UNVERDICTED verdict remains appropriate; my concern is a verification task, not a change of category.","tokens_in":11379,"tokens_out":12882,"duration_ms":131168,"concrete_test":"Open OV25, Theorem 4.17 and verify three claims: (a) its statement includes bimeromorphic models of known LCK manifolds exactly as the survey phrases it, not merely the known LCK manifolds themselves; (b) its proof establishes strict positivity of the Lee-form degree for Oeljeklaus-Toma and Kato manifolds, and not only for LCK manifolds with potential; (c) the degree is defined with respect to an arbitrary Gauduchon metric rather than a specially chosen one. If any of (a)-(c) fails, amend Theorem 4.5 to match the original statement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central value of this survey is that its cited incompatibility results can be relied on as stated. The most load-bearing such result is Theorem 4.5, the only one covering all presently known LCK manifolds. It is imported from [OV25, Theorem 4.17] with no proof and no statement of the auxiliary fact that Remark 4.6 says 'is behind' it: strict positivity of the degree of the Lee form with respect to any Gauduchon metric. Three things are not checked in the survey: (1) the notion of degree of a 1-form is never defined; (2) for Oeljeklaus-Toma and Kato manifolds, neither of which is LCK with potential, the positivity is a recent, non-obvious result; (3) the theorem's hypothesis says 'bimeromorphic to any of the known LCK manifolds', which requires an additional justification about bimeromorphic invariance beyond the positivity statement itself. If any of these fails, or if the survey has broadened the original statement of OV25, then the claimed global incompatibility between LCK and balanced metrics is not supported. This is not an allegation that OV25 is wrong; it is an identification of the precise point where the survey's reliability depends on an unchecked transcription.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This survey reviews special Hermitian metrics on compact complex manifolds, focusing on locally conformally Kähler (LCK) metrics (including Vaisman, LCK with potential, Kato, and Oeljeklaus–Toma manifolds), Gauduchon and k-Gauduchon metrics, pluriclosed, astheno-Kähler, balanced, locally conformally balanced, Hermitian-symplectic, and locally conformally symplectic structures. It presents a Lie-algebraic recipe for constructing nilmanifold examples, collects results on the incompatibility of different metric classes on the same manifold, and surveys stability under blow-up and deformation. The central message is that, apart from special cases such as complex tori or explicit coexistence examples, these metric classes are largely mutually incompatible.","tokens_in":11671,"tokens_out":8642,"duration_ms":77891,"significance":"If the cited results are accurate, this survey is a useful and well-organized reference that collects the main known incompatibility statements (Theorems 4.1, 4.5, 4.7, 4.8, 4.11) and the blow-up/deformation results (Theorems 5.1–5.4) in one place. It includes valuable caveats, such as Remark 4.10, which explicitly notes that Theorem 4.8 fails without left-invariance for the k-Gauduchon metric. The exposition of the nilmanifold construction recipe in Section 3 is a helpful didactic contribution. The paper contains no original proofs, so its value as a reliable reference depends entirely on the faithful transcription of the cited theorems; this is where the main risk lies, particularly for statements imported from the authors' own recent papers.","major_comments":[{"comment":"Theorem 4.5 is the survey's most general LCK/balanced incompatibility statement, covering all known LCK manifolds up to bimeromorphism. The supporting fact described in Remark 4.6, the strict positivity of the degree of the Lee form with respect to any Gauduchon metric, is not stated precisely, and the 'degree of a 1-form' is never defined in the paper. Moreover, the theorem as written extends the cited [OV25, Theorem 4.17] to manifolds bimeromorphic to the known LCK classes, yet the bimeromorphic invariance of the obstruction is not discussed. Since no proofs are included, the reader must rely on the transcription being exactly correct. Please add a formal definition of the degree of a 1-form, state the positivity theorem as a clearly labeled assertion (with a reference), and either justify the bimeromorphic invariance or quote the original theorem verbatim without broadening it.","section":"Section 4, Theorem 4.5 and Remark 4.6"},{"comment":"The statement that LCK and k-Gauduchon conditions are 'mutually incompatible in a given conformal class' is ambiguous, because the k-Gauduchon condition is not conformally invariant. The theorem should be formulated for a single Hermitian metric rather than for a conformal class. As written, the reader could incorrectly infer that no metric in any conformal class can satisfy both conditions, which is also contradicted by Kähler metrics. Please rephrase, for example: 'A Hermitian metric on a compact complex manifold cannot be both LCK and k-Gauduchon unless it is Kähler', and similarly for the balanced/k-Gauduchon claim.","section":"Section 4.1, Theorem 4.1"}],"minor_comments":[{"comment":"The name 'Calaby–Eckmann' should be 'Calabi–Eckmann'.","section":"Remark 4.12"},{"comment":"The phrase 'on the m' is incomplete and should read 'on the manifold'.","section":"Remark 4.6"},{"comment":"The cohomology group 'H^{n−1,n−21}_{BC}' appears to be a typo; the intended bidegree should be stated correctly, likely H^{n−1,n−2}_{BC} or H^{n−2,n−1}_{BC} depending on the convention.","section":"Theorem 5.4(iii)"},{"comment":"The reference [AI03] is labeled with the year 2003 but cites Differ. Geom. Appl. 14 (2001); please harmonize the year in the citation key and the bibliographic entry.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The survey relies heavily on the authors' own recent papers for several load-bearing statements (Theorem 4.5 from OV25, Theorem 4.8 from OOS23, Remark 2.2 from OV24b). Without checking those originals, I cannot be certain that every transcription is faithful. The ambiguities noted in the major comments should be fixable, but the authors should carefully compare their Theorem 4.5 and Theorem 4.1 statements against the original sources and add the missing definitions and qualifications."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the short version: this is a survey, not a research paper. It doesn't prove anything new, but it does a decent job of organizing a messy literature on special non-Kähler metrics — balanced, LCK, pluriclosed, Gauduchon-type, etc. — and it gives a clean recipe for constructing nilmanifold examples via left-invariant structures. If you need a quick orientation to this corner of complex geometry, it works.\n\nWhat's genuinely useful: the paper collects known incompatibility results in one place, especially the nilmanifold theorems (Theorems 4.7 and 4.8), and it flags assumptions clearly — Remark 4.10 correctly notes that left-invariance matters and that the k-Gauduchon part fails without it. The attributions are honest, and the self-citations are legitimate because the cited papers are published. The blow-up and deformation stability sections are a helpful bibliography.\n\nSoft spots, in order of severity. First, the stress-test concern is real: Theorem 4.5 is the broadest LCK/balanced incompatibility, covering all known LCK manifolds, but its proof is not given, and Remark 4.6 says the key ingredient is the strict positivity of the degree of the Lee form — a notion never defined in the survey, and a result that is recent and nontrivial for Oeljeklaus–Toma and Kato manifolds. The theorem also says 'bimeromorphic to any of the known LCK manifolds,' which needs a separate invariance argument. That doesn't mean OV25 is wrong; it means the survey's strongest claim is a transcription, and the reader can't verify it from the text. A survey should either prove the auxiliary fact or at least state it precisely with a definition of degree. Second, there are minor typos — 'Calaby' for Calabi in Remark 4.12, a few broken words, and the 'n−1,n−21' in Theorem 5.4(iii) is garbled. These are easy fixes.\n\nOverall the paper is honest and coherent, and the mathematics it reports is almost certainly correct. It's the kind of survey that belongs in the literature after minor revision. If I were refereeing, I'd ask for a remark on the degree of the Lee form and a clarification that Theorem 4.5 is imported from OV25. I'd send it to peer review; it's not a desk reject.","headline":"A careful survey that compiles known incompatibility results; its strongest claim leans on an unstated positivity theorem from OV25, so treat it as a map, not a proof.","tokens_in":12173,"tokens_out":2724,"would_cite":false,"duration_ms":27229,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C55","22E25","32J18"],"pacs":[],"model":"deepseek-v4-flash","headline":"This survey reports that on compact nilmanifolds with left-invariant complex structure, balanced, pluriclosed, and LCK metrics cannot coexist unless the manifold is a complex torus.","keywords":["Hermitian metric","balanced metric","locally conformally Kähler metric","pluriclosed metric","astheno-Kähler metric","Gauduchon metric","nilmanifold","coexistence of special metrics"],"falsifier":"A compact complex nilmanifold that is not a torus, with a left-invariant complex structure and both a balanced and a pluriclosed metric compatible with that structure, would disprove Theorem 4.7; the Section 3 recipe, prescribing rational structure constants for a 2-step nilpotent Lie algebra, offers a concrete search space for such an example.","tokens_in":11178,"feed_emoji":"📐","tokens_out":7920,"duration_ms":69001,"temperature":0.7,"pith_summary":"This survey assembles the known results about special non-Kähler Hermitian metrics—balanced, pluriclosed, locally conformally Kähler (LCK), Gauduchon, astheno-Kähler, and Hermitian-symplectic—and focuses on the question of which pairs can coexist on the same compact complex manifold. Its central message is that incompatibility is the norm: on compact nilmanifolds with left-invariant complex structure, a balanced metric together with a pluriclosed metric forces the manifold to be a complex torus, and the same collapse happens if an LCK metric is paired with a balanced, left-invariant k-Gauduchon, or pluriclosed metric. The paper additionally records how these metric classes behave under blow-up and small deformations, and describes a general recipe for building explicit examples from structure equations on nilpotent Lie algebras. A sympathetic reader should care because these coexistence and stability results provide some of the few structural organizing principles in the otherwise vast non-Kähler landscape.","feed_headline":"Non-Kähler metrics cannot coexist on nilmanifolds without a torus","feed_subtitle":"A survey of special Hermitian metrics shows LCK, balanced, and pluriclosed structures force the torus on compact nilmanifolds.","key_machinery":"The machinery is the left-invariant reduction: on a compact nilmanifold, if a special metric exists, an averaging argument produces a left-invariant one; then a classification of left-invariant complex structures on nilmanifolds turns the coexistence question into linear algebra on structure equations. For LCK metrics, the Lee 1-form and its strict positivity with respect to any Gauduchon metric carries the obstruction. The survey also presents a general recipe for constructing 2-step nilmanifold examples by prescribing rational structure constants, extending them to a compact quotient, and then writing down left-invariant fundamental forms.","core_discovery":"On its own terms, the survey's central discovery is the collection of incompatibility theorems it reports. Theorem 4.7 states that a compact complex nilmanifold with a left-invariant complex structure carrying both a balanced and a pluriclosed metric compatible with that structure must be a complex torus. Theorem 4.8 extends the same conclusion to LCK metrics: under the same left-invariant setting, an LCK metric cannot coexist with a balanced, a left-invariant k-Gauduchon, or a pluriclosed metric unless the manifold is a torus. At a broader level, Theorem 4.5 says that no known non-Kähler LCK manifold admits a balanced metric, and Theorem 4.11 excludes k-Gauduchon, Hermitian-symplectic, and balanced metrics on compact LCK manifolds with parallel Lee form. The survey also presents stability results: pluriclosed metrics survive blow-up, Hermitian-symplectic metrics survive point blow-up and small deformations, while the full LCK class is not stable under deformation.","pith_inferences":["If the nilmanifold incompatibility theorems reflect a deeper cohomological principle, a testable extension is to check the same pairings on solvmanifolds, where left-invariant structures are harder to classify.","The structure-equation recipe in Section 3 could be used to systematically search for the first known counterexample or for new coexisting pairs among 2-step nilmanifolds, since existence is reduced to linear algebra on the constants.","The strict positivity of the Lee-form degree behind Theorem 4.5 suggests that any future LCK manifold with non-positive Lee degree would be a candidate to break the balanced obstruction, so computing this degree for new LCK examples would directly test the scope of the result."],"forward_implications":["On a compact complex nilmanifold with left-invariant complex structure, a non-torus manifold cannot admit both a balanced and a pluriclosed metric compatible with that structure.","The same obstruction applies to LCK metrics: together with a balanced, left-invariant k-Gauduchon, or pluriclosed metric, they force a complex torus.","None of the currently known classes of non-Kähler LCK manifolds admits a balanced metric.","Compact LCK manifolds with parallel Lee form of dimension at least three exclude all k-Gauduchon and Hermitian-symplectic metrics, and in any dimension exclude balanced metrics.","Pluriclosed and Hermitian-symplectic metrics survive blow-up in the stated situations, while the full LCK class does not survive small deformations."],"supporting_citations":[{"why":"Proves the balanced-plus-pluriclosed incompatibility on nilmanifolds (Theorem 4.7), the central obstruction result.","marker":"[FV16]"},{"why":"Establishes the LCK incompatibility theorems on nilmanifolds and supplies coexistence examples such as LCB-plus-pluriclosed.","marker":"[OOS23]"},{"why":"Provides the balanced obstruction for known LCK manifolds via positivity of the Lee-form degree.","marker":"[OV25]"},{"why":"Gives the compact LCK-with-parallel-Lee-form obstructions to k-Gauduchon, Hermitian-symplectic, and balanced metrics.","marker":"[AO23]"},{"why":"Proves the same-metric incompatibilities between LCK and k-Gauduchon and between balanced and k-Gauduchon.","marker":"[IP13]"},{"why":"Supplies the averaging argument reducing special metrics on nilmanifolds to left-invariant ones.","marker":"[Bel00]"},{"why":"Provides the structure theorem for non-Kähler nilmanifolds with left-invariant complex structures used in Theorem 4.8.","marker":"[Sa07]"},{"why":"Establishes existence and uniqueness of Gauduchon metrics, used throughout as the conformal reference.","marker":"[Gau78]"}],"fun_headline_variants":["Nilmanifolds with balanced and pluriclosed metrics must be tori","LCK and balanced metrics can't share a nilmanifold unless torus","Coexistence of special Hermitian metrics on nilmanifolds implies torus","Non-Kahler metrics only coexist on toroidal nilmanifolds","Survey: Balanced, pluriclosed, LCK force torus on nilmanifolds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The survey's conclusions rest on the cited theorems being correct and accurately transcribed; in particular, the claim that known LCK manifolds cannot carry balanced metrics depends on an unproved technical positivity property of a certain 1-form associated to the metric, not derived in this survey.","fun_headline_variants_meta":{"raw":{"variants":["Nilmanifolds with balanced and pluriclosed metrics must be tori","LCK and balanced metrics can't share a nilmanifold unless torus","Coexistence of special Hermitian metrics on nilmanifolds implies torus","Non-Kahler metrics only coexist on toroidal nilmanifolds","Survey: Balanced, pluriclosed, LCK force torus on nilmanifolds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001023,"raw_usage":{"total_tokens":4245,"prompt_tokens":809,"completion_tokens":3436,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":425,"completion_tokens_details":{"reasoning_tokens":3331}},"tokens_in":425,"tokens_out":3436,"duration_ms":24266,"temperature":1.0,"reasoning_tokens":3331,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:09:30.517674+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A compact complex nilmanifold that is not a torus, with a left-invariant complex structure and both a balanced and a pluriclosed metric compatible with that structure, would disprove Theorem 4.7; the Section 3 recipe, prescribing rational structure constants for a 2-step nilpotent Lie algebra, offers a concrete search space for such an example.","supporting_citations":[],"review_version":1}