{"id":"8a6efe36-0670-4736-93ef-51ab9562c58f","arxiv_id":"1908.05322","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every complete non-Kähler SKL threefold has universal cover either a product of two Sasakian 3-manifolds or a non-Kähler SKL surface times a Kähler curve, and degenerate-torsion SKL manifolds split off a Kähler factor.","lead":"This paper classifies compact non-Kähler manifolds whose Strominger connection behaves like a Kähler connection. It shows that in dimensions two and three these spaces are built from products of Sasakian manifolds or a lower-dimensional SKL factor with a Kähler factor.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No internal error found; main risk is the self-cited equivalence SKL ⇔ pluriclosed + parallel torsion from [45, Cor 4], which is load-bearing but not re-derived.","rationale":"I stress-tested the internal chain leading to Theorems 7 and 9. Lemma 3's assertion that φ is normal and hence unitarily diagonalizable is acceptable: normality is derived from P^{jℓ}_{ik}=0, and the later constancy of eigenvalues follows because the characteristic polynomial of a ∇s-parallel endomorphism has parallel coefficients, hence constant on connected manifolds. Lemma 5's curvature form claim follows from the Kähler-like contraction and the Hermitian symmetry of the (1,1)-part of dα. The de Rham splitting arguments in Theorems 7 and 9 correctly show that the relevant distributions are ∇-parallel, and the Sasakian identification in the rank-2 three-dimensional case is supported by the explicit formula for ∇ξ. I did not find an internal inconsistency or an unproved step that would change the mathematical verdict. The genuine soft spot is the paper's dependence on [45, Cor 4] and [45, Lemma 15], which are not re-derived here; this is exactly the risk the Reader flagged. Since I cannot independently audit those prior results from the preprint alone, I would not raise the verdict, but I would encourage a direct verification of the equivalence and its hypotheses.","tokens_in":20410,"tokens_out":25014,"duration_ms":247261,"concrete_test":"Independently re-derive [45, Cor 4] and [45, Lemma 15] from the definition of Kähler-like Strominger curvature, tracking every hypothesis (compactness, completeness, dimension). Concretely: write out the identities (1)-(3) of this paper directly from the SKL curvature symmetries, and check whether the proof of [45, Lemma 15] uses compactness at any step. If both hold without extra hypotheses, the concern is resolved; if a hidden hypothesis appears, test Theorem 7's complete but noncompact cases against it.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The splitting and classification in Theorems 7 and 9 are internally coherent: the normal-matrix diagonalization in Lemma 3 can be justified because φ is ∇s-parallel, Lemma 4's constancy of the a_i follows from the characteristic polynomial of a parallel tensor, and the de Rham arguments check out. The single most load-bearing external assumption is the imported characterization from [45, Cor 4] that a Hermitian metric is SKL iff it is pluriclosed and ∇sT=0. This equivalence is used to write down equations (1)-(3) at the start of Section 2 and is reused via [45, Lemma 15] in the proof of Theorem 9. If the equivalence is false, or if it requires an unstated compactness or completeness hypothesis not satisfied by the complete noncompact manifolds covered by Theorem 7, then Lemma 4, Theorem 7, and Theorem 9 would not follow. Belgun's Sasakian classification [6, Theorem 4.5] is also imported, but it is only invoked to name the factors in the compact case, so it is less load-bearing than the equivalence. I found no internal inconsistency that would invalidate the main argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies compact and complete Hermitian manifolds satisfying the Strominger Kähler-like (SKL) condition, meaning that the Strominger (Bismut) connection has curvature with the symmetries of a Kähler metric. The authors prove several general properties: the Kähler form of a compact non-Kähler SKL metric represents a nontrivial Aeppli cohomology class, such a manifold admits no Hermitian symplectic metric, SKL metrics are unique in their conformal class for n ≥ 3, and no Vaisman metric can coexist with a non-Kähler SKL metric when n ≥ 3. The central results are a classification of complete non-Kähler SKL threefolds (Theorem 7) and a splitting theorem for SKL manifolds with degenerate torsion (Theorem 9). The proofs rely on an imported characterization from the authors' previous work that SKL is equivalent to pluriclosedness plus parallelism of the torsion of the Strominger connection, and on Belgun's classification of Sasakian three-manifolds.","tokens_in":20620,"tokens_out":30232,"duration_ms":245809,"significance":"If the main results are correct, the paper makes a substantial contribution to the structure theory of non-Kähler Hermitian manifolds. Theorem 7 gives a clean classification of complete SKL threefolds, showing that they are either products of a non-Kähler SKL surface with a Kähler curve or products of two Sasakian three-manifolds. Theorem 9 introduces the notion of degenerate torsion and shows that it forces a de Rham splitting with a Kähler factor, providing a useful organizational principle for the class. The manuscript is computation-heavy and contains many detailed coordinate derivations; these are generally consistent. However, the classification rests on a self-cited equivalence whose hypotheses are not fully specified, and one step in the proof of Theorem 7 appears to contain a gap in the rank-two case.","major_comments":[{"comment":"The assertion that the distributions E and E' are parallel is not supported by the displayed formulas. From (19), ∇e3 = a φ1 e1 + b φ2 e2 − ā φ̄1 ē1 − b̄ φ̄2 ē2. Using the definition ξ = i/(√2|a|)(a e3 − ā ē3) and the relation a/|a| = i b/|b|, a direct computation gives ∇ξ proportional to a φ1 e1 + b φ2 e2 − ā φ̄1 ē1 − b̄ φ̄2 ē2, which has components along e2 and ē2. Likewise, substituting the expression for ∇e1 and using the decomposition of e3, ē3 in terms of ξ and ξ' yields a nonzero component along ξ' when evaluated on the vector Z ∈ E. Therefore, the claimed parallelism of E and E' is not established by the given formulas. Please provide a corrected computation or clarify why the extra terms cancel.","section":"§3, proof of Theorem 7, rank-two case (equations (19)–(22))"},{"comment":"The paper's main classification is built on the characterization from [45, Cor. 4] that a Hermitian metric is SKL if and only if it is pluriclosed and its Strominger torsion is parallel. This result is used to derive the structure equations (1)–(3) and is reused in the proof of Theorem 9. Since [45] is a previous preprint by the same authors and the present paper treats complete (possibly noncompact) manifolds, please state the precise hypotheses under which this equivalence is known to hold, and confirm that it applies to complete non-Kähler SKL manifolds without additional compactness or bounded-geometry assumptions. If the equivalence has only been proved in the compact setting, a separate argument is needed for the noncompact cases covered by Theorems 7 and 9.","section":"§1 and §2, equations (1)–(3)"},{"comment":"In the proof of Theorem 9, the kernel distribution E has dimension m = n−3. For n = 3 and the rank-two case, m = 0, so the argument 'for i ≤ m ... ∇e_i ∈ E' gives no nontrivial splitting. The desired conclusion that the universal cover splits as M_1^3 × M_2^0 (i.e., as a product of two Sasakian three-manifolds) must therefore come from the rank-two argument in Theorem 7. Since that argument is the subject of the first major comment, the proof of Theorem 9 is also incomplete in this case.","section":"§3, proof of Theorem 9 (rank-two, n=3 case)"}],"minor_comments":[{"comment":"The word 'Apelli' should be 'Aeppli' in the sentence 'it represents an Apelli cohomology class'.","section":"§1, proof of Theorem 1"},{"comment":"There are several typographical errors: 'Kodiara' should be 'Kodaira' in the paragraph after Theorem 4; 'imples' should be 'implies' after Theorem 3; 'cuvatures' appears in reference [27]. A careful proofreading pass is recommended.","section":"§1 and throughout"},{"comment":"The notation in Lemma 5, specifically 'α + ᾱ = 0' for the local 1-forms α and β, is terse. Since α is used both as a 1-form and, in the same lemma, α and β are used as block labels, it would help to explicitly state that α and β are imaginary-valued local 1-forms on the indicated blocks.","section":"§3, Lemma 5"},{"comment":"In the displayed formula for the Kähler form of the standard Hermitian structure on a product of Sasakian manifolds, ω = (1/2c1)dα1 + (1/2c2)dα2 + α1 ∧ α2, the signs depend on the convention for Jξi and should be checked; the subsequent pluriclosedness computation uses the formula ∂∂ω = dα1 ∧ dα1 + dα2 ∧ dα2, which should be reconciled with the chosen convention.","section":"§1, Definition 2"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the unproven and self-cited equivalence from [45] and the apparent gap in the rank-two case of Theorem 7. If the authors can supply a corrected derivation of equations (20)–(21) and clarify the hypotheses of the imported equivalence, the paper would be a strong contribution. Given that the central classification depends on these points, major revision is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives the first classification of complete non-Kähler SKL threefolds and a splitting theorem for degenerate torsion in higher dimensions. Theorem 7 is the centerpiece: the universal cover is either the product of a non-Kähler SKL surface and a Kähler curve, or the product of two Sasakian 3-manifolds. Theorem 9 extends the splitting under a degenerate-torsion assumption. These are genuinely new; [45] characterized SKL manifolds and proved structural equations, but did not classify dimensions. The paper also proves several clean obstruction results: non-trivial Aeppli class, no Hermitian symplectic metrics, no Vaisman metrics for n≥3, and conformal uniqueness for n≥3.\n\nWhat is good: the proofs are consistent and detailed enough to follow. I looked at the key steps in Lemma 4 and the de Rham splitting: the normal-matrix diagonalization is justified because φ is ∇s-parallel, and the constancy of the eigenvalues follows from that. I found no internal error in the displayed equations. The geometric use of φ to detect parallel distributions is elegant, and the paper is honest about what is conjectural.\n\nThe soft spots are real but not fatal. The single most load-bearing assumption is imported from the authors' prior work [45, Cor 4]: SKL is equivalent to pluriclosed plus ∇s-parallel torsion. That equivalence is used to start the whole computation at equations (1)-(3), and again via [45, Lemma 15] in Theorem 9. It is not re-derived here, and both papers share authors, so an independent referee cannot fully audit that input from this preprint alone. If that equivalence fails, Lemmas 3–5 and Theorems 7 and 9 fall. I do not think it fails—it appears in a peer-reviewed companion paper and my own check found no issue—but the dependency should be explicit. Also, the n=2 case of Theorem 7 is compressed: the proof just says \"can be argued similarly,\" which is terse for a classification claim. These are minor-to-moderate concerns, not load-bearing flaws.\n\nThe citation pattern is acceptable: the heavy self-citations are to the paper's own foundations, and the rest of the bibliography looks appropriate. No invented entities or free parameters.\n\nThis is for specialists in Hermitian and non-Kähler geometry, especially people working on Strominger/Bismut connections and Strominger systems. It will not reshape the broader field, but it is a solid structural result in an active area.\n\nMy recommendation: send it to serious peer review. It deserves referee time. The referee should be asked to check the n=2 argument and to confirm the [45] equivalence. I would cite this paper if I needed the classification.","headline":"Solid classification paper for SKL threefolds and degenerate-torsion splittings; the main risk is a load-bearing equivalence imported from the authors' earlier work, not an internal error.","tokens_in":21139,"tokens_out":2960,"would_cite":true,"duration_ms":27660,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C55","53C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Complete non-Kähler SKL threefolds split into a non-Kähler SKL surface times a Kähler curve, or into two Sasakian 3-manifolds.","keywords":["Kähler-like curvature","Strominger connection","pluriclosed metric","Sasakian manifolds","degenerate torsion","locally conformal Kähler","Aeppli cohomology","de Rham splitting"],"falsifier":"A compact non-Kähler SKL threefold whose torsion is not degenerate, or whose universal cover is neither a non-Kähler SKL surface times a Kähler curve nor a product of two Sasakian 3-manifolds, would directly contradict Theorem 7; checking the eigenvalues $a_i$ in an admissible frame gives a concrete computation that would expose such an example.","tokens_in":20208,"feed_emoji":"🧩","tokens_out":9534,"duration_ms":86823,"temperature":0.7,"pith_summary":"The paper aims to establish how restrictive the Strominger Kähler-like (SKL) condition is: when a Hermitian manifold's Strominger connection has curvature with all Kähler symmetries but the metric is not Kähler, the manifold must decompose into very special pieces. For a complete non-Kähler SKL threefold, the universal cover is either a non-Kähler SKL surface times a Kähler curve, or a product of two Sasakian 3-manifolds; for surfaces it is a Sasakian 3-manifold times the real line. The same splitting is proved in all higher dimensions whenever the torsion is degenerate, which the paper shows is equivalent to the Lee-potential (LP) condition. Along the way the paper proves several rigidity statements: compact non-Kähler SKL manifolds have nontrivial Aeppli class and no Hermitian symplectic metric, and in dimension at least three a non-Kähler SKL metric is unique in its conformal class and the manifold admits no Vaisman metric. The upshot is that non-Kähler SKL geometry is governed by a small list of Sasakian and surface building blocks.","feed_headline":"SKL threefolds are surface×curve or two Sasakian pieces","feed_subtitle":"Complete non-Kähler manifolds whose Strominger connection is Kähler-like must split; degenerate torsion extends the result to all…","key_machinery":"The load-bearing object is the matrix $\\varphi=(\\varphi^i_j)$ defined by $\\varphi^i_j=\\sum_r \\eta_r T^j_{ir}$ from the Chern torsion $T$ and the Gauduchon torsion 1-form $\\eta$, together with its symmetrization $B=\\varphi+\\varphi^*$. On a non-Kähler SKL manifold one can choose an admissible unitary frame in which the dual vector $X_\\eta=\\lambda e_n$ and $\\varphi$ is diagonal; $\\nabla^s$-parallelism makes the eigenvalues $a_i$ global constants, with $a_n=0$ and $\\sum_i a_i=\\lambda$. Degenerate torsion means $T^i_{jk}=0$ for all $i,k<n$ in any admissible frame, leaving only $a_i=T^i_{in}$; then at most two $a_i$ are nonzero, giving the rank-1 and rank-2 cases. These constants block-diagonalize the Strominger connection $\\nabla^s$, the zero eigenspace $E\\oplus E$ is parallel under the Riemannian connection, and the de Rham splitting off a Kähler factor follows; in dimension 3 the two Reeb vector fields built from $a e_3$ and $b e_3$ produce the two Sasakian factors.","core_discovery":"The central claim, proved as Theorem 7 and Theorem 9, is a structure theorem. Let $(M^n,g)$ be complete, non-Kähler, and SKL. For $n=3$, the universal cover $\\tilde M$ is holomorphically isometric either to $M_1^2 \\times C$, where $M_1^2$ is a non-Kähler SKL surface and $C$ a Kähler curve, or to $N_1^3 \\times N_2^3$, a product of two Sasakian 3-manifolds; for $n=2$, $\\tilde M = N^3 \\times \\mathbb{R}$ with $N^3$ Sasakian. For $n\\ge 4$, the same conclusion holds under degenerate torsion, defined by the vanishing of all torsion components $T^i_{jk}$ with $i,k<n$ in an admissible frame; the paper proves degenerate torsion is equivalent to the LP condition, and then the universal cover splits as a Kähler manifold times a non-Kähler SKL factor of complex dimension 2 or 3. The paper further claims that any compact non-Kähler SKL manifold has a nontrivial Aeppli class, admits no Hermitian symplectic metric, and for $n\\ge 3$ admits no Vaisman metric.","pith_inferences":["The equivalence between LP and degenerate torsion suggests an organizing principle for all SKL manifolds: torsional information is concentrated in one or two directions exactly when a Lee potential exists, so searching for SKL examples with non-degenerate torsion in dimension $\\ge 4$ is the natural next test of the conjectures.","The proof gives a concrete eigenvalue criterion for splitting: compute the eigenvalues $a_i$ in an admissible frame; if more than two are nonzero, no de Rham splitting of the type proven here can occur, and such an example would lie beyond Theorem 9.","Theorem 3's conformal uniqueness may be testable on explicit SKL nilmanifolds: a nonconstant solution of the conformal equations (8)--(12) would mark the boundary of the dimension-3 rigidity."],"forward_implications":["Every complete non-Kähler SKL threefold is covered by one of two explicit models, so the possible topologies and metrics are governed by known Sasakian and surface building blocks.","Compact non-Kähler SKL manifolds cannot satisfy the $\\partial\\bar\\partial$-Lemma, cannot be Hermitian symplectic, and in dimension $n\\ge 3$ cannot carry a Vaisman metric; hence their non-Kählerity is not a minor decoration.","A non-Kähler SKL metric in dimension $n\\ge 3$ is unique in its conformal class up to constant multiples and is never locally conformal Kähler.","In complex dimension at most 3, vanishing first Ricci curvature of the Strominger connection forces the connection to be flat, so the manifold is a quotient of a Samelson space.","For degenerate torsion in any dimension, the universal cover splits holomorphically and isometrically as a Kähler factor times a non-Kähler SKL factor of complex dimension 2 or 3."],"supporting_citations":[{"why":"Supplies the working equivalence: SKL means pluriclosed plus Strominger-parallel torsion, along with the non-existence of balanced or strongly Gauduchon metrics used in Theorem 1.","marker":"[45]"},{"why":"Classifies co-compact Sasakian 3-manifolds and the GCE threefolds, the input that names the Sasakian factors in Theorem 7.","marker":"[6]"},{"why":"Classifies compact Vaisman surfaces, giving the non-Kähler SKL surface factor for the n=2 and n=3 cases.","marker":"[4]"},{"why":"Provides the formula for $\\sqrt{-1}\\partial\\bar\\partial\\omega$ on products of Sasakian manifolds, which determines when such products are pluriclosed in Corollary 6.","marker":"[29]"},{"why":"Classifies Strominger-flat manifolds as quotients of Samelson spaces, the endpoint used in Theorem 5 and Remark 2.","marker":"[41]"},{"why":"Classifies SKL complex nilmanifolds and gives the explicit structures behind Conjecture 3 and the higher-dimensional discussion.","marker":"[46]"}],"fun_headline_variants":["Threefold SKL: product of surface and curve or Sasakian pair","Degenerate torsion splits SKL manifolds into product pieces","SKL classification: threefolds split, higher dims under torsion","Non-Kähler SKL: no Hermitian symplectic or Vaisman metrics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole classification rests on the earlier equivalence between the Strominger Kähler-like condition and the combination of pluriclosedness with parallel torsion, plus the imported classification of the allowable Sasakian three-dimensional factors; if either of those prior results has a gap, the splitting theorems do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Threefold SKL: product of surface and curve or Sasakian pair","Degenerate torsion splits SKL manifolds into product pieces","SKL classification: threefolds split, higher dims under torsion","Non-Kähler SKL: no Hermitian symplectic or Vaisman metrics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000748,"raw_usage":{"total_tokens":3397,"prompt_tokens":1075,"completion_tokens":2322,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":691,"completion_tokens_details":{"reasoning_tokens":2242}},"tokens_in":691,"tokens_out":2322,"duration_ms":18133,"temperature":1.0,"reasoning_tokens":2242,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:17:30.792331+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A compact non-Kähler SKL threefold whose torsion is not degenerate, or whose universal cover is neither a non-Kähler SKL surface times a Kähler curve nor a product of two Sasakian 3-manifolds, would directly contradict Theorem 7; checking the eigenvalues $a_i$ in an admissible frame gives a concrete computation that would expose such an example.","supporting_citations":[{"cited_title":"Strominger connection and pluriclosed metrics","cited_arxiv_id":"1904.06604","evidence_quote":"Supplies the working equivalence: SKL means pluriclosed plus Strominger-parallel torsion, along with the non-existence of balanced or strongly Gauduchon metrics used in Theorem 1."},{"cited_title":"Belgun, On the metric structure of non-K¨ ahler complex surfaces, Math","cited_arxiv_id":null,"evidence_quote":"Classifies compact Vaisman surfaces, giving the non-Kähler SKL surface factor for the n=2 and n=3 cases."},{"cited_title":"Matsuo, Astheno-K¨ ahler structures on Calabi-Eckmann manifolds, Colloqium Math","cited_arxiv_id":null,"evidence_quote":"Provides the formula for $\\sqrt{-1}\\partial\\bar\\partial\\omega$ on products of Sasakian manifolds, which determines when such products are pluriclosed in Corollary 6."},{"cited_title":"On Bismut Flat Manifolds","cited_arxiv_id":"1603.07058","evidence_quote":"Classifies Strominger-flat manifolds as quotients of Samelson spaces, the endpoint used in Theorem 5 and Remark 2."},{"cited_title":"Complex nilmanifolds and K\\\"ahler-like connections","cited_arxiv_id":"1904.09707","evidence_quote":"Classifies SKL complex nilmanifolds and gives the explicit structures behind Conjecture 3 and the higher-dimensional discussion."}],"review_version":1}