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On Strominger K\"ahler-like manifolds with degenerate torsion

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Pith's one-line read Complete non-Kähler SKL threefolds split into a non-Kähler SKL surface times a Kähler curve, or into two Sasakian 3-manifolds.

desk verdict 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. read the letter →

arxiv 1908.05322 v3 pith:P4BNBZOZ submitted 2019-08-14 math.DG

Shing-Tung Yau , Quanting Zhao , Fangyang Zheng This is my paper · ORCID
classification math.DG MSC 53C5553C05
keywords Kähler-likecurvatureStromingerconnectionpluriclosedmetricSasakianmanifoldsdegeneratetorsionlocallyconformalKählerAepplicohomologydeRhamsplitting
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [§3, proof of Theorem 7, rank-two case (equations (19)–(22))] 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.
  2. [§1 and §2, equations (1)–(3)] 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.
  3. [§3, proof of Theorem 9 (rank-two, n=3 case)] 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.
minor comments (4)
  1. [§1, proof of Theorem 1] The word 'Apelli' should be 'Aeppli' in the sentence 'it represents an Apelli cohomology class'.
  2. [§1 and throughout] 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.
  3. [§3, Lemma 5] 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.
  4. [§1, Definition 2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the classification is built on imported external theorems, not on its own conclusion.

full rationale

I walked the derivation chain from the definition of SKL through equations (1)-(3), Lemma 3, Lemma 4, Theorems 7, 8, and 9. The classification does not assume its own conclusion: the SKL condition is defined geometrically by the Kähler-like curvature of the Strominger connection, and the equivalence with pluriclosedness plus parallel torsion is imported from [45] as an external theorem, not rederived in this paper. This is self-citation because Zhao and Zheng are coauthors, but the cited result is parameter-free, its hypotheses do not include the threefold classification or the splitting theorem, and it is not constructed from the statements being proved. The claims that degenerate torsion forces the rank-1 or rank-2 normal forms, that the eigenvalues a_i are globally defined constants, and that the kernel distribution E ⊕ E is Riemannian-parallel are derived from the structure equations and de Rham arguments rather than from the target classification. Belgun's Sasakian classification is used only to name the factors in the compact case. No fitted parameter is relabeled as a prediction, and no equation in the paper is identical to its input by construction. The heavy reliance on [45] is an audit and reproducibility concern, but it is a correctness risk, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

There are no fitted numerical parameters. The constants lambda and rho that appear are geometric invariants of the torsion tensor, not quantities chosen to fit a conclusion. The main external inputs are theorems from prior literature, several by the same authors, which the paper cites but does not reproduce.

assumptions (4)
  • domain assumption SKL iff pluriclosed and ∇s-parallel torsion
    Imported from [45, Cor 4]; used in Corollary 6 and throughout Sections 2-3 as the working definition; not re-proved in this paper.
  • domain assumption Belgun's classifications of Vaisman surfaces and co-compact Sasakian 3-manifolds
    Used in the statements and proof of Theorem 7 to identify universal covers and to name the three standard Sasakian Lie groups.
  • standard math de Rham decomposition theorem for complete Riemannian manifolds
    Used in Theorems 7 and 9 to convert parallel distributions on the universal cover into product decompositions.
  • domain assumption Belgun's and Matsuo's formulas for products of Sasakian manifolds
    Used in Corollary 6 to assert the examples: parallel Strominger torsion and pluriclosedness for products with n_i <= 1.

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Cite this review

Pith. "Pith review of On Strominger K\"ahler-like manifolds with degenerate torsion." pith.science (2026). https://pith.science/paper/P4BNBZOZ

@misc{pith2026190805322,
  author       = {Pith},
  title        = {Pith review of: On Strominger K\"ahler-like manifolds with degenerate torsion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P4BNBZOZ}},
  note         = {Machine review of arXiv:1908.05322}
}
abstract

In this paper, we study a special type of compact Hermitian manifolds that are Strominger K\"ahler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is K\"ahler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a K\"ahler manifold. Previously, we have shown that any SKL manifold $(M^n,g)$ is always pluriclosed, and when the manifold is compact and $g$ is not K\"ahler, it can not admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when $n=2$, the SKL condition is equivalent to the Vaisman condition. In this paper, we give a classification for compact non-K\"ahler SKL manifolds in dimension $3$ and those with degenerate torsion in higher dimensions. We also present some properties about SKL manifolds in general dimensions, for instance, given any compact non-K\"ahler SKL manifold, its K\"ahler form represents a non-trivial Aeppli cohomology class, the metric can never be locally conformal K\"ahler when $n\geq 3$, and the manifold does not admit any Hermitian symplectic metric.

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