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Locally conformally K\"ahler manifolds with constant Levi-Civita or Bismut holomorphic sectional curvature

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Pith's one-line read A compact locally conformally Kähler manifold with constant Levi-Civita or Bismut holomorphic sectional curvature must be Kähler, except for the Bismut case where the only non-Kähler possibility is an isosceles Hopf manifold with zero curva

desk verdict New and mostly convincing: LCK with constant Levi-Civita or Bismut holomorphic sectional curvature are classified, but the strict case rests on a preprint's uniformization theorem. read the letter →

arxiv 2608.01893 v1 pith:XDM3RKPV submitted 2026-08-03 math.DG

Shuwen Chen , Fangyang Zheng This is my paper · ORCID
classification math.DG MSC 53C5553C05
keywords locallyconformallyKählerLevi-CivitaholomorphicsectionalcurvatureBismutBochner-KählermetricsisoscelesHopfmanifoldcomplexspaceformChernKamishimauniformization
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

This paper takes up two conjectures on compact Hermitian manifolds and proves them for the large class known as locally conformally Kähler (LCK) manifolds. It shows that if such a manifold has constant Levi-Civita holomorphic sectional curvature, it must actually be Kähler and therefore a complex space form. For constant Bismut holomorphic sectional curvature, the only non-Kähler possibility is the exceptional case where the curvature constant is zero and the manifold is an isosceles Hopf manifold. This settles the Levi-Civita and Bismut analogues of the Chern constant-curvature conjecture within the LCK class, using a reduction to Bochner-Kähler metrics.

What carries the argument

The key object is the injective symmetrization map $L_g$ from Hermitian symmetric (1,1)-tensors to curvature-like 4-tensors, defined by $L_g(P)_{i\bar{j}k\bar{l}} = P_{i\bar{j}}g_{k\bar{l}}+P_{k\bar{l}}g_{i\bar{j}}+P_{i\bar{l}}g_{k\bar{j}}+P_{k\bar{j}}g_{i\bar{l}}$. Lemma 3 proves injectivity. Proposition 1 shows that if a Hermitian metric with constant Chern, Levi-Civita, or Bismut holomorphic sectional curvature is conformal to a Kähler metric, that Kähler metric is Bochner-Kähler. The proof then invokes the Kamishima–Fried uniformization (Theorem 6): a compact strict LCK manifold whose associated Kähler metric is Bochner-Kähler is, up to scaling, holomorphically isometric to $(\mathbb{C}^n\setminus\{0\}, h_0)$ with deck t

What would settle it

Compute the full Bismut holomorphic sectional curvature of an isosceles Hopf manifold (standard Hopf metric on $(\mathbb{C}^n\setminus\{0\})/\langle \gamma \rangle$ with equal moduli eigenvalues). If any value is non-zero, the exceptional-case conclusion of Theorem 3 is wrong. Alternatively, construct a compact strict LCK manifold with constant Levi-Civita holomorphic sectional curvature; Theorem 2 predicts none exists.

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Extended reading notes

Core claim

The paper's central assertion is that for a compact LCK manifold, a constant Levi-Civita or Bismut holomorphic sectional curvature forces the associated Kähler metric (obtained by conformally rescaling on the universal cover) to be Bochner-Kähler, meaning its curvature tensor lies in the image of the map $L_g$. From there, in the globally conformally Kähler case, compactness and a parallel tensor argument rule out every non-Kähler alternative. In the strict LCK case, an imported uniformization theorem identifies the cover with punctured complex Euclidean space with similarity deck transformations; a pluriharmonic-function argument then eliminates the Levi-Civita case and, for the Bismut case,

Load-bearing premise

The proof for strict LCK manifolds depends on the imported theorem that any compact strict LCK manifold whose associated Kähler metric is Bochner-Kähler is uniformizable as a quotient of punctured complex Euclidean space by similarity transformations; if that uniformization fails, the conclusions in the strict case could break down.

Editorial extensions

If this is right

  • For every compact locally conformally Kähler manifold, a constant Levi-Civita holomorphic sectional curvature forces the metric to be Kähler, resolving Conjecture 2 within this class.
  • For constant Bismut holomorphic sectional curvature, the non-Kähler possibility is completely classified: it occurs only when the constant is zero and the manifold is an isosceles Hopf manifold.
  • Any compact LCK manifold with constant Levi-Civita holomorphic sectional curvature is a complex space form, so its universal cover is CP^n, C^n, or CH^n with the standard metric.
  • Any compact LCK manifold with non-zero constant Bismut holomorphic sectional curvature must be Kähler, confirming Conjecture 3 within this class.
  • The constants for the Bismut case are constrained: if the exceptional isosceles Hopf case occurs, the curvature constant is exactly zero.

Reading between the lines

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

  • A direct corollary not stated in the paper is that compact LCK manifolds with constant Levi-Civita holomorphic sectional curvature have no strict LCK examples, giving a necessary condition for constructing non-Kähler LCK metrics with any constant curvature.
  • The same Bochner-Kähler reduction may apply to other conformally Kähler settings, suggesting a route to classify constant-curvature metrics on manifolds whose associated Kähler metric is Bochner-Kähler.
  • The isosceles Hopf family is likely to be the full set of compact LCK manifolds with zero Bismut holomorphic sectional curvature, since the proof shows any strict LCK with constant Bismut HSC must fall into this family.
  • The pluriharmonic argument in the strict case may be adapted to test other constant-curvature Hermitian connections, since only the curvature symmetrization changes.
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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 / 5 minor

Summary. The paper studies compact locally conformally Kähler (LCK) manifolds with constant Levi-Civita or Bismut holomorphic sectional curvature. Theorem 2 claims that a compact LCK manifold with constant Levi-Civita holomorphic sectional curvature must be Kähler, hence a complex space form. Theorem 3 claims that a compact LCK manifold with constant Bismut holomorphic sectional curvature c is either Kähler or has c=0 and is an isosceles Hopf manifold. The proof strategy is to pass to the associated Kähler metric on the universal cover, show that this Kähler metric is Bochner-Kähler via explicit conformal-change computations (Proposition 1), then use uniformization in the strict LCK case (Theorem 6, imported from Huang-Wan) and a maximum principle / algebraic argument to rule out the non-Kähler possibilities. In the globally conformally Kähler case, a separate argument is given (Proposition 2), with the Levi-Civita case omitted as analogous.

Significance. If the main theorems are correct, they resolve Conjectures 2 and 3 within the LCK class and extend the recent Huang-Wan resolution of the Chern case. The conformal-change identities in Proposition 1 are clean and potentially useful, and the reduction to Bochner-Kähler metrics is conceptually appealing. However, the strict LCK parts of both theorems hinge entirely on Theorem 6, which is quoted from an unreviewed preprint and not proved here. This makes the central claim conditional rather than self-contained. The paper also omits the Levi-Civita half of Proposition 2 and compresses the final linear-algebra step of Theorem 3, so as written the proof has gaps that need to be repaired or explicitly delegated.

major comments (3)
  1. [Section 6, Theorem 6] The strict LCK cases of Theorems 2 and 3 rest entirely on Theorem 6, quoted from [11, Prop. 5.1] and not proved here. This uniformization is a strong, non-obvious global assertion: it identifies the universal cover of a compact strict LCK manifold with Bochner-Kähler associated metric with (C^n minus {0}, h0) and deck transformations gamma(z)=r U z. Because [11] is an arXiv preprint and Theorem 6 is not established in this manuscript, the central conclusion is conditional. Please either supply a proof or a precise published reference, and in any case state the dependence explicitly in the statements of Theorems 2 and 3.
  2. [Section 5, Proposition 2] The Levi-Civita case of Proposition 2 is omitted with the statement that the proof is exactly analogous. This is load-bearing for Theorem 2 in the globally conformally Kähler case. The analogy is plausible but the details (the eigenvalue splitting for A_g, and the maximum/minimum argument for w with beta depending on c, w, and |partial w|^2) must be written out; as written, Theorem 2 is not fully proved.
  3. [Section 6, Proof of Theorem 3 after (6.4)] The final linear-algebra step is too compressed and contains ambiguous notation. The term written as 4 times a product should be 4 z^T S times the conjugate of S z, or the calculation should be explained. Also, the step from (6.4)-(6.5) to (6.6) is not derived; it involves collecting (2,0) and (1,1) terms. Please expand so that the deduction of (alpha+c)q=0 and -alpha c I = 4 S Sbar is checkable. Additionally, the conclusion that the manifold is an isosceles Hopf manifold requires showing the deck group has a finite-index cyclic subgroup generated by a diagonal contraction with equal moduli; this is not shown and should be stated or proved.
minor comments (5)
  1. [Abstract and Section 1] Typo: 'answers similar questions' should be 'answer similar questions'. Also 'linear Hope manifold' in Section 1 should be 'linear Hopf manifold'.
  2. [Introduction] The assertion that isosceles Hopf manifolds have vanishing Bismut holomorphic sectional curvature is stated without proof or reference. Since this grounds the c=0 exceptional case in Theorem 3, a short computation or a citation should be provided.
  3. [Section 5, Proposition 2, Case 1] The implication dF wedge omega = 0 implies dF = 0 requires n at least 2. The proposition should state n at least 2 or handle n=1 separately.
  4. [Section 6, end of Theorem 3] The notation involving alpha pi-star omega_g in the last display is unclear or mistyped: the Kähler form of the pulled-back metric should be a constant multiple of (sqrt(-1)/|z|^2) partial partial-bar |z|^2. Please clarify.
  5. [References] In the reference list, the page range 495-518 for [10] appears detached from the entry; check formatting.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the LC/Bismut constancy conditions are reduced by direct computation to Bochner–Kähler and then to an external uniformization theorem; no claimed prediction reduces to an input.

full rationale

The derivation chain is not circular. The paper first records standard curvature identities (Lemma 2, Corollary 4) and then proves Proposition 1: if h=e^{2u}g is Kähler and g has constant Chern, Levi-Civita, or Bismut holomorphic sectional curvature, then h is Bochner–Kähler. This is done by explicit computation of A^b and A^g in (4.6)–(4.9); the conclusion is not assumed. In the globally conformally Kähler case, Proposition 2 uses the parallel-tensor argument and maximum principle to force f or w to be constant, hence g Kähler; the Levi-Civita half is stated as analogous and omitted, which is a proof gap but not a circular step. In the strict LCK case, Theorems 2 and 3 rely on Theorem 6, imported from Huang–Wan [11], to pass from the Bochner–Kähler property of the associated metric to the uniformization (C^n\{0}, h0). That theorem is external to this paper and is not authored by the present authors; it is also about a different hypothesis (Bochner–Kähler associated metric), not about the target constancy. After uniformization, the proofs solve the PDEs βh0−χ_w=0 and αh0−χ_f=0, derive contradictions (Theorem 2) or the isosceles Hopf metric with c=0 (Theorem 3). The constant c is not fitted and the Hopf conclusion is derived, not used as an input. The only unproved assertions are Theorem 6 itself, the omitted analogous Levi-Civita half of Proposition 2, and the introductory statement that an isosceles Hopf manifold has vanishing Bismut holomorphic sectional curvature; these are external-support or completeness issues, not circular reductions. The authors' self-citations ([7], [8], [16], [20], [24], [25]) are background or standard formulas and are not load-bearing for the main theorems.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper contributes no fitted constants and introduces no new objects. Its central claims rest on standard curvature identities, on two external classification theorems (Kamishima-Bryant, Huang-Wan uniformization), and on an unproved fact about Hopf manifolds. The largest uncredited load is the Huang-Wan theorem, which is a black box from an unreviewed preprint.

assumptions (9)
  • standard math Lemma 1: H^D constant c iff symmetrized curvature tensor equals c/2 G
    Stated in §2 as well-known; used to translate constant holomorphic sectional curvature into a tensor equation.
  • standard math Lemma 2 curvature relations: formulas (2.5), (2.6) and Corollary 4 (2.7)
    Standard identities for Chern, Levi-Civita and Bismut curvature; proof sketched in §2.
  • standard math Conformal change formulas (4.1), (4.2)
    Chern torsion and curvature under conformal metric change; cited to [24], used in Proposition 1.
  • standard math Lemma 3 injectivity of the map L_g
    Proved in §3 via trace identities; used to conclude A=0 on flat models.
  • domain assumption Kamishima-Bryant: complete Bochner-Kähler manifolds are locally symmetric products M^p_kappa times M^(n-p)_(-kappa)
    External Theorem 5, used in Proposition 2 to obtain parallel A^b or A^g.
  • domain assumption Theorem 6 (Huang-Wan): strict compact LCK with Bochner-Kähler associated metric uniformizes to (C^n minus {0}, h0) with deck transformations r U
    External theorem from [11], load-bearing for §6; not proved here.
  • standard math Hartogs extension and strong maximum principle
    Used to extend pluriharmonic functions across the origin and reach contradictions in §6.
  • standard math Takagi factorization for complex symmetric matrices
    Used without proof in the final algebra of Theorem 3.
  • domain assumption Isosceles Hopf manifolds have vanishing Bismut holomorphic sectional curvature
    Stated in §1 without proof; it justifies the c=0 exceptional family in Theorem 3.

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Pith. "Pith review of Locally conformally K\"ahler manifolds with constant Levi-Civita or Bismut holomorphic sectional curvature." pith.science (2026). https://pith.science/paper/XDM3RKPV

@misc{pith2026260801893,
  author       = {Pith},
  title        = {Pith review of: Locally conformally K\"ahler manifolds with constant Levi-Civita or Bismut holomorphic sectional curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XDM3RKPV}},
  note         = {Machine review of arXiv:2608.01893}
}
read the original abstract

An old conjecture in non-K\"ahler geometry states that any compact Hermitian manifold with constant Chern holomorphic sectional curvature must be either K\"ahler or Chern flat. The conjecture is known to be true in dimension 2 but still open in dimensions 3 or higher, except for several special classes of Hermitian manifolds. For the important class of locally conformally K\"ahler manifolds, the conjecture was proved by H. Chen, L. Chen, and Nie in 2021 when the constant holomorphic sectional curvature is non-positive and the remaining case was solved recently by Huang and Wan using the result of Kamishima on Bochner-K\"ahler manifolds. In this article, we use their technique to answers similar questions for locally conformally K\"ahler manifolds with constant Levi-Civita or Bismut holomorphic sectional curvature.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Compact balanced threefolds and LCK manifolds with constant holomorphic sectional curvature

    math.DG 2026-08 accept novelty 7.0 of 10

    Compact balanced threefolds with nonpositive constant Chern holomorphic sectional curvature are Chern flat (c=0) or Kähler (c<0), and constant-curvature LCK manifolds are Kähler or Hopf-covered.

Reference graph

Works this paper leans on

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